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From Taylor to quadratic Hermite-Pade polynomials.

Abstract. Taylor polynomials, Pade approximants, and algebraic Hermite-Pade approximants form a hierarchy of approximation concepts of growing complexity. In the present contribution we climb this ladder of concepts by reviewing results about the asymptotic behaviour of polynomials that are connected with the three concepts. In each case the concepts are used for the approximation of the exponential function. The review starts with a classical result by G. Szego about the asymptotic behaviour of zeros of the Taylor polynomials, it is then continued with asymptotic results by E.B. Saff and R.S. Varga about the asymptotic behaviour of zeros and poles of Me approximants, and in the last part, analogous results are considered with respect to quadatic Hermite-Pad6 polynomials. Here, known results are reviewed and some new ones are added. The new results are concerned with the non-diagonal case of quadatic Hermite-Pad6 polynomials.

Key words. Taylor series, Me approximants, Hermite-Pade polynomials

AMS subject classifications. 41A21, 41A58, 41A63, 30B10

1. Introduction. The present contribution is an extended version of a talk given at the Conference 'Constructive Functions Tech-04' in honor of Eduard B. Saff's 60th birthday. It is concerned with the asymptotic behavior of Taylor, Padd, and Hermite-Pads polynomials (of type I) associated with the exponential function.

The investigations in all three cases are based on a rescaling method that has been introduced by G. Szego in [39] in the 20th of the last century for the study of Taylor polynomials. The method has then been extended in [29], [32], [34], and [35] by E.B. Saff and R.S. Varga for the study of Padd polynomials and Padd approximants to the exponential function, and lately the method has also been used by the author of the present contribution in [37] and [38] for an investigation of the asymptotic behavior of quadratic Hermite-Pad6 polynomials of type I and their zeros in the diagonal case. These last investigations will be extended to non-diagonal ray sequences in Section 6 of the present contribution.

The investigations in [29] - [35] deal with Padd approximants in the whole Pade table. Thus, also non-diagonal ray sequences have been considered, and among many other interesting things, it is quite instructive to see how Szego's earlier results about Taylor polynomials reappear as a limiting case in [35]. Inspired by this aspect of the investigations in [35], an extension of results from [37] to the non-diagonal case has been included in the present contribution.

In the next section we start with formal definitions of all three types of polynomials. In Section 3 we then turn to Szego's result, followed by a review of E.B. Saff's and R.S. Varga's results in Section 4. Results from [37] and [38] about the asymptotic behavior of diagonal quadratic Hermite-Pade polynomials of type I are reviewed in Section 5, and in Section 6 results from [37] are extended to non-diagonal ray sequences of quadratic Hermite-Pade polynomials.

2. From Taylor Polynomials to Hermite-Pade Polynomials in Three Steps. Let the function f be given by its power series development

f(z) = [c.sub.0] + [c.sub.1]z + [c.sub.2][z.sup.2] + ...

at the origin.

2.1. Taylor Polynomials. For each [member of] N the Taylor polynomial of the function f is defined as

(2.1) [t.sub.n](z) : = [c.sub.0] + [[c.sub.1]z + ... + [c.sub.n][z.sup.n,

and the remainder term satisfies

(2.2) f(z) - [t.sub.n(z) = 0([z.sup.n+1]) as z [right arrow] 0.

The polynomial [t.sub.n] is uniquely determined by relation (2.2).

2.2. Pade Polynomials and Pade Approximants. For each pair (m, n) [member of] [N.sup.2] of indices there exist two polynomials [P.sub.mn]= [member of] [P.sub.m] and [q.sub.mn] [member of] [P.sub.n]\{0} such that

(2.3) [q.sub.mn](z)f(z) - [p.sub.mn](z) = O([z.sup.m+n+1]) as z [right arrow] 0,

and they are called Pade polynomials associated with the function f and the indices m and n. By [P.sub.n] we have denoted the set of all polynomials of degree at most n [member of] N.

The existence of the pair ([p.sub.mn], [q.sub.mn]) of polynomials follows immediately from considering relation (2.3), which is equivalent to m + n + 1 homogenous linear equations in the coefficients of the two unknown polynomials [p.sub.mn] and [q.sub.mn]. Pade polynomials are not unique; in any case, they can be multiplied by a non-zero constant, but more substantial non-uniqueness may exist.

Contrary to the Pade polynomials, the Pade approximant

(2.4) [m/n] = [m/n]f : = [p.sub.mn]/[q.sub.mn]

is unique for each pair of indices (m, n). We note that in general we have

(2.5) f(z) - [m/n](z) [not equal to] O([z.sup.m+n+1]) as z [right arrow] 0,

which shows that relation (2.2) does not generalize directly to Pade approximants. Indeed, (2.5), the proper generalization of (2.2) is relation (2.3), and it is called the linearized version of the error relation.

2.3. Hermite-Pade Polynomials. Hermite-Pade polynomials appear in two versions that are known as type I and type II. There exists a duality between both types. Contrary to Taylor and Pade polynomials, where we deal with a single function f, the Hermite-Pade polynomials are associated with a whole system

(2.6) f = (1, [f.sub.1],..., [f.sub.m]), m [greater than or equal to] 1,

of functions. We assume that each of the m functions [f.sub.j], j = 1,..., m, is analytic in a neighborhood of the origin. Each set of Hermite-Pade polynomials is associated with a multi-index n = ([n.sub.0],..., [n.sub.m]) [member of] [N.sup.m+1] with m + 1 components.

DEFINITION 2.1. Hermite-Pade Polynomials of Type I: For any multi-index n = ([n.sub.0],..., [n.sub.m]) [member of] [N.sup.m+1] there exists a vector of polynomials ([p.sub.0],..., [p.sub.m]) [member of] [Pn.sub.0]-1 X [Pn.sub.1] -1 X ... X [Pn.sub.m]-1\{(0,...,0)} such that

(2.7) [m.summation over (j=0)] [p.sub.j](z)fj(z) = O([z.sup.|n|-1]) as z [righat arrow] 0,

where |n| := no + ... + [n.sub.m]. The vector ([p.sub.0],..., [p.sub.m]) is called Hermite-Pade form of type I, and its elements [p.sub.j] = [p.sub.jn], j = 0,..., m, n [member of] [N.sub.m+1], are the Hermite-Pade polynomials of type I of multi-index n associated with the system of functions f.

DEFINITION 2.2. Hermite-Pade Polynomials of Type H. For any multi-index n = ([n.sub.0],..., [n.sub.m) [member or] [N.sub.m+1] there exists a vector of polynomials ([p.sub.0,..., [p.sub.m]) [member or] P[N.sub.0]\{0} x P[N.sub.1] x ... x P[N.sub.m] with

(2.8) [N.sub.j] := |n|-[n.sub.j], j = 0,..., m,

such that

(2.9) [p.sub.0](z)[f.sub.j](z) - [p.sub.j](z) = 0([z.sub.|n|+1]) as z [right arrow] 0 for j = 1,...., m.

The vector ([p.sub.0],..., [p.sub.m]) is called Hermite-Pade form of type II, and its elements [p.sub.j] = [p.sub.jn], j = 0,..., m, n [member of] [N.sub.m+1], are the Hermite-Pade polynomials of type II of multi-index n associated with the system of functions f.

Counting the number of linear restrictions implied by relation (2.7) or by the relations in (2.9) shows that Hermite-Pade polynomials of both types always exist. But the polynomials are not unique, the situation is similar to that of Pade polynomials only that now the possibilities for non-uniqueness are more manifold.

From both relations (2.7) and (2.9), one gets the defining relation (2.3) of the Pade polynomials as a special case if one chooses m = 1. Indeed, in (2.7) one has to take m = 1, [f.sub.1] = -f, and ([n.sub.0], [n.sub.1]) = (n+1, m+1), while in (2.9) the choice has to be m = 1, [f.sub.1] = f, and ([n.sub.0], [n.sub.1) = (m,n). Thus, Hermite-Pade polynomials of both types can be seen as generalizations of Pade polynomials in a quite analogous way as Pade polynomials can be seen as a generalization of Taylor polynomials.

2.4. Hermite-Pade Approximants. The ideas behind the definition of the two types of Hermite-Pade polynomials are two different approximation concepts, of which each one uses one of the two types of polynomials as basic building blocks. The two concepts will be introduced next.

Let f be a single function that is assumed to be analytic at the origin. We define a system of functions (2.6) by

(2.10) f := (1,f,..,[f.sup.m]).

DEFINITION 2.3. Algebraic Hermite-Pade Approximants: For a given multi-index n [member of] [N.sup.m+1] let [p.sub.0],..., [p.sub.m] [member of] P[n.sub.m] -1 x... x P[n.sub.m] -1\{(0,...,0)} be the Hermite-Pade polynomials of type I defined by (2.7) and the system (2.10). Let then the algebraic function y = y (z) be defined by

(2.11) [m.summation over (j=0)] [p.sub.j](z)y[(z).sub.j] [equivalent to] 0.

From the m branches of y we select the branch y = [y.sub.n] which has the highest contact with f at the origin. This branch [y.sub.n], is called the algebraic Hermite-Pade approximant to f of degree m associated with the multi-index n.

The Pade approximants defined in (2.4) are algebraic Hermite-Pade approximants of degree m = 1, and in an analogous sense the Taylor polynomials can be seen as algebraic approximants of degree m = 0. Thus, the different concepts of approximants considered so far form a hierarchy of increasing complexity and generality.

Like the Hermite-Pade polynomials of type I lead to algebraic Hermite-Pade approximants, so in an analogous way the Hermite-Pade polynomials of type II lead to simultaneous rational Hermite-Pade approximants. Now, we start from a system (2.6) with [f.sub.0] [equivalent to] 1 and m independent functions [f.sub.1],..., [f.sub.m], that are assumed to be analytic in a neighborhood of the origin.

DEFINITION 2.4. Simultaneous Rational Hermite-Pade Approximants: For a given multi-index n [member of] [N.sub.m+1] let [p.sub.0],..., [p.sub.m] be the Hermite-Pade polynomials of type II define by (2.9). Then the vector of rational functions

(2.12) ([p.sub.1]/[p.sub.0](z),..., [p.sub.m]/[p.sub.0](z))

with common denominator polynomial [p.sub.0] is called simultaneous rational (Hermite-Pade) approximant to the (reduced) system of functions [f.sub.red] = ([f.sub.1],..., [f.sub.m]) associated with the multi-index n.

For m = 1, in (2.12) we have the Pade approximant to [f.sub.1] with numerator and denominator degrees (no, nl).

Although algebraic Hermite-Pade and simultaneous rational approximants are two rather different concepts, they have at least one aspect in common: in both cases the approximant has m simultaneous components. In case of simultaneous rational approximants the assertion is immediate; in case of algebraic Hermite-Pade approximants, the components are the m branches of the algebraic function [y.sub.n]. If the function f in (2.10) has branch points, then a very interesting question is whether and how far the m branches of the algebraic approximants [y.sub.n], follow branches of the function f around its branch points.

2.5. Historic Remarks. Pade's thesis [21] is generally seen as the birth certificate of Pade approximants. Of course, important aspects of the concept had already been studied earlier in the framework of continued fractions, most notably the investigations by G. Frobenius in [10]. Now-a-days, the situation is somewhat reversed, and large parts of the analytic theory of continued fractions are seen as part of the theory of Pade approximation. A great part of Pade's research were concerned with the exponential function, which incidentally is also the central topic of the present contribution.

The research by G. Frobenius [10] is seen as an important forerunner of Pade's contributions since there already the emphasis has been put on the whole table of approximants, and there the algorithmically important identities between neighboring approximants or between numerators and denominators of neighboring approximants have been studied for the first time.

The introduction of Hermite-Pade polynomials is perhaps most famous for its role in C. Hermite's proof of the transcendency of the number e (cf. [11]). A very readable account of this historic achievement is contained in the appendix of [13]. In the further development of the theory of Hermite-Pade polynomials one can distinguish two different lines of research: One direction is concerned with systems of special functions, like the exponential, the binomial, or the logarithmic function, etc. We mention here the publications [16], [17], and [18] by K. Mahler and the publications [12] and [5] by authors who are closely connected with Mahler's research. In the other direction one is concerned with polynomials associated with general classes of functions and systems of functions. The situation is comparable with that of the theory of orthogonal polynomials, where also a split into the same two orientations can be observed. As representatives for the second direction we mention the studies of Angelesco and Nikishin systems, and for an introduction to this direction of research we mention the book [19] or the survey articles [1].

3. Szergo's Result. In an investigation published in [39] G. Szego has studied the asymptotic distribution of the zeros of the Taylor polynomials

(3.1) [t.sub.n](w) = 1+Z+...+ [w.sup.n]/n!

to the exponential function [e.sup.w].

Since the exponential function is entire, and since it is different from zero everywhere in C, all zeros of the polynomials [t.sub.n] tend to infinity as n [arrow right] [infinity], and consequently all informative characteristics of the asymptotic distribution of the zeros gets lost during the process. In order to avoid such an unfavorable effect, G. Szego reorganized the independent variable w in such a way that the zeros of the transformed polynomials [T.sub.n] have finite cluster points in C that are not reduced to a single point. An appropriate transformation is given by

(3.2) z = [z.sub.n] = w/n.

With (3.2) the Taylor polynomial [t.sub.n] transforms into

(3.3) [T.sub.n](z) = [t.sub.n](nz) = 1 + nz +...+ [n.sup.n]/n! [z.sup.n],

and Stirling's formula shows that the n-th root of the leading coefficient of [T.sub.n] converges to a value different from 0 and [infinity]. Actually, it converges to e, i.e., we have

(3.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Let [Z.sub.n] = Z([T.sub.n]) denote the set of all zeros of the polynomial [T.sub.n], and let further [W.sub.n] denote the set of all zeros of the transformed remainder term

(3.5) [E.sub.n](z) := [e.sup.nz] - [T.sub.n](z).

In [39] the following theorem has been proved.

THEOREM 3.1. Let the curve So be defined by

(3.6) [S.sub.0] := {z [member of] C | |z|[e.sup.1-Re(z) = 1}.

Then the sets [Z.sub.n] of zeros of the transformed polynomials [T.sub.n] cluster on the subcurve [S.sub.0] [intersection] D of [S.sub.0] as n [right arrow] [infinity], and the sets [W.sub.n] of zeros of the remainder terms (3.5) cluster on [S.sub.0] \ D as n [right arrow] [infinity].

The curve [S.sub.0] is nowadays known as the Szego curve. It is shown in Figure 3.1 together with the zeros of the polynomial [T.sub.36] and some zeros of the remainder term [E.sub.36]. We note that for zeros of [E.sub.36] close to infinity the asymptotic distance between two neighboring zeros is 2[pi]/36 = 0.175. In Figure 3.1 the 6 zeros are not far enough away from the origin since the distance between them is still somewhat larger than 0.175.

The curve [S.sub.0] can be interpreted as the intersection of the plane defined by { (x, y, z) [member or] [R.sup.3] | z = x - 1 } in [R.sup.3] with the logarithmic cone { (x, y, z) [member of [R.sup.3] | log | [x.sup.2] + [y.sup.2| = 2 z }. A short proof of Theorem 3.1 can be based on this fact together with a clever use of Cauchy's integral formula. Many related results about the distribution of zeros of sections of power series to the exponential function and other entier functions can be found in [9].

With potential-theoretic means it is possible to describe the asymptotic density of the zeros of [T.sub.n]; cf. [25], [26]. Estimates of the speed with which the zeros of [T.sub.n] converge towards [S.sub.0] [intersection] D have been proved in [4]. Recently, R.S. Varga has also investigated the discrepancy between the sets [Z.sub.n] and their asymptotic distribution on [S.sub.0] [intersection] D, and he has spoken about this topic at the same conference in honor of E.B. Saff's 60th birthday, to which also the present contribution is dedicated.

[FIGURE 3.1 OMITTED]

Szego's result, which have just been reviewed, was an important inspiration for the research by E.B. Saff and R.S. Varga about the convergence and asymptotic behavior Pade approximants and Pade polynomials, which will be reviewed in the next Section.

4. E.B. Saff's and R.S. Varga's Results. In Subsection 2.2 it has already been noted that Pade approximants are a natural generalizations of Taylor polynomials into the field of rational functions. The Pade polynomials [p.sub.mn] [member of] [P.sub.m, [q.sub.mn] [member of] [P.sub.n], and the Pade approximants [m/n] have been defined in (2.3) and (2.4), respectively. In case of the exponential function there exist explicit formulae for the Pade polynomials (cf. [35], formula (1.3), or [2], Section 1.2). For instance, for the diagonal approximants [n/n], we have

(4.1) [p.sub.nn](w) = (2n n) + (2n-1 n-1)w/1!+...+(n 0)[w.sup.n]/n!, [q.sub.nn(w)=[p.sub.nn](-w).

Questions about convergence are basic for any type of approximants; for Pade approximants and the exponential function basic answers have already been given in Pade's thesis [21] (cf. also [22], [23], [24]). Locally uniform convergence holds true throughout C for any sequence of numerator and denominator degrees ([m.sub.j], [n.sub.j]) satisfying min([m.sub.j, [n.sub.j]) [right arrow] [infinity] as j [right arrow] [infinity]. In case of ray sequences, we have the surprising result that the numerator and denominator polynomials [p.sub.mn] and [q.sub.mn] converge even individually. Indeed, we have

(4.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for ray sequences {([m.sub.j], [n.sub.j])} with

(4.3) [n.sub.j]/[m.sub.j] [right arrow] [lambda] [member of] (0,[infinity]) as j [right arrow] [infinity].

From (4.2) it follows that all zeros of [p.sub.mn] as well as those of [q.sub.mn] converge to infinity if min(m, n) tends to infinity. This is an impressive property, which is quite unusual for the general convergence theory of Pade approximants. As in the case of Taylor polynomials, one would like to know more about the specific manner in which poles and zeros of the Pade approximants tend to infinity.

In a series of papers ([29], [30], [32], [33], [34], [35]) E.B. Saff and R.S. Varga have investigated the location of zeros and poles of Taylor polynomials and Pade approximants to the exponential function. Following the example set by G. Szego with his rescaling method, which has been reviewed in the last section, they carried out an analogous study of the asymptotic behavior of the zeros and poles of Pade approximants. We follow here in our review the treatment in [35].

Since Pade approximants [m/n] contain m + n + 2 free parameters, E.B. Saff and R.S. Varga used

(4.4) z = w/m+n

as new independent variable for the rescaling, which then leads to the transformed (rescaled) Pade polynomials

(4.5) [P.sub.mn](z) := [p.sub.mn]((m + n) z) and [Q.sub.mn](z) := [q.sub.mn((m + n) z).

One of the most interesting results in [35] is an explicit description of the arcs on which the zeros of the rescaled polynomials [P.sub.mn] and [Q.sub.mn] and also the zeros of the transformed remainder term

(4.6) [E.sub.mn](z) := [Q.sub.mn](z)[e.sup.(m+n)z] - [P.sub.mn](z)

cluster if the sequence of indices {(m, n)} = {(mj, nj)} tends to infinity along a ray sequence satisfying (4.3). These arcs are the analogue of the Szego curve [S.sub.0] from (3.6) for the Pade case.

The situation is now somewhat more complicated, and we need some auxiliary definitions before we can specify the arcs of the asymptotic relations. We follow very closely the terminology introduced in Section 1 of [35].

Depending on the parameter [lambda] of the ray sequence in (4.3), the two points [z.sup.+.sub.[lambda]], [z.sup.-.sub.[lambda]] [member of] [delta]D are defined as

(4.7) z [+ or -][lambda] := 1-[lambda]/1+[lambda][+ or -] i2 + [square root of ([lambda])]/1+[lambda],

and the function ga is defined as

(4.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The function g[lambda] has branch points at [z.sup.+.sub.[lambda]] and [z.sup.-.sub.[lambda]] and its natural domain of definition is the two-sheeted Riemann surface [R.sub.[lambda]] defined by the relation

(4.9) [R.sub.[lambda]]: [w.sup.2] = 1 + [z.sup.2]-2z (1-[lambda]/1+[lambda]).

[FIGURE 4.1 OMITTED]

Let the two half-lines [L.sub.[lambda].sup.[+ or -] be defined by [L.sub.[lambda].sup.[+ or -] := { z [member of] C | Re(z) = Re(z.sup.+.sub.[lambda]]), [+ or -] Im(z) [greater than or equal to] [+ or -] Im (z.sub.[lambda].sup.[+ or -])}, then the Riemann surface [R.sub.[lambda] can be represented by two sheets [S.sup.+] and [S.sup.-] that are identical copies of C \ ([L.sub.[lambda].sup.+][union] [L.sub.[lambda].sup.-]) glued together in the usual crosswise fashion along [L.sub.[lambda].sup.+] and [L.sub.[lambda].sup.-]. In the sequel we identify the sheet [S.sup.+] with C \ ([L.sub.[lambda].sup.+] [union] [L.sub.[lambda].sup.-]) and make the assumption that the sign of the square root in (4.8) is positive on [S.sup.+] near z = 0, i.e., g[lambda] (0) = 1. It is not difficult to verify that the two expressions 1 [+ or -] z + g [lambda](z) do not vanish in

C \ ([L.sub.[lambda].sup.+][union] [L.sub.[lambda].sup.-]) for 0 < [lambda] < [infinity]. Consequently, the function

(4.10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

is well defined for all 0 < [lambda] < [infinity] and z [member of] C \ ([L.sub.[lambda].sup.+] [union] [L.sub.[lambda].sup.-]).

By [S.sub.[lambda] we denote the sector

(4.11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In [35] it has been proved that

(4.12) [D.sub.[lambda]] := {z [member of] D [intersection] [S.sub.[lambda]| |[w.sub.[lambda](z)|=1} and

(4.13) [E.sub.[lambda]] := {z [member of] D\[S.sub.[lambda]| |[w.sub.[lambda](z)|=1}

are two analytic Jordan arcs, each connecting the two points [z.sup.+.sub.[lambda]] and [z.sup.+.sub.[lambda]] in D, and further that

(4.14) [F.sub.[lambda] := {z [member of] C\D | |[w.sub.[lambda]](z)|=1}

consists of two disjoint Jordan arcs each connecting one of the two points [z.sup.+.sub.[lambda]] and [z.sup.-.sub.[lambda]] with infinity. We have [F.[lambda] [subset] [S.sub.[lambda] [intersection] { z [member of] C | Re(z) [greater than or equal to] Re([z.sup.+.sub.[lambda]]) } for 0 < [lambda] [less than or equal to] 1 and [F.sub.[lambda]] [subset] { z [member of] C | Re(z) [less than or equal to] Re (z.sup.+.sub.[lambda]]) } \ [S.[lambda] for 1 [less than or equal to] [lambda] < [infinity]. In Figure 4.1, the two arcs [D.sub.[lambda]] and [E.sub.[lambda]] are shown together with pieces of the two arcs that constitute [F.sub.[lambda]] for A = 1/3. In Figure 4.2 the same arcs are shown for the symmetric case [lambda] = 1.

The result from [35], which is of central interest in the present review, is formulated in the next theorem.

THEOREM 4.1. ([35], Theorem 2.2) Let the ray sequence {[[m.sub.j]/[n.sub.j]]} of Pade approximants satisfy (4.3) with parameter [lambda] [member of] (0, [infinity]), then the poles and zeros of the rescaled Pade approximants [[m.sub.j]/[n.sub.j]] = P[m.sub.j][n.sub.j]/Q[m.sub.j][n.sub.j] and the zeros of the rescaled remainder terms (4.6) have the following properties:

(i) All zeros of [[m.sub.j]/[n.sub.j]] cluster on [D.[lambda] for j [right arrow] [infinity]. (ii) All poles of [[m.sub.j]/[n.sub.j]] cluster on [E.[lambda] for j [right arrow] [infinity]. (iii) All zeros of the transformed remainder terms E[m.sub.j][n.sub.j] (z) = Q[m.sub.j][n.sub.j](z)x [e.sup.(m.sub.j]+[n.sub.j]) z - [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (z) introduced in (4.6) cluster on [F.sub.[lambda]] for j [right arrow] [infinity].

In Figure 4.1 the rescaled poles and zeros of the Pade approximant [24/8] are plotted together with the two arcs [D.sub.1/3] and [E.sub.1/3] on which these objects cluster with growing degrees. Further, those zeros of the remainder term [E.sub.24,8] that fit to the window of Figure 4.1 are plotted together with the parts of the two arcs of [F.sub.1/3] that belong to the same window. Note that the pair of indices (24, 8) is an element of a ray sequence with [lambda] = 1/3.

The plot in Figure 4.1 shows that already for the degrees m = 24 and n = 8 a good agreement between the points and their asymptotic loci can be observed. The density of the poles and zeros as well as their asymptotic agreement with the arcs is poorest near the two bifurcation points [z.sup.+.sub.[lambda]] and [z.sup.-.sub.[lambda]].

With some imagination one can recognize that the configuration shown in Figure 4.1 turns into that of Figure 3.1, i.e., into the Szego curve [S.sub.0] from (3.6), if [lambda] tends to 0. This observation can be made solid by investigating the limit of the expression in (4.10) for [lambda] [right arrow] 0, which has been done in formula ('2.5) of [35]. In this respect, Szego's result is a special case of the results proved by E.B. Saff and R.S. Varga in [35].

In case of diagonal Pade approximants [n/n], we have [lambda] = 1, [z.sub.1.sup.[+ or -] = [+ or -]i, and the union [D.sub.1] [union] [E.sub.1] [union] [F.sub.1] of the arcs [D.sub.1], [E.sub.1], and [F.sub.1] from (4.12), (4.13), and (4.14) forms a configuration that is symmetric with respect to the y- and x-axis. In Figure 4.2, these arcs are plotted together with the zeros and poles of the rescaled Pade approximants [3232] and some of the zeros of the rescaled remainder term [E.sub.32,32] from (4.6).

[FIGURE 4.2 OMITTED]

In the symmetric case [lambda] = 1, the expressions in (4.8) and (4.10) become especially simple. We have

(4.15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(4.16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Using (4.12), (4.13), and (4.14) as before, one gets the arcs [D.sub.1], [E.sub.1], and [F.sub.1] from (4.15) and (4.16) instead of (4.8) and (4.10).

The expressions in (4.15) and (4.16) show in a transparent way that the asymptotic relations for the Pade polynomials [P.sub.nn] and [Q.sub.nn] contain as a main element an algebraic functions of a second degree, which is in the symmetric case [lambda] = 1 the rather simply constructed function [g.sub.1](z) = [square root of (1 + [z.sup.2])]. In the next section we will see that in case of quadratic Hermite-Pade polynomials the asymptotic relations are again associated with an algebraic functions, but there it is a function of third degree.

5. Quadratic Hermite-Pade Polynomials, the Diagonal Case. We now come to the third stage of our considerations: the asymptotic behavior of quadratic Hermite-Pade polynomials. In Section 2.3 it has been shown that Hennite-Pade polynomials of both types can be seen as generalizations of Pade polynomials like the Pade polynomials are generalizations of Taylor polynomials. Quadratic Hermite-Pade polynomials, which are defined by choosing m = 2 in Definition 2.1 and 2.2, are the first step in this process. In [37] and [38] the asymp- totic behavior of quadratic Hermite-Pade polynomials of type I has been studied for diagonal sequences of indices. Parts of these results are direct generalizations of the case [lambda] = 1 in [35].

Let [p.sub.n], [q.sub.n], and [r.sub.n] denote the three quadratic Hermite-Pade polynomials [p.sub.1],n, [p.sub.2], n, [p.sub.3],n [member of] [P.sub.n-1] of type I associated with the diagonal multi-index n = (n + 1, n + 1, n + 1) [member of] [N.sup.3] and the exponential system f = (1, exp, [exp.sup.2]) as introduced in Definition 2.1. Thus, all three polynomials [p.sub.n], [q.sub.n], [r.sub.n] are of degree at most n, and from the defining relation (2.7) in Definition 2.1 we know that they satisfy the relation

(5.1) [p.sub.n](w) + [q.sub.n](w) [e.sub.n] + [r.sub.n](w) [e.sup.2w] = O([w.sup.3n+2]) as [right arrow] 0.

The investigations in [37] and [38] are based on the rescaling method, and

(5.2) z = w/3n

is used as new rescaled independent variable, which then leads to the rescaled Hermite-Pade polynomials

(5.3) [P.sub.n](z) := [p.sub.n](3n z), [Q.sub.n](z) := [q.sub.n](3n z), and [R.sub.n(z) := [r.sub.n](3n z).

Note that the three polynomials [p.sub.n], [q.sub.n], [r.sub.n] have together 3n + 3 free parameters. Together with a multiplication by [e.sup.-3n z], relation (5.1) transform into

(5.4) [E.sub.n](z) := [e.sup.-3n z][P.sub.n](z) + [Q.sub.n](z) + [R.sub.n](z)[e.sup.3nz] = O([z.sup.3n+2]) as z [right arrow] 0

in the new variable z.

In Figure 5.2 the zeros of the rescaled polynomials [P.sub.n], [Q.sub.n], [R.sub.n] are shown for n = 33 together with those zeros of the remainder term [E.sub.33] from (5.4) that lie within the window of Figure 5.2. In a certain sense the configuration in Figure 5.2 is a doubling of the configuration shown in Figure 4.2 for diagonal Pade polynomials. In Figure 5.2 also the arcs are plotted on which the zeros cluster as n [right arrow] [infinity]. The description of these arcs has been one of the main aims of the research in [37].

While the definitions (4.12), (4.13), and (4.14) of the asymptotic arcs for the zeros of the Pade polynomials have been formulated in analogy to Szego's approach in (3.6) of Theorem 3.1, a different approach had to be chosen in [37]. There, the analysis relies strongly on the geometry of a Riemann surface that is associated with an algebraic function of third degree. Details will be given in Subsection 5.1.

The research in [37] and [38] was initiated by investigations of Hermite-Pade polynomials [p.sub.n], [q.sub.n], [r.sub.n] in [3], [7], and [8]. In [3] among other things a 4-term recurrence relation and a very precise asymptotic estimate for the polynomials [p.sub.n], [q.sub.n], [r.sub.n] and for the remainder term [e.sub.n] has been proved. While in [3], like in [37], only the diagonal case has been studied, extensions to the non-diagonal case are contained in [7] and [8]. There, also interesting con- nections with the theory of special functions have played a prominent role. In both papers great interest has been put on the location of the zeros of the polynomials [p.sub.n], [q.sub.n], and [r.sub.n]. Especially, these last aspect has triggered the research in [37] and [38].

Recently, in [14] (cf. also [15]) the asymptotic analysis of quadratic Hermite-Pade polynomials of type I has been incorporated into the Riemann-Hilbert approach for proving strong asymptotic relations for orthogonal polynomials, which is based on Deift's and Zhou's method of steepest descent from [6]. Further, we mention that in [41] and [42] steps of an extension of the analysis to Hermite-Pade polynomials of type I for m > 2 have been done.

5.1. The Riemann Surface R and the Functions [psi] and h. In the present subsection we define an algebraic functions [psi] of third degree together with its associated Riemann surface R in a first step, which then is followed by the definition of an harmonic function h, which will be an important element of the asymptotic relations for the rescaled Hermite-Pade polynomials [P.sub.n], [Q.sub.n], [R.sub.n] and the remainder term [E.sub.n]. In a comparison with the results reviewed in Section 4, the algebraic function [psi] and the Riemann surface R are the pendants of the algebraic function [g.sub.[lambda]] from (4.8) with [lambda] = 1 and the Riemann surface introduced in (4.9).

DEFINITION 5.1. The Riemann surface R together with the bijective mapping [psi]: R [right arrow] C and the canonical projection [pi] : R [right arrow] is defined by the property that the two functions [psi] and

(5.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

satisfy the relation

(5.6) z o [psi]([zeta]) = [pi]([zeta]) for all [zeta] [member of] R.

Relation (5.6) shows that locally the function [psi] is the inverse of the rational function (5.5), and therefore [psi] o [[pi].sup.-1] is an algebraic function of third degree. The surface R has three sheets and four simple branch points [zeta]j, j = 1,..., 4, over the four base points

(5.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Indeed, the derivative

(5.8) z[(v).sup.1] = -[v.sup.4]+1/3 / [v.sup.2][([v.sup.2]-1).sup.2]

has four simple zeros at the roots vj = 4[square root of (-1/3)], j = 1,..., 4, and it then is easy to check that the four points [z.sub.1],..., [z.sub.4] in (5.7) are defined by [z.sub.j ]= [pi] o [[psi].sup.-1]([v.sub.j]) = z([v.sub.j]), j = 1,... , 4. The Riemann surface R is of genus 0.

In the sequel, points on R will be denoted by [zeta], while the associated base points [pi]([zeta]) [member of] C will be denoted by z. For shortness we call the four points [[zeta].sub.j], j = 1, ..., 4, as well as their base points [z.sub.j] = [pi]([zeta].sub.j]) in (5.7) as branch points of R. From (5.5) and (5.6) one easily deduces that

(5.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In the sequel, we assume that the defining relation (5.4) of the Hermite-Pade polynomials [P.sub.n], [Q.sub.n], [R.sub.n], is lifted to R so that a neighborhood of the origin in C corresponds to a neighborhood of the point [[zeta].sub.0] on R defined by

(5.10) [[zeta].sub.0] := [[psi].sup.-1]([infinity]).

From the definition of the Riemann surface R in (5.5) and (5.6) it follows that R can be constructed by gluing together three sheets [S.sub.1], [S.sub.0], [S.sub.n [subset] R in the following way: Let [[GAMMA].sub.-1] and [[GAMMA.sub.1] [subset] C be two disjoint Jordan arcs with the property that [[GAMMA].sub.1] connects the two branch points [z.sub.1] and [z.sub.4], and [[GAMMA].sub.-1] connects the two branch points [z.sub.2] and [z.sub.3]. Let then the three sheets [S.sub.-1], [S.sub.0], and [S.sub.1] be copies of C\[[GAMMA].sub.-1], C\([[GAMMA].sub.-1] [union] [[GAMMA].sub.1), and C\ [[GAMMA].sub.1], respectively, assume that the two sheets [S.sub.1] and [S.sub.0] are glued together along the arc [[GAMMA].sub.1] in the usual cross-wise fashion, and in the same way assume that the two sheets [S.sub.-1] and [S.sub.0] are glued together along the arc [[GAMMA].sub.-1]. The boundary points of [delta][S.sub.j], j = -1, 0, 1, are attributed to the neighboring sheets in such a way that all three restrictions [pi]|[s.sub.j = -1, 0, 1, of the canonical projection [pi] are univalent. At the present moment the two arcs [[GAMMA].sub.-1] and [[GAMMA].sub.1] are still variable and will be fixed later.

Having defined R, which will serve as domain of definition, we are now ready to define the function h.

DEFINITION 5.2. Let the functions h : R [right arrow] R be defined by

(5.11) h([zeta]) := Re u o [psi]([zeta]) fpr [zeta] [member of] R with

(5.12) u(v) := 2[v.sup.2]/[v.sup.2]-1 + log 2/3v([v.sup.2-1).

It follows from (5.9), (5.10), (5.11), and (5.12) that the function his harmonic in R\({[[zeta].sub.0]} [union][[pi].sup.-1] ({infinity})) and subharmonic at [[zeta].sub.0].

5.2. The Definition of Jordan Arcs and a First Result. With the help of the function h from Definition 5.2 we define Jordan arcs that will turn out to be core pieces of the cluster sets of the zeros of the rescaled Hermite-Pade polynomials [P.sub.n], [Q.sub.n], [R.sub.n] and the remainder terms [E.sub.n, if n [right arrow] [infinity].

Starting point for our definition is the function

(5.13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

From Definition 5.2 we know that h is harmonic in R({[[zeta].sub.0]} [union] [[pi].sup.-1] ({[infinity]})) and subharmonic in a neighborhood of [[zeta].sub.0]. From standard knowledge in potential theory (cf. [27], Chapter 2) it then follows that the function [h.sub.max], as the maximum of subharmonic functions, is again subharmonic in C. Further, we know from the same background that [h.sub.max] is harmonic everywhere in C except for those loci z where at least two different branches of the multi-valued function h o [[pi].sup.-1] coincide and are at the same time identical with the maximal value [h.sub.max](z).

As a consequence of the subharmonicity of [h.sub.max] it follows from the Poisson-Jensen formula of potential theory (cf. [27], Theorem 4.5.1) that in any bounded domain in C the function [h.sub.max] can be represented as the sum of a harmonic function and a Green potential. These details and some of its immediate consequences are put together in the next lemma.

LEMMA 5.3. (cf. [37], Lemma 2.6) The function [h.sub.max] is subharmonic in C. There exists a system [GAMMA] of analytic Jordan arcs such that [h.sub.max] is harmonic in C\[GAMMA], but not harmonic in any neighborhood of a point z [member of] [GAMMA]. There exists a positive measure v on [GAMMA] such that for any R > 0 we have

(5.14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

with [h.sub.R] a function that is harmonic in {|z|<R} and satisfies

(5.15) [h.sub.R(z) = [h.sub.max](z) for |z| = R.

Since R > 0 is arbitrary, the measure v in Lemma 5.3 is well defined throughout C and we have supp (v) = [GAMMA].

[FIGURE 5.1 OMITTED]

The main topic in the present subsection is the description of the system [GAMMA] of arcs. In Definition 5.4, Lemma 5.5, and Theorem 5.6, below, we will see that [GAMMA] consists of four subsets [S.sub.j], j = -1, 0, 1, [infinity], that are the asymptotic cluster sets of the zeros of the rescaled Hermite-Pade polynomials [P.sub.n], [Q.sub.n], [R.sub.n], and the remainder term [E.sub.n].

It is immediate from what has been said so far that the system [GAMMA] of analytic Jordan arcs is contained in the larger system of intersection arcs

(5.16) [GAMMA] := {z [member of] C| card {h([zeta]) | [zeta] [member of] [[pi[.sup.-1] ({z})}<3}.

Since the Riemann surface R has three sheets over C, and since h is not identically constant on any open set, it follows that the set { h ([zeta])| [zeta] [member of] [[pi].sup.-1] ({z})} contains three different points for almost all z [member of] C; the only exceptions are the analytic arcs in [GAMMA]. These arcs are the totality of all intersection arcs of the multi-valued function h o [[pi].sup.-1].

Studying the behavior of the functions h and [h.sub.max] in neighborhoods of the four branch points [[zeta].sub.1],..., [[zeta].sub.4] [member of] R and [z.sub.1],..., [z.sub.4] [member of] C, respectively, shows that each branch point is the end point of exactly three different subarcs from [GAMMA]. Three of these arcs connect the two points {[z.sub.1], [z.sub.4]}, and three other ones connect the two points {[z.sub.2], [z.sub.3]}. A formal proof of this last assertion is part of the proof of Lemma 2.6 in [37].

Let now [[GAMMA].sub.1,j], j = -1,1, [infinity], denote the three arcs that connect [z.sub.1] with [z.sub.4], and [[GAMMA].sub.-1,j], j = -1,1, [infinity], the three other arcs that connect [z.sub.2] with [z.sub.3]. It turns out that two arcs, which are denoted by [[GAMMA].sub.1,[infinity]] and [[GAMMA].sub.-1,[infinity]], pass through infinity, while the other four arcs [[GAMMA].sub.ij], i, j = -1,1, are contained in D. All six arcs are shown in Figure 5.1. The unbounded arcs [[GAMMA].sub.1,[infinity]] and [[GAMMA].sub.-1,[infinity]], are represented only in part, and the two arcs [[GAMMA].sub.1,-1] and [[GAMMA].sub.-1,1] are partly represented by dashed lines; the reason for this is that along the dashed lines the values of the intersecting branches of h o [[pi].sup.-1] are no longer maximal, and therefore these dashed parts of [[GAMMA].sub.1,-1] and [[GAMMA].sub.-1,1] do not belong to the set [GAMMA] of Lemma 5.3, but they belong to [GAMMA].

[FIGURE 5.2 OMITTED]

The angles under which the six arcs [[GAMMA].sub.i,j] j = -1,1, [infinity], i = -1,1 end at the four branch points [z.sub.1],..., [z.sub.4] can be calculated from local developments of the functions h and [psi]. Having calculated these initial directions, the arcs themselves can then be calculated by using the property that along each of them two branches of the function h o [[pi].sup.-1] coincide.

Next, we come to the definition of the four sets [S.sub.j], j = -1,0,1, [infinity], that will turn out to be the asymptotic cluster sets of the zeros of the rescaled Hermite-Pade polynomials [P.sub.n], [Q.sub.n], [R.sub.n], and the remainder term [E.sub.n].

DEFINITION 5.4. Let the point [y.sub.1] [member of] i R be defined by [y.sub.1] := [[GAMMA].sub.1,-1][intersection][[GAMMA].sub.-1,1][intersection]{ z | Im (z) > 0 }. Then we set

(5.17) [S.sub.1] := [[GAMMA].sub.1,1],

(5.18) [S.sub.-1] := [[GAMMA].sub.-1,-1],

(5.19) [S.sub.0] := ([[GAMMA].sub.1,-1]{ z | Re (z) [greater than or equal to] 0}) [union] ([[GAMMA].sub.-1,1][intersection]{z | Re(z) [less than or equal to] 0}) [union][-[y.sub.1],[y.sub.1],

(5.20) [S.sub.[infinity] := [[GAMMA].sub.1,[inifinity]] [union][[GAMMA].sub.-1,[infinity]].

The sets [S.sub.j], j = -1,0,1, [infinity], are plotted in Figure 5.1. For the point [y.sub.1] one gets as numerical value [y.sub.1] = 0.621391 i.

LEMMA 5.5. The set [GAMMA] form Lemma 5.2 is given by

(5.21) [GAMMA] = [S.sub.-1][union][S.sub.0][union][S.sub.1][union][S.sub.[infinity]].

If the four branch points [z.sub.1],..., [z.sub.4] are removed from [GAMMA] [subset] C, then [GAMMA]\{[z.sub.1],..., [z.sub.4] consists of four components and the closure of each of them is one of the four sets [S.sub.j], j = -1,0,1, [infinity].

In many respects the next theorem is the analogue to Theorem 4.1 for diagonal quadratic Hermite-Pade polynomials. It is also the main result of the present subsection.

THEOREM 5.6. For the zeros of the rescaled diagonal Hermite-Pade polynomials [P.sub.n], [Q.sub.n], [R.sub.n], and the remainder term [E.sub.n], introduced in (5.4) the following assertions hold true: (i) All zeros of the polynomial [P.sub.n], cluster on [S.sub.-1] for n [right arrow] [infinity]. (ii) All zeros of the polynomial [Q.sub.n], cluster on [S.sub.0] for n [right arrow] [infinity]. (iii) All zeros of the polynomial [R.sub.n], cluster on [S.sub.1] for n [right arrow] [infinity]. (iv) All zeros of the remainder term [E.sub.n] cluster on S[infinity] for n [right arrow] [infinity]

In Figure 5.2 all zeros of the rescaled Hermite-Pade polynomials [P.sub.33], [Q.sub.33], [R.sub.33] and some zeros of the remainder term [E.sub.33] are plotted together with the arcs of [S.sub.-1], [S.sub.0], [S.sub.1], and parts of the arcs of [S.sub.[infinity]] on which the zeros cluster. The picture shows that already for the degree n = 33 there is a very good accordance between the zeros and their asymptotic cluster sets.

A proof of Theorem 5.6 follows rather immediately from a result about strong asymptotics for the [P.sub.n], [Q.sub.n], [R.sub.n] and [E.sub.n] in Theorem 2.9 in [37]. In the present review we will not concentrate on details of the proof, instead we shall discuss some other types of asymptotic relations in the next subsection.

5.3. Further Asymptotic Results. In [37] not only the asymptotic cluster sets of the zeros of the rescaled diagonal quadratic Hermite-Pade polynomials [P.sub.n], [Q.sub.n], [R.sub.n] and the remainder term [E.sub.n], have been studied. Also their asymptotic distributions, and besides of that also asymptotic relations for the polynomials and the remainder term themselves have been addressed. Again, the functions h and [h.sub.max] are the basic input for the asymptotic relations.

We start our review with results concerning the asymptotic distributions of zeros, and for this purpose we introduce four measures [[micro].sub.j], j = -1,0,1, [infinity].

DEFINITION 5.7. Let v be the measure introduced in Lemma 5.3 and [S.sub.j], j = -1,0,1, [infinity], the sets from Definition 5.2. Based on these objects we define

(5.22) [[micro].sub.j] := v|[s.sub.j], j = -1,0,1,[infinity].

Let Z(p) be the (multi-) set of all zeros of a polynomial p with multiplicities taken account of by repetition, let further [v.sub.p] be the zero counting measure of the polynomial p, which puts weight 1 at each zero of p taking account of multiplicities, i.e., [v.sub.p](B) = card(Z(p)[intersection]B) for every Borel set B [subset] C. Let further * [right arrow] denote the weak convergence of measures. The next result has been proved in [37].

THEOREM 5.8. (cf. [371, Theorem 2.1) For the zeros of the rescaled diagonal Hermite-Pade polynomials [P.sub.n], [Q.sub.n], [R.sub.n], and the remainder terms [E.sub.n], we have the following asymptotic relations

(5.23) 1/n [v.sub.P] * [right arrow] [micro]-1 as n [right arrow] [infinity],

(5.24) 1/n [v.sub.Qn] * [right arrow] [micro]0 as n [right arrow] [infinity],

(5.25) 1/n [v.sub.R] * [right arrow] [micro]1 as n [right arrow] [infinity],

(5.26) 1/n [v.sub.E] * [right arrow] [[micro].sub.[infinity]] + 3 [[delta].sub.0] as n [right arrow] [infinity],

where [[delta].sub.0] is the Dirac measure at z = 0.

Theorem 5.8 complements Theorem 5.6 in the sense that we now not only know where the zeros of [P.sub.n], [Q.sub.n] [R.sub.n], and of [E.sub.n] cluster but also how these zeros are asymptotically distributed.

Via Lemma 5.3 and Definition 5.7 the asymptotic distributions [[micro].sub.j], j = -1,0,1, [infinity]], can be traced back to the function [h.sub.max], for which we have a constructive and a numerically efficient definition. More about this can be found in [37], Theorem 2.10 and .2.11. The next proposition is a rather immediate consequence of Lemma 5.3, and it sheds light on the analytic background of the definition of the asymptotic distributions [[micro].sub.j], j = -1,0,1, [infinity]].

PROPOSITION 5.9. (cf. [37], Theorem 2.8) The measures [[micro].sub.j], j = -1,0,1, [infinity]] in (5.22) are absolutely continuous, their supports supp ([[micro].sub.j]) = [S.sub.j], j = -1,0,1,[infinity]] consist of analytic arcs, and for their densities we have the representation

(5.27) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for z [member of] [S.sub.j], j = -1,0,1, [infinity], where [delta]/[delta]n+ and [delta]/[delta]n_ are the normal derivatives to both sides of the arcs in [S.sub.j], ds is the line element on [S.sub.j], and orientations have to be chosen in (5.27) in such a way that all measures [micro], j = -1,0,1, [infinity], are positive.

In [38], Theorem 2.8, results about the asymptotic distributions of the zeros of [P.sub.n], [Q.sub.n], [R.sub.n], and [E.sub.n], have been proved that are a full degree more precise than those given in (5.23) - (5.26) of Theorem 5.8. These stronger relations in [38] are precise enough to distinguish asymptotically individual positions of zeros. Naturally, their formulation is more complicated and involved, and besides of the two functions h and [h.sub.max] there are also branches of the function [psi] o [[pi].sup.-1] needed.

In the second part of the present subsection, we come to the asymptotic relations for the polynomials [P.sub.n], [Q.sub.n], [R.sub.n] and the remainder term [E.sub.n] themselves. We started with some auxiliary definitions.

Near infinity the multi-valued function h o [[pi].sup.-1] has three branches, which we denote by [h.sub.-1], [h.sub.0], and [h.sub.1], and which have the local developments

(5.28) [h.sub.-1] (z) = -3 Re (z) + log |z| + O(1/z),

(5.29) [h.sub.0] (z) = log |z| + log(2) + O(1/z),

(5.30) [h.sub.1] (z) = 3 Re (z) + log |z| + O(1/z),

as z [right arrow][infinity]. The expressions in (5.28) - (5.30) follow rather directly from the defining relations (5.5) and (5.6) in Definition 5.1. In a neighborhood of the origin z = 0 the function h o [[pi].sup.-1] has again three branches; here we are only interested in the branch that has a logarithmic term. We denote this branch by [h.sub.[infinity]], and remark that it corresponds to the function h near the point [[zeta].sub.0] [member of] R that has been introduced in (5.10). For [h.sub.[infinity]] we have the local development

(5.31) [h.sub.[infinity]](z) = 3 log |z| + O(1) as z [right arrow] 0.

DEFINITION 5.10. In the four domains

(5.32) Dj := C\[S.sub.j], j = -1,0,1,[infinity],

we define four functions [h.sub.j], j = -1,0,1, [infinity], by harmonic continuations. In case of the first three functions [h.sub.-1, [h.sub.0], [h.sub.1] we start the continuation with the three function elements (5.28), (5.29), (5.30) at infinity and continue the process throughout the domains [D.sub.-1], [D.sub.0], [D.sub.1], respectively. In the case of the function [h.sub.[infinity]], we start from the origin and use (5.31) as initial function element.

Let us remark that the four branch points [z.sub.1],..., [z.sub.4] belong to the two sets [S.sub.0] and [S.sub.[infinity], and hence the two functions [h.sub.0] and [h.sub.[infinity]] can be continued throughout the two domains [D.sub.0] and [D.sub.[infinity] without any problem. In the cases of the two other functions [h.sub.-1] and [h.sub.1], an inspection of the behavior of the function h on the Riemann surface R shows that also here the process of harmonic continuation can be carried out without hitting a branch point within one of the domains [D.sub.-1] or [D.sub.1].

With the four functions [h.sub.j], j = -1,0,1, [infinity], we can formulate the asymptotic relations for the polynomials [P.sub.n], [Q.sub.n], [R.sub.n] and the remainder term [E.sub.n].

THEOREM 5.11. (cf. [37], Theorem 2.2) For the rescaled diagonal Hermite-Pade polynomials [P.sub.n], [Q.sub.n], [R.sub.n] and the remainder term [E.sub.n] we have the following asymptotic relations:

(5.33) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(5.34) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(5.35) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(5.36) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In Theorem 2.2 of [37] a stronger version of asymptotic relations of the type (5.33) - (5.36) has been proved, which however will not be discussed here.

In the next Section we extend all results that have been reviewed in the present section to the non-diagonal case of quadratic Hermite-Pade polynomials.

6. Asymptotics for Non-Diagonal Quadratic Hermite-Pade Polynomials. The investigations by E.B. Saff and R.S. Varga in [35], which have been reviewed in Section 4, have strongly motivated the research in the present section. In [35] not only diagonal sequences, but the whole range of non-diagonal ray sequences of Pade polynomials has been studied, and among other very interesting results it has been shown how the asymptotic cluster sets of the zeros of non-diagonal Pade polynomials continuously change with the angle of the ray sequences in the Pade table. As a consequence one can see how the typical situation of diagonal Pade approximants transforms step by step into that of Taylor polynomials.

In the present section we study analogous questions for quadratic Hermite-Pade polynomials of type I, i.e., we extend the asymptotic relations that have been presented in the last section to non-diagonal ray sequences. Again, it is possible to see how the asymptotic relations of the diagonal case of quadratic Hermite-Pade polynomials transform step by step into those of Pade approximants.

It turns out that the basic structure of the asymptotic relations is very similar in a topological sense for all non-diagonal ray sequences. Therefore, we can use concepts and notations from the last section, only that several formulae have to be changed, they become, of course, somewhat more complicated and also dependent on parameters. On the other hand, the use of the earlier terminology and that of conceptional analogies allow us to give a rather short presentation of the new results.

In the next subsection we introduce general notations, which includes the introduction of the two parameters [lambda] and K, which specify ray sequences. In Subsection 6.2, a Riemann surface R and two functions [psi] and h are introduced that generalize the corresponding objects from Subsection 5.1. In close analogy to Subsection 5.2, we define two systems [GAMMA] and [GAMMA] of Jordan arcs with the help of the function h from Subsection 6.1. As before, arcs from [GAMMA] will be the building blocks for the asymptotic cluster sets of the zeros of the non-diagonal Hermite-Pade polynomials and the associated remainder term. The description of these objects is done in Subsection 6.3. In Subsection 6.4, additional asymptotic results are given. Here, we use the analogy to results presented in Subsection 5.3 for the diagonal case of quadratic Hermite-Pade polynomials. At last, in Subsection 6.5 we describe and discuss the changes that are necessary in the proofs of [37] in order to adapt and transform them into proofs of the results stated here for the non-diagonal situation.

[FIGURE 6.1 OMITTED]

6.1. Non-Diagonal Ray Sequences of Quadratic Hermite-Pade Polynomials. By [p.sub.m], [q.sub.k], [r.sub.l] we denote the three quadratic Hermite-Pade polynomials of type I in the same way as in Section 5 only that now their degrees m, k, l [member of] N are no longer assumed to be equal. The three polynomials [p.sub.m] [member of] [P.sub.m], [q.sub.k] [member of] [P.sub.k], [r.sub.l] [member of] [P.sub.l] correspond to the three polynomials [p.sub.1,n], [P.sub.2,n], [P.sub.3,n] in Definition 2.1 with multi-index n := (m + 1, k + 1, l + 1) [member of] [N.sup.3], m = 2, and the exponential system f = (1, exp, [exp.sup.2]). Note that each of the three polynomials [p.sub.m], [q.sub.k], [r.sub.l] depends on all three indices m, k, l. The three polynomials satisfy the relation

(6.1) [p.sub.m](w) + [q.sub.k](w) [e.sup.w] + [r.sub.i](w) [e.sup.2w] = O([W.sub.|n|-1]) as w [right arrow] 0,

which is a rewriting of the defining relation (2.7) in Definition 2.1. We have |n| = m + k + l+3.

[FIGURE 6.2 OMITTED]

For the rescaling method we now use as new rescaled independent variable

(6.2) z = w/|n|.

Note that the three polynomials [p.sub.m], [q.sub.k], [r.sub.l] together possess m + k + l + 3 free coefficient. The transformation (6.2) leads to the rescaled Hermite-Pade polynomials

(6.3) [P.sub.m](z) := pm(|n|z), [Q.sub.k](z) := [q.sub.k](|n|z), and [R.sub.l](z) := [r.sub.l(|n|z).

Together with a multiplication by [e.sup.-|n|z, relation (6.1) transform into

(6.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

in the new variable z. Since the two relations (6.1) and (6.4) are homogeneous, we can assume that [P.sub.m] is monic, and since we also know that the exponential system f = (1, exp, [exp.sup.2]) is perfect (cf. [18]), we can make the further going assumption that

(6.5) [P.sub.m](z) = [z.sup.m] + ...

In the five Figures 6.1 - 6.5 the zeros of the rescaled polynomials [P.sub.m], [Q.sub.k], [R.sub.l], and some of the zeros of the remainder term [E.sub.n] are plotted for different choices of degrees. Besides of the zeros, there are also lines, which, however, will only be explained further below. The two first Figures 6.1 and 6.2 should be seen together with the two earlier Figures 5.2 and 4.2. The whole sequence of the Figures 5.2, 6.1, 6.2, and 4.2 illustrates how the configuration of the diagonal case of Hermite-Pade polynomials in Figure 5.2 step by step transforms into that of the diagonal case of Pade polynomials in Figure 4.2.

[FIGURE 6.3 OMITTED]

The three Figures 6.3-6.5 illustrate how the asymptotic relations change when the degree k of the middle polynomial [Q.sub.k] is varied, while the degrees m and l of the two outer polynomial [P.sub.m], and [R.sub.l] are kept fixed.

For the asymptotic analysis it is necessary that the three indices m, k, l [member of] N tend to infinity along ray sequences {([m.sub.j], [k.sub.j], [l.sub.j])}j[member of]N. In a ray sequences {([m.sub.j], [k.sub.j], [l.sub.j)}j[member of]N we assume that by definition the following limits hold true:

(6.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In the case of the diagonal sequence we have [lambda] = K = 1. If the parameters [lambda] and K vary through (0, [infinity])(2), we cover all possible ray sequences in the table of quadratic Hermite-Pade polynomials of type I.

6.2. The Riemann Surface R and two Functions [psi] and h. In analogy to Subsection 5.1, a Riemann surface R and two functions [psi] and h are introduced, which all three are immediate generalizations of the corresponding objects in the Definitions 5.1 and 5.2.

DEFINITION 6.1. For [lambda], K, [member of] (0, [infinity]) , the Riemann surface R and the bijective mapping [psi]: R [right arrow] C are defined by the property that the two functions [psi] and

(6.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

satisfy the relation

(6.8) z o [psi]([zeta]) = [pi]([zeta]) for [zeta] [member of] R

with [pi]: R [right arrow] C the canonical projection of the Riemann surface R.

The surface R as well as the function [psi] depend on the two parameters [lambda], K. It is immediate that Definition 5.1 is a special case of Definition 6.1 (one has only to choose [lambda] = K = 1). Again, [psi] o [[pi].sup.-1] is an algebraic function of 3rd degree. For any combination of the two parameters [lambda], K, [member of] (0, [infinity]), the Riemann surface R has four branch points over the base points [z.sub.1],..., [z.sub.4] [member of] C. The dependence of the points [z.sub.1],..., [z.sub.4] on the two parameters [lambda], K is too complicated for making it worthwhile to give an explicit formula for the points. This is a remarkable difference to formula (5.7). The four branch points [z.sub.1],..., [z.sub.4] form two conjugated pairs {[z.sub.1], [z.sub.4] and {[z.sub.2], [z.sub.3]}, and they can be calculated in each concrete situation by solving the equation z[(v).sup.l] = 0 in the v-plane and using then the mapping [psi]. The procedure has been described after (5.7).

As before, the Riemann surface R can be broken down in three sheets [S.sub.-1], [S.sub.0], [S.sub.1], which are glued together along two Jordan arcs [[GAMMA].sub.1] and [[GAMMA].sub.-1] in the same way as this has been described after (5.10). As in (5.10) the point [[zeta].sub.0] [member of] R is defined by

(6.9) [[zeta].sub.0] := [[psi].sup.-1] ([infinity]),

and the defining relation (6.4) of the Hermite-Pade polynomials [P.sub.m], [Q.sub.k], [R.sub.l] is lifted from C to a neighborhood of [[zeta].sub.0] on R.

After the Riemann surface R and the function [psi] have been defined for the new situation, we come to the introduction of the function h on R, which is done in the same way as in Definition 5.2.

DEFINITION 6.2. For [lambda], K [member of] (0, [infinity]), the functions h : R [right arrow] R is defined by

(6.10) h([zeta]) := Re u o [psi]([zeta]) for [zeta] [member of] R

and the function u in (6.10) is now given by

(6.11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Again, it is immediate that the earlier Definition 5.2 is a special case of Definition 6.2. As the Riemann surface R and the function [psi], so also the function h depends on the two parameters [lambda] and K in an essential way. As before in Subsection 5.1, it follows from (6.7), (6.8), (6.10), and (6.11) that for all [lambda], K [member of] (0, [infinity]) the function h is harmonic in R\({[[zeta].sub.0]} [union] [[pi.sup.-1] ({[infinity]})) and subharmonic at [[zeta].sub.0].

[FIGURE 6.4 OMITTED]

6.3. The Definition of Jordan Arcs and a First Result. The definition of the asymptotic cluster sets [S.sub.-1], [S.sub.0], [S.sub.1], and [S.sub.[infinity] for the zeros of the polynomials [P.sub.m], [Q.sub.k], [R.sub.l], and the remainder term [E.sub.n],, respectively, is central for all other asymptotic relations in the present investigation. Starting point for a definition is the function

(6.12) [h.sub.max].(z) := max { h([zeta]) |[zeta] [member of] R, [pi]([zeta]) = z} for z [member of] C,

which now depends on the two parameters [lambda] and K. With the same arguments from potential theory as those mentioned after (5.13) in Subsection 5.2, it follows that [h.sub.max] is subharmonic in C and harmonic outside of a system [GAMMA] of analytic Jordan arcs. The consequences of this observation have been formulated in Lemma 5.3, and this lemma holds also true in the new situation.

LEMMA 6.3. The conclusions of Lemma 5.3 hold true for any combination of parameters [lambda], K [member of] (0, [infinity]). The system [GAMMA] of analytic Jordan arcs in C and a measure v with supp (v) = [GAMMA] depend now on the two parameters A and &

The analytic Jordan arcs in r belong to the larger set

(6.13) [GAMMA] := {z [member of] C|card{h ([zeta]) | [zeta] [member of] [[pi].sup.-1({z})} < 3}

of intersection arcs of the branches of the function h o [[pi].sup.-1]. The arcs in [GAMMA] are the building blocks of the asymptotic cluster sets [S.sub.-1], [S.sub.0], [S.sub.1], and [S.sub.[inifinity], and it follows from Lemma 6.3 and (6.13) that the set [GAMMA] consists of those subarcs of [GAMMA] on which the intersecting branches of h o [[pi].sup.-1] have the same value as that of the function [h.sub.max]. Like the functions h and [h.sub.max], so also the sets [GAMMA] and [GAMMA] depend on the two parameters [lambda] and K.

[FIGURE 6.5 OMITTED]

A good way of calculating the arcs of [GAMMA] is to start from the four branch points [z.sub.1],..., [z.sub.4]. There exist six Jordan arcs in [GAMMA] that end at these four points. They are denoted by [[GAMMA].sub.ij], i = -1,1, j = -1,1, [infinity] as before. Three of them connect the two points [z.sub.1] and [z.sub.4], we denote them by [[GAMMA].sub.1,j], j = -1,1, [infinity], and the other three connect the two points [z.sub.2] and [z.sub.3], and we denote them by [[GAMMA.sub.-1,j], j = -1,1, [infinity]. In each of the two groups, there exists one arc that passes through infinity, another one that has an intersection with (-[infinity], 0), and a third one that has an intersection with (0, [infinity]). According to these three property we denote the three arcs by [[GAMMA].sub.i,[infinity]], [[GAMMA].sub.i,-1], [[GAMMA].sub.i,1] in each group i = -1,1, respectively. For a better orientation one may consult Figure 5.1.

The angle with which the six arcs [[GAMMA].sub.i,j], i = -1,1, j = -1,1, [infinity], end at the four points [z.sub.1],..., [z.sub.4] can be calculated from local developments of the branches of the function h o [[pi].sup.-1] near the four points [z.sub.1],..., [z.sub.4].

The two Jordan arcs [[GAMMA].sub.1,-1] and [[GAMMA].sub.-1,1] intersect in exactly two points [y.sub.1], [y.sub.2] [member of] C, and we have [y.sub.1] = [y.sub.2]. In the symmetric situation of Section 5, these two intersection points [y.sub.1] and [y.sub.2] had to lie on the imaginary axis, this is no longer the case in the general situation. The intersection points [y.sub.1] and [y.sub.2] are important for the definition of the cluster set [S.sub.0] in Definition 6.4. The set [S.sub.0] is constructed in the following way: Starting from the four points [z.sub.1],..., [z.sub.4], one follows the arcs [[GAMMA].sub.1,-1] and [[GAMMA].sub.-1,1 up to the two intersection points [y.sub.1] and [y.sub.2]. Then one stops following the arcs [[GAMMA].sub.1,-1] and [[GAMMA].sub.-1,1], and instead one follows an intersection arc from [GAMMA]\([[GAMMA].sub.1,-1][union] [[GAMMA].sub.-1,1) that connects the two intersection points [y.sub.1] and [y.sub.2]. Details are given in the next definition.

The two arcs [[GAMMA].sub.1,[infinity]] and [[GAMMA].sub.-1,[infinity] are essential for the definition of the cluster set [S.sub.[infinity]], but contrary to the symmetric situation in Subsection 5.2, now the two arcs may have two intersection points in C, which has consequences for the construction of the set [S.sub.[infinity]]. If such intersection points exist, then the procedure is analogous to that applied in the definition of the set [S.sub.0].

After these preparations we can define the four asymptotic cluster sets [S.sub.j = [S.sub.j]([lambda],K), j = -1,0,1, [infinity], for the zeros of non-diagonal sequences of the rescaled Hermite-Pade polynomials [P.sub.m], [Q.sub.l], [R.sub.k] and the associated remainder term [E.sub.n]. The definition is a generalization of Definition 5.4.

DEFINITION 6.4. For [lambda], K [member of] (0, [infinity]), let the point [y.sub.1] [member of] C be defined by [y.sub.1] := [[GAMMA].sub.1,-1] [intersection] [[GAMMA].sub.-1,1] [intersection] {z | Im(z) > 0} and set [y.sub.2] = [y.sub.1]. Then we set

(6.14) [S.sub.1] = [S.sub.1] ([lambda], K.) := [[GAMMA].sub.1,1],

(6.15) [S.sub.-1] = [S.sub.-1] ([lambda], K.) := [[GAMMA].sub.-1,-1].

Let [[z.sub.1], [y.sub.1]] denote the section of the arc [[GAMMA].sub.1,-1], that connects the two points [z.sub.1] and [y.sub.1], and let the three sections [[z.sub.4], [y.sub.2]] [subset] [[GAMMA].sub.1,-1] [[z.sub.2], [y.sub.1]] [subset] [[GAMMA].sub.-1,1] [[z.sub.3], [y.sub.2]] [subset] [[GAMMA].sub.-1,1] b analogous way. Let, further, [[y.sub.1], [y.sub.2]] be the shortest section of the arc or curve in [[GAMMA]\([[GAMMA].sub.1,-1] [union] [[GAMMA].sub.-1,1]) that connects the two points [y.sub.1] and [y.sub.2]. With these definitions we set

(6.16) [S.sub.0] = [S.sub.0] ([lambda], K]) := [[z.sub.1], [y.sub.1]] [union] [[z.sub.2], [y.sub.1]] [union] [[z.sub.3], [y.sub.2]] [union] [[z.sub.4], [y.sub.2]] [union] [[y.sub.1], [y.sub.2]].

If the two arcs [[GAMMA].sub.1,[infinity]] and [[GAMMA].sub.-1,[infinity]], are disjoint in C, then we set

(6.17) [S.sub.[infinity] = [S.sub.0]([lambda], K) := [[GAMMA].sub.1,[infinity]] [union] [[GAMMA].sub.-1,[infinity]],

otherwise, the set [S.sub.[infinity]] is put together from sections of the arcs [[GAMMA].sub.1,[infinity]] and [[GAMMA].sub.-1,[infinity]] that connect the points [z.sub.1],..., [z.sub.4] with the intersection points, and in addition two sections from [GAMMA]\([[GAMMA].sub.1,[infinity]] [union] [[GAMMA].sub.-1,[infinity]]) that connect the two intersection points with infinity. The situation is analogous to the definition of [S.sub.0], and Figure 6.4 is an illustrative example for this construction.

The next lemma can be helpful for the understanding of the definition of the sets [S.sub.-1], [S.sub.0], [S.sub.1], and [S.sub.[infinity]].

LEMMA 6.5. For every pair of parameters ([lambda], K.) [member of] (0, [infinity]) the following three assertions hold true:

(i) If from the set [GAMMA] of Lemma 6.2 the four branch points [z.sub.1],..., [z.sub.4] are removed, then [GAMMA]\{[z.sub.1],..., [Z.sub.4]} consists of four components, and the closure of each of these components is one of the four sets [S.sub.j]= [S.sub.j] ([lambda], K), j = -1,0,1, [infinity], from Definition 6.4.

(ii) The system [GAMMA] consists of those arcs in [GAMMA] on which the values of the intersecting branches of h o [[pi].sup.-1] are identical with the value of [h.sub.max].

(iii) We have

(6.18) [GAMMA] = [S.sub.-1] [union] [S.sub.0] [union] [S.sub.1] [union] [S.sub.[infinity]].

Examples of the union [[GAMMA] = [S.sub.-1] [union] [S.sub.0] [union] [S.sub.1] [union] [S.sub.[infinity]] are shown (full lines) for different constellations of the parameters [lambda] and K in the five Figures 6.1-6.5. Since in these figures the whole set [GAMMA] has been represented by the same type of lines, the four cluster sets [S.sub.-1], [S.sub.0], [S.sub.1], [S.sub.[infinity] have to be identified and distinguished by the type of zeros that are lying close to them. The set [GAMMA] is contained in [GAMMA], and the arcs in [GAMMA] \ [GAMMA] are represented by dashed lines in the five Figures 6.1-6.5.

The next theorem generalizes Theorem 5.6, and it is the analogue to Theorem 4.1, which is concerned with non-diagonal Pade polynomials.

THEOREM 6.6. Let the ray sequence of multi-indices {(mj + 1, [k.sub.j] + 1,[l.sub.j] + 1)} satisfy conditions (6.6) with parameters [lambda], K [member of] (0, [infinity]), then the zeros of the rescaled quadratic Hermite-Pade polynomials [P.sub.mj], [Q.sub.kj], [R.sub.lj] and the associated remainder terms [E.sub.nj], [n.sub.j] = ([m.sub.j] + 1, [k.sub.k] + 1, [l.sub.j] + 1), as introduced in (6.4) satisfy the following asymptotic relations:

(i) All zeros of the polynomials [P.sub.mj] cluster on [S.sub.-1] ([lambda], K) for j [right arrow] [infinity].

(ii) All zeros of the polynomials [Q.sub.kj] cluster on [S.sub.0] ([lambda], K) for j [right arrow] [infinity].

(iii) All zeros of the polynomials [R.sub.lj] cluster on [S.sub.1] ([lambda], K) for j [right arrow] [infinity]

(iv) All zeros of the remainder terms [E.sub.nj] cluster on [S.sub.[infinity]] ([lambda], K) for j [right arrow] [infinity].

The sets [S.sub.-1], [S.sub.0], [S.sub.1], and [S.sub.[infinity]] have been introduced in Definition 6.4.

In the five Figures 6.1-6.5, zeros of the rescaled non-diagonal Hermite-Pade polynomials and their associated remainder terms are plotted together with the union [GAMMA] of their asymptotic cluster sets [S.sub.-1], [S.sub.0], [S.sub.1], and parts of [S.sub.[infinity]] for various constellations of degrees m, k, l, and consequently parameters [lambda], K [member of] (0, [infinity]). The pictures show that there is a very good accordance between the zeros and their asymptotic cluster sets already for the chosen moderate degrees. These pictures also show how the configuration of the asymptotic cluster sets [S.sub.-1], [S.sub.0], [S.sub.1], and [S.sub.[infinity] varies with the parameters [lambda], K. [member of] (0, [infinity]).

Figure 6.5 deserves a special remark. In this figure the two branch points [z.sub.1] = 0.00225+ 0.58856 i and [z.sub.2] = -0.00225 + 0.58856 i as well as two branch points [z.sub.3] = -0.00225- 0.58856 i and [z.sub.4] = 0.00225 - 0.58856 i are laying so close together that they cannot be distinguished in Figure 6.5. Near to these points lie also the intersection points [y.sub.1], [y.sub.2] of [[GAMMA].sub.1,-1] with [[GAMMA].sub.-1,1] and the two intersection points of [[GAMMA].sub.1,[infinity]] and [[GAMMA].sub.-1,[infinity]]. These points can also not be distinguished from their neighbors. As a consequence, the sets [S.sub.0] and [S.sub.[infinity]] appear in Figure 6.5 as straight lines, but in reality there are bifurcations at both ends.

6.4. Further Asymptotic Results. In the present Subsection the results from [37] that have been reviewed in Subsection 5.3 are extended to non-diagonal ray sequences. The structure of presentation is practically identical with that of Subsection 5.3, and remarks from there remain valid in the new circumstances. Changes and adaptations have become necessary only because of peculiarities connected with non-diagonal ray sequences, which are characterized, as before, by the two parameters [lambda], K [member of] (0, [infinity]) from (6.6).

Like in Definition 5.7, we introduce four measures [[micro].sub.j], j = -1,0,1, [infinity], which will be the asymptotic distributions of the zeros of the rescaled non-diagonal quadratic Hermite-Pade polynomials [P.sub.m], [Q.sub.l], [R.sub.k] and the associated remainder terms [E.sub.n].

DEFINITION 6.7. From Lemma 5.3 and Lemma 6.3 we know that there exists a measure v for very parameter constellation [lambda], K [member of] (0, [infinity]). With the sets [S.sub.j] = [S.sub.j] ([lambda],K), j = -1,0,1, [infinity], from Definition 6.2 we define

(6.19) [[micro].sub.j] := v\[s.sub.j], j = -1,0,1,[infinity].

The next theorem can be seen as a completion of Theorem 6.6 since now we have not only the cluster sets of the zeros of [P.sub.m], [Q.sub.l], [R.sub.k], and [E.sub.n], but also their asymptotic distributions.

THEOREM 6.8. Let the ray sequence of multi-indices {([m.sub.j] + 1, [k.sub.j + 1,[l.sub.j] + 1)} satisfy conditions (6.6) with parameters [lambda], K [member of] (0, [infinity]), let [[micro].sub.j], j = -1,0,1, [infinity], be the four measures introduced in (6.19), and let further [[delta].sub.0] be the Dirac measure at z = 0. Then for the zeros of the rescaled non-diagonal Hermite-Pade polynomials [P.sub.mj], [Q.sub.kj], [R.sub.lj] and the associated remainder terms [E.sub.nj], [n.sub.j] = ([m.sub.j] + 1, [k.sub.j] + 1, [l.sub.j] + 1) from (6.4) we have the following asymptotic relations:

(6.20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(6.21) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(6.22) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(6.23) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The measures [[micro].sub.-1], 1/[lambda] [[micro].sub.0] and 1/K [[micro].sub.1] are probability measures, while [[micro].sub.[infinity]] is a measure with infinite mass.

Proposition 5.9 of Subsection 5.3 holds also in the new situation, and it provides means to calculate the density of the asymptotic distributions [[micro].sub.j], j = -1,0,1, [infinity].

In the second part of the present subsection, we present asymptotic relations for the rescaled non-diagonal polynomials [P.sub.mj], [Q.sub.kj], [R.sub.lj] and the associated remainder terms [E.sub.nj]. As in (5.28)-(5.29) we first consider the three branches of the multi-valued function h o [[pi].sup.-1] near infinity. They are denoted by [h.sub.-1], [h.sub.0], and [h.sub.1], and we have

(6.24) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(6.25) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(6.26) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

as z [right arrow] [infinity]. The expressions in (6.24-6.26) follow from the defining relations (6.7), (6.8), (6.10) and (6.11) in Definitions 6.1 and 6.2 in the same way as their forerunners (5.28-5.30) followed from the analogous relations. In a neighborhood of the origin z = 0 we need the branch of the function h o [[pi].sup.-1], which possesses a logarithmic term. It is denoted by [h.sub.[infinity]], and we have

(6.27) [h.sub.[infinity]](z) = (1 + [lambda] + K) log |z| + O(1) as z [right arrow] 0.

DEFINITION 6.9. In the four domains

(6.28) [D.sub.j] := C\[S.sub.j], j=-1,0,1, [infinity],

we defined four functions [h.sub.j], j = -1,0,1,[infinity], by harmonic continuations. In the cases of the first three functions [h.sub.-1], [h.sub.0], [h.sub.1] we start the continuation at infinity with the three function elements (6.24), (6.25), (6.26) and continue this process throughout the domains [D.sub.-1], [D.sub.0], [D.sub.1], respectively. In the case of the function [h.sub.[infinity], we start from the origin and use as initial function element (6.27).

With the four functions [h.sub.j, j = -1,0,1,[infinity], we can formulate the asymptotic relations for the polynomials [P.sub.n], [Q.sub.k], [R.sub.l] and the remainder term [E.sub.n].

THEOREM 6.10. Let the ray sequence of multi-indices {([m.sub.j] + 1,[k.sub.j] + 1,[l.sub.j] + 1)} satisfy conditions (6.6) with parameters [lambda], K [member of (0, [infinity]), then for the rescaled quadratic Hermite-Pade polynomials [P.sub.mj], [Q.sub.kj], [R.sub.lj] and the associated remainder terms [E.sub.nj] = ([m.sub.j] + 1,[k.sub.j] + 1, [l.sub.j] + 1) from (6.4) we have the following asymptotic relations:

(6.29) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(6.30) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(6.31) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(6.32) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The objects on the right-hand side of (6.29)-(6.32) can be calculated efficiently. The most difficult part is perhaps the selection of the right branch of the function h o [[pi].sup.-1] in each concrete situation.

6.5. Basic Ideas of the Proofs. The new results of the present subsection are natural extensions of those reviewed in Section 5. The basic structure has remained very much the same, and so the proofs of the earlier results from [37] can be transplanted without principle difficulties. Having this in mind and also the limitation and readability of the present contribution, it seems to be appropriate not to repeat too many details from [37], and instead to concentrate on the basic ideas of the proofs, which then will make plausible the necessary changes in procedure of [37].

The basic tool for the proofs in [37] is a saddle point method for the evaluation of certain integrals. The analysis starts from explicit representations for the rescaled Hermite-Pade polynomials [P.sub.m], [Q.sub.k], [R.sub.l] and the associated remainder term [E.sub.n], n = (m + 1,k + 1,l + 1), which are given for the diagonal case in Section 1.3 of [37]. For arbitrarily degrees m,k,l [member of] N we have

(6.33) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(6.34) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(6.35) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(6.36) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where the integration paths [C.sub.-1], [C.sub.0], [C.sub.1], [C.sub.[infinity], are closed curves encircling the point -1,0,1, or [infinity], respectively. In (6.33)-(6.36) normalization (6.5) has been taken into account.

If we assume that the three degrees m = [m.sub.j], k = [k.sub.j],l = [l.sub.j] belong to a ray sequence [{([m.sub.j] + 1,[k.sub.j]; + 1,[l.sub.j] + 1)}.sub.j[member of]N] that satisfies (6.6) with parameters [lambda], K [member of] (0, [infinity]), then it is rather immediate from (6.33)-(6.36) that the asymptotic analysis boils down to an asymptotic evaluation of integrals of the form

(6.37) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In order to apply the saddle point method, we have to rewrite (6.37) in the form

(6.38) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

with O(1/[m.sub.j]) a Landau symbol for j [right arrow] [infinity] that holds uniformly for z belonging to a compact set in C. Comparing (6.38) with (6.37) and using the limits in (6.6) together with Stirling's formula one arrives after some calculations (for details see [37, Sections 4.1 and 4.2]) to the conclusion that the functions g and G in (6.38) are necessary of the form

(6.39) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(6.40) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Let now z [member of] C be fixed. From the saddle point method (cf. [20], [40], or [37], Section 4.1) it follows that for [m.sub.j] [member of] N large the value of the integral (6.38) is dominated by the value [g(v)G(v, z).sup.mj] with v [member of] C one of the critical points that are defined by the condition

(6.41) [delta]/partial derivative]v G(v,z) = 0.

Since partial derivative]/partial derivative]v G(v,z) = G(v,z) partial derivative]/partial derivative]v log G(v,z) it follows from (6.39) that equation (6.41) is equivalent to the equation

(6.42) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

From (6.42) we deduce that for every z [member of] C there exist exactly three critical points [v.sub.j] = [v.sub.z,j] [member of] C, j = 1,2,3. As long as z ?? [GAMMA], with [GAMMA] introduced in Lemma 6.3, the value of each of the four integrals in (6.33)-(6.36) depends only on one of the three critical points [v.sub.z,j], j = 1,2,3, and the interaction between the integration paths [C.sub.-1], [C.sub.0], [C.sub.1], [C.sub.[infinity] and the location of the critical points [v.sub.z,j], j = 1,2,3, decides which of the three critical points is relevant for each of the four integrals.

If the three critical points [v.sub.z,j], j = 1,2,3, in the v-plane are pulled back over the z-plane, then this defines the Riemann surface R of Definition 6.1. Indeed, the function z(v) in (6.7) is an immediate consequence of (6.42), and its inverse defines the function [psi]. The role of the critical points in the saddle point method is then also the reason for the specific definition of the function u in (6.11) of Definition 6.2. Indeed, the value of u is log|G([v.sub.z,j], z)| with j [member of] {1,2,3}, and the index j indicates on which sheet of the Riemann surface R the function h lives.

Zeros of the polynomials [P.sub.m], [Q.sub.k], [R.sub.l], and the remainder term [E.sub.n], in (6.33)-(6.36) can only appear when more than one critical point [v.sub.j] becomes effective in the evaluation of the integral in question. Because of this reason the zeros of the polynomials and the remainder term cluster on intersection arcs of branches of the multi-valued function [h o [[pi].sup.-1].

In [37] the whole analysis is given in full detail, but of course limited to the diagonal case [lambda] = K = 1. In the review of results in Section5 a detailed account has been given with respect to the correspondence between the lemmas, theorems, and the proposition in Section 5 and the relevant proofs in [37]. Because of the structural identity between the Sections 5 and 6, this correspondence is also valid for the extended results in the present section, and we will not go into further details here.

Dedicated to Ed Saff on the occasion of his 60th birthday

* Received September 8, 2005. Accepted for publication December 22, 2005. Recommended by D. Lubinsky.

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HERBERT STAHL TFH-Berlin/FB II; Luxemburger Strasse 10; 13353 Berlin; Germany (stahl@tfh-berlin.de). The research has been supported in part by INTAS Research Network NeCCA 03-51-6637, and by the Deutsche Forschungsgemeinschaft (DFG), grant STA 299/13-1. at the origin.
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