1. Introduction

The four Appell functions ^[10] come from the natural extension of the well-known Gauss hypergeometric function from one to two variables. At the same time, Appell functions are special cases of more general hypergeometric functions ^[11]. They have various applications in many branches of mathematics and physics, especially in quantum mechanics and field theory ^[8]. For example, as proved by Kniehl ^[2], they appear in the evaluation of some Feynman’s integrals. These functions have many representations. Schlosser ^[9], in his survey, explains how they are related to some Euler’s elliptic and double integrals, and solutions of some partial differential equations. In what follows, we will confine our work for the first Appell function in a particular case and use the power series expansion as a starting definition. A truncated part of this Taylor’s expansion will be taken as the standard estimation of We will compare it with the value given by the Padé approximant defined by Baker ^[1].

2. Notations

For complex values of the parameters and the variables and , the function we investigate is given by

where

and is the Pochhammer symbol defined by if if and if It is known that is the convergence region of this double series.

Here and in the following, let and be the sets of complex numbers and positive integers, respectively, and let We generalize so, the results on considered by Borwein ^[7], the case examined by Cuyt ^[6], and those on some pseudo-multivariate functions investigated by Zhou ^[5]. To compute we will consider the multivariate Padé approximation and compare it with the estimation given by Taylor’s expansion, truncated at a certain order.

We introduce some basics, in order to define the multivariate Padé approximant.

For every we begin by construct the following subsets of

as illustrated by Figure 1. The elements of these sets are classified with respect to a triangular numbering, since it is compatible with Taylor’s expansion and allows us to express the error introduced by the truncation. So, each point of will take the rank given by In the opposite sense, the element will have the following coordinates:

and

where is the diagonal containing this point, given by

Our purpose is to find two polynomials

which satisfy the following equation lattice adopted by Cuyt ^[4]:

To determine , one must resolve the homogeneous system

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Figure 1. Sets defined by triangular numbering:

where is a point of

The coefficients of the numerator polynomial are then obtained using

for each The general multivariate Padé approximant will be the rational fraction

The fact that satisfies the inclusion property, together with the imposed condition take care of the Padé approximation property, namely

3. A New General Padé Approximant

To make the resolution of the homogeneous system above easier, we will define a new equation lattice with the following sets represented in Figure 2, in a similar way as in Golub ^[3].

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Figure 2. A solution with rectangular

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Figure 3. A solution with shifted

and

where is the diagonal which contains the of and is the number of diagonals of , also given by

Here we assume here that such that the new set is inside . We could already remark that for enough great values of takes only two values 1 or 2. The new problem is to find the parameters and which define the rectangular set , by solving the equation:

which implies that the number of unknowns in the homogeneous system is exactly the number of independent equations

The new equation lattice to be satisfied is

Here, we have a simple solution of the corresponding homogeneous system, for relatively great values of n.

4. Asymptotic Case for Great Enough n

For m fixed, the precision of the approximation is better for great values of For instance, if will be contained in at most the two diagonals defined by In what follows, we assume, without losing generality, that With this hypothesis, we find and A solution of the homogeneous system, according to the new equation lattice is given by:

which defines the explicit expression of the denominator polynomial by

and the corresponding numerator polynomial over

Let us remark that if and all the coefficients are constant, which means that is a rational fraction.

5. A Solution with Shifted Denominator

In this section, we will try to improve the results presented in previous sections, by reducing the number of operations we need to compute the coefficients of the numerator polynomial For this, define the partial function by: for each integer thus, we have .

Firstly, examine the error introduced by the truncation of Taylor’s expansion.

with

is a real such that the point is on the line segment joining and .

Thus, it is not easy to evaluate the error of truncation, since we have no idea about the real

We now try to express the error introduced by the Padé approximation.

which can be written as follows:

and finally,

To reduce the time and number of operations we need for the computation, we will shift the rectangular set to the right side as far as possible (see Figure 3).

We write now the equation lattice just above after replacing by where is the diagonal containing the last point of given by

But, if we multiply by and translate the shift of we get

which defines a new Padé approximant with respect to the equation lattice above expressed by

This result shows that it is not necessary to compute the numerator polynomial over , but only over the smaller set In the end, notice that this Padé approximant, together with Taylor’s estimation, satisfies the same accuracy-through-order condition. We believe, our approach is more accurate than Taylor’s development, since it needs less calculations and avoids errors introduced by the computer.

To compare these two approximations, we took the case of for which we have the exact expression, namely The results are illustrated by Figure 4.

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Figure 4. Padé (Blue and Red), Taylor (Green) and exact (Yellow)

6. Concluding Remarks

The approach considered here has an advantage of being simple. It seems to give better results than the classical Taylor’s expansion as we can see in Figure 4. However, some difficulties occur in the evaluation of the error. To improve the precision of our method, one could consider the cases where the set is contained in more than two diagonals but a difficulty arises due to the fact that the homogeneous system to solve is bigger and contains more unknowns. Also, the solution can’t be expressed with a simple formula like the one found with presented in section 4 as an asymptotic case. To reduce the computation-time and the number of operations needed, we have shifted the denominator set, to the right side. There is also the possibility of shifting upward, along the axis. There could be an optimal position of to find.

Acknowledgments

The author would like to express his gratitude to the referees for their careful reading of the manuscript and for giving me some of their precious time. I also thank Hassane Allouche for suggesting me to work in this direction.

References

[1]	G. A. Jr. Baker, P. Graves-Morris, Padé Approximants, Second Ed.,Cambridge University Press, Cambridge, 1996.
	In article		PubMed
[2]	B. Kniehl, O. V. Tarasov, Finding new relationships between hypergeometric functions by evaluating Feynman Integrals,Nucl. Phys. B 854 [PM] (2012) 841.
	In article
[3]	A. P. Golub, L. O. Chernestka, Two Dimensional Generalized Moment representations and Padé Approximant for some Humbert series. Ukr. Math. ZH. - 2013.-V. 65, N◦ 10.-P. 1315-1331.
	In article
[4]	A. Cuyt, How well can the concept of Padé approximant be generalized to the multivariate case? J. Comp. Math, 105 (1999), 25-50, MR1690577, (2000e:41029).
	In article
[5]	P. Zhou, A. Cuyt, and J. Tan. General Order Multiple Padé Approximant for Pseudomultavariate Functions. Math. Comp., 78(268):2137-2155, 2009.
	In article		View Article
[6]	A. Cuyt, K. Driver, J. Tan and B. Verdonk, Exploring multivariate Padé approximant for multiple hypergeometric series. Adv. Comput. Math., 10:29-49, 1999.
	In article		View Article
[7]	P. B. Borwein, A. Cuyt, P. Zhou, Explicit Construction of General Multivariate Padé Approximant to an Appell function. Adv. Comput. Math. 2005. V. 22, N◦ 3.- P. 249-273.
	In article
[8]	H. M. Srivastava, P. W. Karlsson, Multiple Gaussian Hypergeometric Series in Mathematics and Its Applications, Ellis Horwood, Halsted Press, New York, 1985.
	In article
[9]	M. J. Schlosser, Multiple Hypergeometric Series- Appell Series and beyond. arXiv: 1305.1966v1 [Math. CA]
	In article
[10]	P. Appell, Sur les fonctions hypergéométriques de plusieurs variables, In Mémoir. Sci. Math. Paris: Gautier Villars, 1925.
	In article
[11]	P. Appell; J. Kampé de Fériet, Fonctions Hypergéométriques et Hypersphériques: Polynômes d’Hermite, Paris: Gautier Villars, 1926.
	In article
[12]	Erdélyi A., Magnus W., Oberhettinger F. and Tricomi F. G., Higher Transcendental Functions, New York : Krieger, Vol. 1, pp. 222-224, 1981.
	In article