Stieltjes interlacing of zeros of Jacobi polynomials from different sequences.

1. Introduction. It is well known that if [{[p.sub.n]}.sup.[infinity].sub.n=0] is a sequence of orthogonal polynomials, the zeros of [p.sub.n] are real and simple, and if [x.sub.1,n] < [x.sub.2,n], < ... < [x.sub.n,n] n are the zeros of [p.sub.n] while [x.sub.1,n - 1] < [x.sub.2,n - 1] < ... < [[x.sub.n - 1, n - 1] are the zeros of [p.sub.n - 1], then

[x.sub.1,n] < [x.sub.1,n - 1] < [x.sub.2,n] < [x.sub.2,n - 1] < ... < [x.sub.n - 1, n - 1] < [x.sub.n,n],

a property called the interlacing of zeros. Another classical result on interlacing of zeros of orthogonal polynomials is due to Stieltjes who proved that if m < n, then between any two successive zeros of [p.sub.m] there is at least one zero of [p.sub.n], a property called Stieltjes interlacing [13, Theorem 3.3.3]. Clearly, if m < n - 1, there are not enough zeros of [p.sub.m] to interlace fully with the n zeros of [p.sub.n]. Nevertheless, using the same argument as Stieltjes, one can show that for m < n - 1, provided [p.sub.m] and [p.sub.n] have no common zeros, there exist m open intervals, with endpoints at successive zeros of [p.sub.n], each of which contains exactly one zero of [p.sub.m]. Moreover, in , Beardon shows that for each m < n - 1, if [p.sub.m] and [p.sub.n] are co-prime, there exists a real polynomial [S.sub.n - m - 1] of degree n - m - 1 whose real simple zeros provide a set of points that completes the interlacing picture. An important feature of the polynomials [S.sub.n - m - 1] is that they are completely determined by the coefficients in the three term recurrence relation satisfied by the orthogonal sequence [{[p.sub.n]}.sup.[infinity].sub.n=0]. The polynomials [S.sub.n - m - 1] are the dual polynomials introduced by de Boor and Saff in  or, equivalently, the associated polynomials analyzed by Vinet and Zhedanov in .

The interlacing property of zeros of polynomials is important in numerical quadrature applications, and in , Segura proved that interlacing of zeros holds, under certain assumptions, within sequences of classical orthogonal polynomials even when the parameter(s) on which they depend lie outside the value(s) required to ensure orthogonality. He also considered the interlacing of zeros of polynomials [p.sub.n - 1] and [p.sub.n + 1] in any orthogonal sequence [{[p.sub.n]}.sup.[infinity].sub.n=0] and showed that interlacing of zeros occurs to the left and to the right of a specified point [12, Theorem 1]. Segura identified this point in terms of the coefficients in the three term recurrence relation satisfied by [{[p.sub.n]}.sup.[infinity].sub.n=0]; equivalently, it is the zero of the linear de Boor-Saff polynomial [3, Theorem 3]. Stieltjes interlacing was studied for the zeros of polynomials from different sequences of one-parameter orthogonal families, namely, Gegenbauer polynomials [C.sup.[lambda].sub.n] and Laguerre polynomials [L.sup.[alpha].sub.n] in  and , respectively, and associated polynomials analogous to the de Boor-Saff polynomials were identified in each case. Related work in which recurrence relations for [sub.2][F.sub.1] functions are considered can be found in .

In a generalization that is complementary to that of Segura in , it was proved in  that the zeros of [P.sup.[alpha], [beta].sub.n] interlace with the zeros of polynomials from some different Jacobi sequences, including those of [P.sup.[alpha]-t,[beta]+k.sub.n] and [P.sup.[alpha]-t,[beta]+k.sub.n-1] for 0 [less than or equal to] t, k [less than or equal to] 2, thereby confirming and extending a conjecture made by Richard Askey in . Numerical examples were given to illustrate that, in general, if t or k is greater than 2, interlacing of zeros need not necessarily occur.

In this paper, we investigate the extent to which Stieltjes interlacing holds between the zeros of two Jacobi polynomials if each polynomial belongs to a sequence generated by a different value of the parameter [alpha] and/or [beta]. We also identify, in each case, a polynomial that plays the role of the de Boor-Saff polynomial [3, 4], in the sense that its zeros provide a (non-unique) set of points that complete the interlacing process.

2. Results. We recall that, for [alpha], [beta] -1, the sequence of Jacobi polynomials [{[P.sup.[alpha], [beta].sub.n]}.sup.[infinity].sub.n=0] is orthogonal with respect to the weight function w(x) = [(1 - x).sup.[alpha]] [(1 + x).sup.[beta]] on (-1, 1) and satisfies the three term recurrence relation 

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

Our first four results consider cases when Stieljes interlacing occurs between the zeros of Jacobi polynomials from different sequences whose degrees differ by two.

THEOREM 2.1.

(i) If [P.sup.[alpha]+t,[beta].sub.n - 1] and [P.sup.[alpha],[beta].sub.n+1] are co-prime, then

(a) the zeros of [P.sup.[alpha]+t,[beta].sub.n-1] and [[[beta].sup.2] - [[alpha].sup.2] + t([beta] - [alpha] + 2n (n + [beta] + 1))]/[(2n + [alpha] + [beta] + t)(2n + [alpha] + [beta] + 2)] interlace with the zeros of [P.sup.[alpha],[beta].sub.n+1] for fixed t [member of] {0, 1, 2};

(b) the zeros of [P.sup.[alpha]+3,[beta].sub.n-1] and [n(n + [alpha] + [beta] + 2) + ([alpha] + 2)(n - [alpha] + [beta])]/[(n + [alpha] + 2)(2 + [alpha] + [beta] + 2)] interlace with the zeros of [P.sup.[alpha],[beta].sub.n+1];

(c) the zeros of [P.sup.[alpha]+4,[beta].sub.n-1] and [2n(n + [alpha] + [beta] + 3) + ([alpha] + 3)([beta] - [alpha])]/[2n(n + [alpha] + [beta] + 3)([alpha] + 3)([alpha] + [beta] + 2)] interlace with the zeros of [P.sup.[alpha],[beta].sub.n+1].

(ii) If [P.sup.[alpha]+t,[beta].sub.n-1] and [P.sup.[alpha],[beta].sub.n+1] are not co-prime, they have one common zero located at the respective points identified in (i) (a) to (c) and the n - 1 zeros of [P.sup.[alpha]+t,[beta].sub.n-1] interlace with the remaining n (non-common) zeros of [P.sup.[alpha],[beta].sub.n+1].

REMARK 2.2. A theorem due to Gibson  proves that if [{[p.sub.n]}.sup.[infinity].sub.n=0] is any orthogonal sequence, the polynomials [p.sub.n+1] and [p.sub.m], m = 1, 2, ..., n - 1 can have at most min{m, n - m} common zeros. Theorem 2.1 (ii) extends Gibson's result to Jacobi polynomials of degree n - 1 and n + 1 from different orthogonal sequences.

REMARK 2.3. The case t = 0 in Theorem 2.1 (i) was proved by Segura [12, Section 3.1]. For completeness and the convenience of the reader, we provide an alternative proof of this case.

Since Jacobi polynomials satisfy the symmetry property [10, p. 82, Equation (4.1.1)]

(2.2) [P.sup.[alpha],[beta].sub.n](x) = [(-1).sup.n] [P.sup.[beta],[alpha].sub.n] (-x),

we immediately obtain the following Corollary of Theorem 2.1.

COROLLARY 2.4.

(i) If [P.sup.[alpha],[beta]+t.sub.n-1] and [P.sup.[alpha],[beta].sub.n+1] are co-prime, then

(a) The zeros of [P.sup.[alpha],[beta]+t.sub.n-1] and [[[beta].sup.2] - [[alpha].sup.2] - t([alpha] - [beta] + 2n (n + [alpha] + 1))[/[(2n + [alpha] + [beta] + t)(2n + [alpha] + [beta] + 2)] interlace with the zeros of [P.sup.[alpha],[beta].sub.n+1] for fixed t [member of] {1, 2};

(b) The zeros of [P.sup.[alpha],[beta]+3.sub.n-1] and - [n(n + [alpha] + [beta] + 2) + ([beta] + 2)(n - [beta] + [alpha])]/[(n + [beta] + 2)(n + [alpha] + [beta] + 2)] interlace with the zeros of [P.sup.[alpha],[beta].sub.n+1];

(c) The zeros of [P.sup.[alpha],[beta]+4.sub.n-1] and - [2n(n + [alpha] + [beta] + 3) + ([beta] + 3)([alpha] - [beta])]/[2n(n + [alpha] + [beta] + 3)([beta] + 3)([alpha] + [beta] + 2)] interlace with the zeros of [P.sup.[alpha],[beta].sub.n+1].

(ii) If [P.sup.[alpha],[beta]+t.sub.n-1] and [P.sup.[alpha],[beta].sub.n+1] are not co-prime, they have one common zero located at the respective points identified in (i) (a) to (c) and the n - 1 zeros of [P.sup.[alpha],[beta]+t.sub.n-1] interlace with the remaining n (non-common) zeros of [P.sup.[alpha],[beta].sub.n+1].

Numerical experiments suggest that results analogous to those proved in Theorem 2.1 and its Corollary also hold as t varies continuously between 0 and 4.

CONJECTURE 2.5. For t [member of] (0,2), if [P.sup.[alpha]+t,[beta].sub.n-1] and [P.sup.[alpha],[beta].sub.n+1] are co-prime, the zeros of [P.sup.[alpha]+t,[beta].sub.n-1] and [[[beta].sup.2] - [[alpha].sup.2] + t([beta] - [alpha] + 2n (n + [beta] + 1))]/[(2n + [alpha] + [beta] + t)(2n + [alpha] + [beta] + 2)] interlace with the zeros of [P.sup.[alpha],[beta].sub.sub.n-1].

Our next two results prove that Stieltjes interlacing of the zeros of Jacobi polynomials from different sequences also holds when both the parameters [alpha] and [beta] change within certain constraints.

THEOREM 2.6.

(i) For each fixed j, k [member of] {1, 2}, if [P.sup.[alpha]+j,[beta]+k.sub.n-1] and [P.sup.[alpha],[beta].sub.n+1]

(a) are co-prime, then the zeros of [P.sup.[alpha]+j,[beta]+k.sub.n-1] and [[beta] - [alpha] - n(k - j)]/[[alpha] + [beta] + 2 + n(4 - j - k)] interlace with the zeros of [P.sup.[alpha],[beta].sub.n+1];

(b) are not co-prime, they have one common zero located at the point identified in (i) (a) and the n - 1 zeros of [P.sup.[alpha]+j,[beta]+k.sub.n-1] interlace with the n remaining (non-common) zeros of [P.sup.[alpha],[beta].sub.n+1].

(ii) If [P.sup.[alpha]+3,[beta]+1.sub.n-1] and [P.sup.[alpha],[beta].sub.n+1].

(a) are co-prime, then the zero of [P.sup.[alpha]+3,[beta]+1.sub.n-1] and [[n.sup.2] + n([alpha] + [beta] + 3) - ([alpha] + 2)([alpha] - [beta])]/[[n.sup.2] + n([alpha] + [beta] + 3) + ([alpha] + 2)([alpha] + [beta])] interlace with the zeros of [P.sup.[alpha],[beta].sub.n+1];

(b) are not co-prime, then they have one common zero located at the point identified in (ii) (a) and the n - 1 zeros of [P.sup.[alpha]+j,[beta]+k.sub.n-1] interlace with the n remaining (non-common) zeros of [P.sup.[alpha],[beta].sub.n+1].

(iii) If [P.sup.[alpha]+1,[beta]+3.sub.n-1] and [P.sup.[alpha],[beta].sub.n+1]

(a) are co-prime, then the zeros of [P.sup.[alpha]+1,[beta]+3.sub.n-1] and [- [n.sup.2] - n([alpha] + [beta] + 3) - ([beta] + 2)([alpha] - [beta])]/[[n.sup.2] + n ([alpha] + [beta] + 3) + ([beta] + 2)([alpha] + [beta] + 2)] interlace with the zeros of [P.sup.[alpha],[beta].sub.n+1];

(b) are not co-prime, then they have one common zero located at the point identified in (iii) (a) and the n - 1 zeros of [P.sup.[alpha]+1,[beta]+3.sub.n-1] interlace with the n remaining (non-common) zeros of [P.sup.[alpha],[beta].sub.n+1].

THEOREM 2.7.

(i) If the respective pairs of polynomials are co-prime, then

(a) the zeros of [P.sup.[alpha]-1,[beta]+1.sub.n-1] and [[alpha] + [beta]]/[2n + [alpha] + [beta]] interlace with the zeros of [P.sup.[alpha],[beta].sub.n+1];

(b) the zeros of [P.sup.[alpha]-1,[beta]+ 2.sub.n - 1] and [-n + [beta] + 1]/[n + [beta] + 1] interlace with the zeros of [P.sup.[alpha], [beta].sub.n + 1];

(c) the zeros of [P.sup.[alpha]+1,[beta]-1.sub.n - 1] and [-[alpha] - [beta]]/[2n + [alpha] + [beta]] interlace with the zeros of [P.sup.[alpha], [beta].sub.n + 1];

(d) the zeros of [P.sup.[alpha]+2,[beta]+1.sub.n - 1] and [n - [alpha] - 1]/[n + [alpha] + 1] interlace with the zeros of [P.sup.[alpha], [beta].sub.n + 1].

(ii) If the respective pairs of polynomials in (i) (a) to (d) are not co-prime, then they have one common zero located at the points identified in (i) (a) to (d) and the n - 1 zeros of the respective polynomial of degree n 1 in each case interlace with the n (non-common) zeros of (b) the zeros of [P.sup.[alpha] - 1, [beta] + 2.sub.n - 1] and - n + [beta] + 1/n + [beta] + 1 interlace with the zeros of [P.sup.[alpha], [beta].sub.n + 1].

Remark 2.8. Restrictions on the ranges of t and k are required in our theorems since, in general, Stieltjes interlacing is not retained between the zeros of Jacobi polynomials from different sequences, whose degrees differ by two. Using Mathematica, we see that

When n = 5, [alpha] = 20.7 and [beta] = 0.5, the zeros of [P.sup.[alpha],[beta].sub.6] and [P.sup.[alpha]+5,[beta].sub.4] or [P.sup.[alpha],[beta]-1.sub.4] do not interlace, illustrating that Stieltjes interlacing does not hold in general for t > 4, k = 0 or t = 0, k < 0.

When t = k = 1 and n, a and / are chosen as in the example above, the zeros of [P.sup.[alpha]-1,[beta]-1.sub.4] and [P.sup.[alpha],[beta].sub.6] do not interlace.

The zeros of [P.sup.[alpha], [beta].sub.n + 1] and those of [P.sup.[alpha] + 4, [beta] + 1.sub.n - 1] or [P.sup.[alpha]+3, [beta]+2.sub.n - 1] do not interlace when n = 7, [alpha] = -0.9 and [beta] = 329.3.

We now state a general result for Stieltjes interlacing between the zeros of [P.sup.[alpha],[beta].sub.n + 1] and the n - k zeros of the kth derivative of [P.sup.[alpha],[beta].sub.n].

THEOREM 2.9. Let [P.sup.[alpha],[beta].sub.n], [alpha],[beta] > -1, n [member of] N, denote the Jacobi polynomial of degree n.

(i) For each k [member of] {1, 2, ..., n - 1}, there exist polynomials [G.sub.k] and [H.sub.k] of degree k such that

(2.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the Pochhammer symbol [10, p. 8, Equation (1.3.6)].

(ii) Let k [member of] {1,2, ..., n - 1}, k fixed. If [P.sup.[alpha],[beta].sub.n + 1] and [P.sup.[alpha]+k, [beta] + k.sub.n - k] are co-prime, then the zeros of the kth derivative of [P.sup.[alpha], [beta].sub.n], together with the k real zeros of [G.sub.k], interlace with the zeros of [P.sup.[alpha], [beta].sub.n + 1].

(iii) Let k [member of] {1,2, ..., n - 1}, k fixed. If [P.sup.[alpha], [beta].sub.n + 1] and [P.sup.[alpha] + k, [beta] + k.sub.n - k] have r common zeros, then the (n - 2r) non-common zeros of the product [G.sub.k][P.sup.[alpha] + k, [beta] + k.sub.n - k], together with the r common zeros of [P.sup.[alpha], [beta].sub.n + 1] and [P.sup.[alpha] + k, [beta] + k.sub.n - k], interlace with the (n + 1 - r) non-common zeros of [P.sup.[alpha], [beta].sub.n + 1].

3. Proofs. Jacobi polynomials are linked with the [sub.2][F.sub.1] Gauss hypergeometric polynomials via the following identity [1, p. 99]

(3.1) [P.sup.[alpha], [beta].sub.n] (x) = [[([alpha] + 1).sub.n]/n!][sub.2][F.sub.1](- n, n + [alpha] + [beta] + 1; [alpha] + 1; [1 - x]/2).

In our proofs, we make use of this connection between Jacobi and [sub.2][F.sub.1] hypergeometric polynomials, as well as the following contiguous function relations satisfied by [sub.2][F.sub.1] polynomials.

Lemma 3.1. Let [F.sub.n] = [sub.2][F.sub.1](-n, b; c; z) and denote [sub.2][F.sub.1](- n - 1, b + 1; c; z) by [F.sub.n+1](b+), [sub.2][F.sub.1](- n + 1, b + 1; c - 3; z) by [F.sub.n - 1](b +, c - 3) and so on. Then,

(3.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3 6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3.10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where a = (b + 1)(b + 2)(b - c)(c + n + 1)(c + n + 2)(1 + b + n)[z.sup.4]n.

Proof. For each j = 1, 2, ..., n, the coefficient of [z.sup.j] on the left-hand side of (3.2) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

while the coefficient of [z.sup.j] on the right-hand side of (3.2) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A straightforward calculation shows that these coefficients are equal and the result follows. The other identities can be proved in the same way by comparing coefficients.

REMARK 3.2. The identities in Lemma 3.1 follow from the contiguous relations for [sub.2][F.sub.1] hypergeometric polynomials [11, p. 71]. A useful algorithm in this regard is available as a computer package .

The following Lemma simplifies the proofs of Theorem 2.1 and Theorem 2.6.

LEMMA 3.3. Let [{[p.sub.n]}.sup.[infinity].sub.n=0] be a sequence of polynomials orthogonal on the (finite or infinite) interval (c, d). Let [g.sub.n - 1] be any polynomial of degree n - 1 that for each n [member of] N satisfies

(3.11) [g.sub.n - 1](x) = [a.sub.n](x)[p.sub.n + 1](x) - (x - [A.sub.n])[b.sub.n](x)[p.sub.n](x)

for some constant [A.sub.n] and some functions [a.sub.n](x) and [b.sub.n](x), with [b.sub.n](x) [not equal to] 0 for x [member of] (c, d). Then, for each n [member of] N,

(i) the zeros of [g.sub.n - 1] are all real and simple and, together with the point [A.sub.n], they interlace with the zeros of [p.sub.n + 1] if [g.sub.n - 1] and [p.sub.n + 1] are co-prime;

(ii) if [g.sub.n - 1] and [p.sub.n + 1] are not co-prime, they have one common zero located at x = [A.sub.n] and the n - 1 zeros of [g.sub.n - 1] interlace with the n (non-common) zeros of [p.sub.n + 1].

Proof. Let [w.sub.1] < [w.sub.2] < ... < [w.sub.n + 1] denote the zeros of [p.sub.n + 1].

(i) Since [p.sub.n] and [p.sub.n + 1] are always co-prime, and by assumption [b.sub.n](x) [not equal to] 0 for x [membe r of] (c, d) and [p.sub.n + 1] and [g.sub.n - 1] are co-prime, we deduce from (3.11) that [A.sub.n] [not equal to] [w.sub.k] for any k [member of] {1, 2, ..., n + 1}. Evaluating (3.11) at wk and [w.sub.k + 1], we obtain

(3.12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for each k [member of] {1, 2, ..., n}. Since [w.sub.k] and [w.sub.k+1] [member of] (c, d) while [b.sub.n] does not change sign in (c, d), we know that [b.sub.n]([w.sub.k])[b.sub.n]([w.sub.k+1]) > 0. Hence, the right-hand side of (3.12) is positive if and only if [A.sub.n] [not member of] ([w.sub.k],[w.sub.k+1]). Since pn([w.sub.k])pn([w.sub.k+1]) < 0 for each k [member of] {1, 2,..., n} because the zeros of pn and [p.sub.n + 1] are interlacing, we deduce that, provided An [not member of] ([w.sub.k],[w.sub.k+1]), [g.sub.n - 1] has a different sign at consecutive zeros of [p.sub.n + 1] and therefore has an odd number of zeros (counting multiplicity) in each interval ([w.sub.k], [w.sub.k+1]), k [member of] {1, 2,..., n}, apart from one interval that may contain the point An. It follows from the Intermediate Value Theorem that for each n [member of] N the n - 1 real simple zeros of [g.sub.n - 1], together with the point [A.sub.n], interlace with the n + 1 zeros of [p.sub.n + 1].

(ii) If [p.sub.n + 1] and [g.sub.n - 1] have common zeros, it follows from (3.11) that there can only be one common zero at x = [A.sub.n] since [p.sub.n] and [p.sub.n + 1] are co-prime. For x [not equal to] [A.sub.n] we can rewrite (3.11) as

(3.13) [G.sub.n - 2](x) = [a.sub.n](x)[P.sub.n](x) - [b.sub.n](x)[p.sub.n](x),

where (x - [A.sub.n])[G.sub.n - 2](x) = [g.sub.n - 1] (x) and (x - [A.sub.n])Pn(x) = [p.sub.n + 1](x). Note that the zeros of [P.sub.n] are exactly the n (non-common) zeros of [p.sub.n + 1], say [v.sub.1] < ... < [v.sub.n], and at most one interval of the form ([v.sub.k], [v.sub.k+1]), k [member of]{1, ..., n - 1}, can contain the point [A.sub.n]. Evaluating (3.13) at [v.sub.k] and [v.sub.k+1], for each k [member of]{1, ..., n - 1} such that [A.sub.n] [not member of] ([v.sub.k], [v.sub.k+1]), we obtain

[G.sub.n - 2]([v.sub.k]) [G.sub.n - 2] ([v.sub.k+1]) = [b.sub.n]([v.sub.k])[b.sub.n] ([v.sub.k+1])[p.sub.n] ([v.sub.k])[p.sub.n] ([v.sub.k + 1]) < 0,

and it follows that [G.sub.n - 2] has an odd number of zeros in each interval ([v.sub.k], [v.sub.k+1]), k [member of] {1, 2, ..., n}, that does not contain [A.sub.n]. Since there are at least n - 2 of these intervals and deg([G.sub.n - 2]) = n - 2, there are at most n - 2 such intervals and we deduce that [A.sub.n] = [w.sub.j] where j [member of] {2, ..., n} and the zeros of [G.sub.n - 2], together with the point [A.sub.n], interlace with the n zeros of [P.sub.n]. The stated result is then an immediate consequence of the definitions of [G.sub.n - 2] and [P.sub.n].

Proof of Theorem 2.1

(i) (a) If t = 0, the result follows from (2.1) and Lemma 3.3 (i). For t = 1, we use (3.2) with b = [alpha] + [beta] + n + 1 and c = [alpha] + 1, together with (3.1), and then apply Lemma 3.3 (i). For t = 2, the stated result follows from (3.3) and (3.1) together with Lemma 3.3 (i).

(b) Replacing b by n + [alpha] + [beta] + 1, c by a + 1 and z by 1 - x/2 in (3.6) and using

(3.1), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where A(x) = n(n + [beta])(x - 1) + 2([alpha] + 1)([alpha] + 2). Lemma 3.3 (i) then yields the result.

(c) From (3.10) and (3.1) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[C.sub.n] = 2n(n + [alpha] + [beta] + 3) +([alpha] + 3)([alpha] + [beta] + 2),

[D.sub.n] = (2n + [alpha] + [beta] + 2)(n + [beta])(n + [alpha] + [beta] + 2)(n + [alpha] + [beta] + 3), and B(x) is a polynomial of degree 2 in x which depends on n, [alpha], and [beta].

The result follows from Lemma 3.3 (i).

(ii) This follows immediately from Lemma 3.3 (ii) and the proofs of Theorem 2.1 (i)(a) to (c).

Proof of Theorem 2.6

(i) (a) The case when j = k = 1 will be proved in Theorem 2.9. For j = k = 2, (3.8) and (3.1) yield

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and C(x) is a polynomial of degree 2 in x which depends on n, [alpha] and [beta]. The result follows from Lemma 3.3 (i).

For j = 1, k = 2, the mixed recurrence relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is obtained from (3.1) together with (3.4). Lemma 3.3 (i) then yields the stated result.

For j = 2, k = 1, the result follows from the symmetry property (2.2).

(b) From (3.1) and (3.9), we obtain the mixed recurrence relation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and Lemma 3.3 (i) then yields the stated result.

(c) This follows directly from the symmetry property (2.2).

(ii) This follows from Lemma 3.3 (ii) and the proofs of Theorem 2.6 (i) (a) to (c).

We omit the proof of Theorem 2.7 which follows exactly the same reasoning as the proofs of Theorems 2.1 and 2.6.

Proof of Theorem 2.9

(i) We use the mixed recurrence relations

(3.14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(3.15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which can be obtained from (3.1), (3.5), and (3.7). We prove our result by induction on k.

For k = 1, equation (2.3) is the same as equation (3.14) with [H.sub.0](x) = - 1, [G.sub.1](x) = 1/2 ((2n + [alpha] + [beta] + 2)x + [alpha] - [beta]) and [Q.sub.n,1] = 1/4(2n + [alpha] + [beta] + 2). Therefore, (2.3) holds for k = 1.

Next, we assume that the result holds for m = 1, 2, ..., k, i.e we assume that (3.16)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

with [G.sub.m] and [H.sub.m] polynomials of degree m and [Q.sub.n,m] = [[(n + [alpha] + [beta] + 2).sub.m - 1] (2n + [alpha] + [beta] + 2)]/[2.sup.2m] for m = 1, 2, ..., k.

For m = k +1, the left-hand side of (2.3) is equal to

[(1 [x.sup.2]).sup.k+1] [Q.sub.n, k + 1] [P.sup.[alpha] + k + 1, [beta] + k + 1.sub.n - k - 1] (x),

and, applying (3.14) and (3.15), a straightforward calculation shows that this equals

[G.sub.k + 1](x)[P.sup.[alpha], beta].sub.n] (x) + (n + 1) [H.sub.k] (x) [P.sup.[alpha], beta].sub.n + 1] (x)

with

[H.sub.k] (x) = -n/2 (x - [[alpha] - [beta]]/[2n + [alpha] + [beta] 1 + 2]) [H.sub.k - 1](x) - [n + [alpha] + [beta] + 2]/[2n + [alpha] + [beta] + 2] [G.sub.k] (x)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which is the right-hand side of (2.3) for m = k + 1. It follows that (3.16) holds for m = k + 1, and the result follows by induction on k.

(ii) We note that [D.sup.k] [[P.sup.[alpha], [beta].sub.n]] = 1/[2.sup.k][(n + [alpha] + [beta] + 1).sub.k] [P.sup.[alpha] + k, [beta] + k.sub.n - k], where [D.sup.k] denotes the k-th derivative [13, p. 63]. From (2.3), provided [P.sup.[alpha], [beta].sub.n + 1] (x) [not equal] 0, we have

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

Now, if [w.sub.1] < [w.sub.2] < ... < [w.sub.n + 1] are the zeros of [P.sup.[alpha], [beta].sub.n + 1], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [A.sub.j] > 0 for each j [member of] {1, ..., n + 1} [13, Theorem 3.3.5]. Therefore (3.17) can be written as

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

Since [P.sup.[alpha], [beta].sub.n + 1] and [P.sup.[alpha], [beta].sub.n] are always co-prime while [P.sup.[alpha], [beta].sub.n + 1] and [P.sup.[alpha] + k, [beta] + k.sub.n - k] are co-prime by assumption, it follows from (2.3) that [G.sub.k]([w.sub.j]) [not equal to] 0 for any j [member of] {1, 2, ..., n + 1}. Suppose that [G.sub.k] does not change sign in [I.sub.j] = ([w.sub.j], [w.sub.j + 1]) where j [member of] {1, 2, ..., n}. Since [A.sub.j] > 0 and the polynomial [H.sub.k - 1] is bounded on [I.sub.j] while the right hand side of (3.18) takes arbitrarily large positive and negative values, it follows that [P.sup.[alpha] + k, [beta] + k.sub.n - k] must have an odd number of zeros in each interval in which [G.sub.k] does not change sign. Since [G.sub.k] is of degree k, there are at least n - k intervals ([w.sub.j], [w.sub.j + 1]), j [member of] {1,..., n} in which [G.sub.k] does not change sign, and so each of these intervals must contain exactly one of the n k real, simple zeros of [P.sup.[alpha] + k, [beta] + k.sub.n - k]. We deduce that the k zeros of [G.sub.k] are real and simple and, together with the zeros of [P.sup.[alpha] + k, [beta] + k.sub.n - k], interlace with the n + 1 zeros of [P.sup.[alpha], [beta].sub.n + 1].

(iii) Assume that [P.sup.[alpha], [beta].sub.n + 1] and [P.sup.[alpha] + k, [beta] + k.sub.n - k] have r common zeros. From (2.3), it follows that if [P.sup.[alpha] + k, [beta] + k.sub.n - k] and [P.sup.[alpha], [beta].sub.n + 1] have any common zeros, these must also be zeros of [G.sub.k] since [P.sup.[alpha], [beta].sub.n] and [P.sup.[alpha], [beta].sub.n + 1] are co-prime. It follows that r [less than or equal to] min{k, n - k} and there are at least (n - 2r) open intervals [I.sub.j] = ([w.sub.j], [w.sub.j + 1]) with endpoints at successive zeros [w.sub.j] and [w.sub.j + 1] of [P.sup.[alpha], [beta].sub.n + 1] where neither [w.sub.j] or [w.sub.j + 1] is a zero of [P.sup.[alpha] + k, [beta] + k.sub.n - k] or [G.sub.k](x). If [G.sub.k] does not change sign in an interval [I.sub.j] = ([w.sub.j], [w.sub.j + 1]), it follows from (3.18), since [A.sub.j] > 0 and [H.sub.k - 1] is bounded while the right hand side takes arbitrarily large positive and negative values for x [member of] [I.sub.j], that [P.sup.[alpha] + k, [beta] + k.sub.n - k] must have an odd number of zeros in that interval. Since this applies to at least (n - 2r) intervals [I.sub.j] and [P.sup.[alpha] + k, [beta] + k.sub.n - k] has exactly (n - k - r) simple zeros that are not zeros of [P.sup.[alpha], [beta].sub.n + 1] while [G.sub.k] has at most (k - r) zeros that are not zeros of [P.sup.[alpha], [beta].sub.n + 1], it follows that there must be exactly (n - 2r) intervals Ij = ([w.sub.j], [w.sub.j + 1]) with endpoints at successive zeros [w.sub.j] and [w.sub.j + i] of [P.sub.n + 1] where neither [w.sub.j] or [w.sub.j + 1] is a zero of PnQ+kk,1+k. This implies that the common zeros of [P.sup.[alpha], [beta].sub.n + 1] and [P.sup.[alpha] + k, [beta] + k.sub.n - k] cannot be two consecutive zeros of [P.sup.[alpha], [beta].sub.n + 1], and the stated result now follows using the same argument as in (ii).

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K. DRIVER ([dagger]), A. JOOSTE ([double dagger]), and K. JORDAAN ([double dagger])

* Received May 5, 2011. Accepted for publication September 28, 2011. Published online October 28, 2011. Recommended by F. Marcellan. Research by K. D. is supported by the National Research Foundation of South Africa under grant number 2053730. Research by A. J. and K. J. is supported by the National Research Foundation under grant number 2054423.

([dagger]) Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag X3, Rondebosch 7701, Cape Town, South Africa (kathy.driver@uct.ac.za)

([double dagger]) Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, 0002, South Africa (alta.jooste@up.ac.za, kjordaan@up.ac.za).
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