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Exploring multivariate Padé approximants for multiple hypergeometric series

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Abstract

We investigate the approximation of some hypergeometric functions of two variables, namely the Appell functions F i , i = 1,...,4, by multivariate Padé approximants. Section 1 reviews the results that exist for the projection of the F i onto ϰ=0 or y=0, namely, the Gauss function 2 F 1(a, b; c; z), since a great deal is known about Padé approximants for this hypergeometric series. Section 2 summarizes the definitions of both homogeneous and general multivariate Padé approximants. In section 3 we prove that the table of homogeneous multivariate Padé approximants is normal under similar conditions to those that hold in the univariate case. In contrast, in section 4, theorems are given which indicate that, already for the special case F 1(a, b, b′; c; x; y) with a = b = b′ = 1 and c = 2, there is a high degree of degeneracy in the table of general multivariate Padé approximants. Section 5 presents some concluding remarks, highlighting the difference between the two types of multivariate Padé approximants in this context and discussing directions for future work.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Antwerp, B-2610, Antwerp-Wilrijk, Belgium

    Annie Cuyt & Brigitte Verdonk

  2. University of Witwatersrand, South-Africa

    Kathy Driver

  3. Hefei University of Technology, P.R, China

    Jieqing Tan

Authors
  1. Annie Cuyt
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  2. Kathy Driver
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  3. Jieqing Tan
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  4. Brigitte Verdonk
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Cuyt, A., Driver, K., Tan, J. et al. Exploring multivariate Padé approximants for multiple hypergeometric series. Advances in Computational Mathematics 10, 29–49 (1999). https://doi.org/10.1023/A:1018918429917

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