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More examples on general order multivariate Pade approximants for pseudo-multivariate functions.

Abstract. Although general order multivariate Pade approximants have been introduced some decades ago, very few explicit formulas have been given so far. We show in this paper that, for any given pseudo-multivariate function, we can compute its (M, N) general order multivariate Pade approximant for some given index sets M, N with the usage of Maple or other software. Examples include a multivariate form of the sine function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

a multivariate form of the logarithm function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

a multivariate form of the inverse tangent function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and many others.

Key words. multivariate Pade approximant; pseudo-multivariate function

AMS subject classification. 41A21

1. Introduction. Multivariate Pade approximants have been extensively investigated in the past few decades. The existence, uniqueness and non-uniqueness for homogeneous and general order multivariate Pade approximants and some convergence theorems have been established [3], [4], [5]. Despite all these activities, there are very few explicit constructions of multivariate Pade approximants. By using the residue theorem and the functional equation method, several researchers have successfully constructed multivariate Pade approximants to some functions which satisfy functional equations [2], [11], [12], [13]. Unfortunately, not many functions satisfy those functional equations. Besides, because the index sets for the numerator and denominator polynomials can not be chosen freely, most numerators of the approximants look complicated. In Cuyt-Tan-Zhou [7], we explicitly construct multivariate Pade approximants to so-called pseudo-multivariate functions, by using the Pade approximants of particular univariate functions which, in most of the cases, are the univariate projections of the pseudo-multivariate functions obtained by letting all but one variable be zero. Examples given in [7] are the general order multivariate Pade approximants for the multivariate form of the exponential function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

the multivariate form of the q- exponential function (see also [1], [12])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

the Appell function (see also [6], [9])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and the multivariate form of the partial theta function (see also [12], [8], [1])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Our aim in this paper is to show that, by using Theorem 2.1 in [7] and Maple, we can compute the (M, N) general order multivariate Pade approximant to any given pseudo-multivariate function for M, N, E defined in Theorem 2.1 in [7] (and stated as Theorem 1.3 in this paper).

DEFINITION 1.1. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

be a formal power series, and let M, N, E be index sets in N x N = [N.sup.2]. The (M, N) general order multivariate Pade approximant to F(x, y) on the set E is a rational function

[[M/N].sub.E](x, y) := P(x, y)/Q(x, y)

where the polynomials

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

are such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and E satisfying the inclusion property

(1.2) (i,j) [member of] E, 0 [less than or equal to] k [less than or equal to] i, 0 [less than or equal to] l [less than or equal to] j [right arrow] (k,l) [member if E.

DEFINITION 1.2. A multivariate function F(x, y) is said to be pseudo-multivariate if the coefficients of its formal power series

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

satisfy

[c.sub.ij] = g(i+j), i,j = 0,1,...,

where g (k) is a certain function of k.

Please see [7] for more details on definitions of the (M, N) general order multivariate Pade approximant to F(x, y) on the set E, which is slightly different from the one given in earlier papers, and the pseudo-multivariate functions.

THEOREM 1.3. (Theorem 2.1 in [7]) Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

be a pseudo-multivariate function. Form, n [member of] N, let

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

be the (m, n) Pade approximant of the function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

let s = max{m, n}, and let

(1.4) N := {(i,j) : 0 [less than or equal to] i,j [less than or equal to] n},

(1.5) M := {(i,j): 0 [less than or equal to] i,j [less than or equal to] s} [intersection] {(i,j) : 0 [less than or equal to] i+j [less than or equal to] m+n},

(1.6) E := {(i,j): 0 [less than or equal to] i + j [less than or equal to] m+ n, i,j [greater than or equal to]0}

be index sets in [N.sup.2]. Then the (M, N) general order multivariate Pade approximant to F(x, y) on the index set E is

[[M/N].sub.E](x, y) = P(x,y)/Q(x,y),

where

(1.7) Q (x, y) := [q.sub.qm,n] (x) [q.sub.m,n] (y),

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

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

The proof of this theorem in [7] uses the fact that if x [not equal to] y,

F (x, y) = xh (x) - yh (y)/x -y,

and if x = y,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

REMARK 1.4. It is easy to see that P (x, y) / Q (x, y) is irreducible, [p.sub.m,n] (z)/[q.sub.m,n] (z) is irreducible.

2. More Explicit Constructions. The importance of finding more explicit formulas of multivariate Pade approximants to some given functions is obvious as very few of them are known so far. For any given pseudo-multivariate function, we can compute its (M, N) general order multivariate Pade approximant on the set E for given m, n, where M, N, E, and m, n are defined in Theorem 1.3. For some pseudo-multivariate functions whose one variable projection functions have general explicit formulas of Pade approximants, like the ones given in [7], we can use Theorem 1.3 to write their general order multivariate Pade approximants. For those pseudo-multivariate functions whose one variable projection functions don't have general explicit formulas of Pade approximants, we can use Maple or other software to compute the (M, N) general order multivariate Pade approximant on the set E. In what follows we present an example of a short procedure in Maple, called mpa, to compute the (M, N) general order multivariate Pade approximant for two variable pseudo-multivariate functions.

In the procedure mpa(f, x, m, n), f is the one variable projection function of the pseudo-multivariate function F, x is the variable of f , m and n are non-negative integers. It computes the (M, N) general order multivariate Pade approximant to F(x, y) on the set E, where M, N, and E are defined in Theorem 1.3.

> with(numapprox):

> mpa:=proc(f,x,m,n)

> local g,px,py,gx,qy,PP QQ,PQ;

> g:=pade(f,x,[m,n]);

> px:=numer(g); qx:=denom(g);

> py:=subs(x=y,px); qy:=subs(x=y,qx);

> PP:=simplify((x*px*qy-y*qx*py)/(x-y));

> QQ:=simplify(gx*qy);

> PQ:=simplify(PP/QQ);

> return (PQ);

> end proc:

EXAMPLE 2.1. The function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

is a pseudo-multivariate function with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Running mpa(sin(x), x, 3, 2) in Maple gives

-140[x.sup.3] + 7[x.sup.3][y.sup.2] + 140[x.sup.2]y + 7[x.sup.2][y.sup.3] - 120x + 140x[y.sup.2] - 1200y + 140[y.sup.3]/3 (20 + [x.sup.2]) (20 + [y.sup.2])

which is the (M, N) general order multivariate Pade approximant to the function S (x, y) on the set E, with

N = {(i,j) : 0 [less than or equal to] i,j [less than or equal to] 2},

M = {(i,j) : 0 [less than or equal to] <i,j [less than or equal to] 3, i+j [less than or equal to] 5},

E = {(i,j) : 0 [less than or equal to] i+j [less than or equal to] 5}.

EXAMPLE 2.2. A multivariate form of the logarithm series is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It is a pseudo-multivariate function with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Running mpa(ln (1 - z),z 3,1) gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which is the (M, N) general order multivariate Pade approximant to the function L (x, y) on the set E, with

N = {(i,j) : 0 [less than or equal to] i, j [less than or equal to] 1},

M = {(i,j) : 0 [less than or equal to] i, j [less than or equal to] 3,i+j [less than or equal to] 4},

E = {(i,j) : 0 [less than or equal to] i+j [less than or equal to] 4}.

EXAMPLE 2.3. The function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

is a pseudo-multivariate function with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Similarly, mpa(sgr (1 + 2z),z,3,2) gives the (M, N) general order multivariate Pade approximant P (x, y)/Q (x, y) to R(x, y) on the index set E for |x|, |y| < 1/2, where

P (x, y) = 4[x.sup.3] + 8[x.sup.3]y + 3[x.sup.3][y.sup.2] + 36[x.sup.2] + 76[x.sup.2]y + 35[x.sup.2][y.sup.2] + 3[x.sup.2][y.sup.3] +48x + 120xy + 76x[y.sup.2] + 8x[y.sup.3] + 48y + 36[y.sup.2] + 4[y.sup.3] + 16,

and

Q (x, y) = (4 + 8x + 3[x.sup.2]) (4 + 8y + 3[y.sup.2]),

with the same M, N, E as given in Example 2.1.

EXAMPLE 2.4. The function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

is a pseudo-multivariate function with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then we can use mpa(arctan(x), x, m, n) to compute the (M, N) general order multivariate Pade approximant to T (x, y) on the index set E for any given positive integers m, n.

EXAMPLE 2.5. The function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

is a pseudo-multivariate function with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then we can use mpa(arcsin(x) - x, x, m, n) to compute the (M, N) general order multivariate Pade approximant to H (x, y) on the index set E for any given positive integers m, n.

For some pseudo-multivariate series, we might not be able to have the explicit functions for the sum of their projection series of one variable. In this case, we can use Maple to write the partial sum up to the degree of m + n + 1 in the variable, and compute the (m, n) Pade approximant to the partial sum and then use mpa(f, x, m, n) to compute the (M, N) general order multivariate Pade approximant to the multivariate series on the index set E for any given positive integers m, n.

EXAMPLE 2.6. The function

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

is a pseudo-multivariate function with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

First, we write a short procedure Sn(n,x) to write the sum of the first n terms of the series in h (z)

> Sn:=proc(n,x)

> local i,s;

> s:=0;

> for i from 0 to n do:

> s:=s+x^i/(factorial(i)+i^3+1); od;

> return (s);

> end proc:

Then running mpa(Sn(6, x),x,3,2) gives the the (M, N) general order multivariate Pade approximant P (x, y) /Q (x, y) to R(x, y) on the index set E, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

Q (x, y) = 1122 (-1937619 + 484160x + 82841[x.sup.2]) (-1937619 + 484160y + 82841[y.sup.2]),

with the same M, N, E as given in Example 2.1. We can use mpa(Sn(m+n + 1, x), x, m, n) to compute the (M, N) general order multivariate Pade approximant to F(x, y) on the index set E for any given positive integers m, n.

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

* Received February 25, 2005. Accepted for publication November 16, 2005. Recommended by D. Lubinsky. Research supported by NSERC of Canada.

REFERENCES

[1] P. B. BORWEIN, Pade approximants for the q-elementary functions, Constr. Approx., 4 (1988), pp. 391-402.

[2] P. B. BORWEIN, A. CUYT, AND P. ZHOU, Explicit construction of general multivariate Pade approximants to an Appell Junction, Adv. Comp. Math., 22 (2005), pp. 249-273.

[3] A. CUYT, How well can the concept of Pade approximant be generalized to the multivariate case?, J. Comp. Appl. Math., 105 (1999), pp. 25-50.

[4] A. CUYT, K. DRIVER, AND D. LUBINSKY,A direct approach to convergence ofmultivariate nonhomogeneous Pade approximants, J. Comp. Appl. Math., 69 (1996), pp. 353-366.

[5] A. CUYT, K. DRIVER, AND D. LUBINSKY, Kronecker type theorems, normality and continuity of the multivariate Pade operator, Numer. Math., 73 (1996), pp. 311-327.

[6] A. CUYT, K. DRIVER, J. TAN, AND B. VERDONK, Exploring multivariate Pade approximants for multiple hypergeometric Series, Adv. Comp. Math., 10 (1999), pp. 29-49.

[7] A. CUYT, J. TAN, AND P. ZHOU, General order multivariate Pade approximants for pseudo-multivariate functions, Math. Comput., 75 (2006), pp. 727-741.

[8] D. S. LUBINSKY AND E. B. SAFE, Convergence of Pade approximants of partial theta functions and the Rogers-Szego polynomials, Constr. Approx., 3 (1987), pp. 331-361.

[9] L. J. SLATER, Generalized Hypergeometric Functions, Cambridge University Press, 1966.

[10] J. W IMP AND B. BECKERMANN, Some explicit formulas for Pade approximants of ratios of hypergeometric functions, WSSIAA 2, pp. 427-434, World Scientific Publishing Company, 1993.

[11] P. ZHOU, Explicit construction of multivariate Pade approximants, J. Comp. Appl. Math., 79 (1997), pp. 117.

[12] P. ZHOU, Multivariate Pade approximants associated with functional relations, J. Approx. Theory, 93 (1998), pp. 201-230.

[13] P. ZHOU, Explicit construction of multivariate Pade approximants for a q-logarithm function, J. Approx. Theory, 103 (2000), pp. 18-28.

PING ZHOU Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, NS, Canada, 132G 2W5 (pzhou@stfx.ca).
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Author:Zhou, Ping
Publication:Electronic Transactions on Numerical Analysis
Date:Dec 1, 2006
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