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General order multivariate Padé approximants for pseudo-multivariate functions

Authors:	Annie Cuyt Jieqing Tan Ping Zhou

Institution:	Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, B-2020 Antwerpen, Belgium ; Institute of Applied Mathematics, Hefei University of Technology, 193 Tunxi Road, 230009 Hefei, People's Republic of China ; Mathematics, Statistics and Computer Science Department, St. Francis Xavier University, Antigonish, Nova Scotia, Canada B2G 2W5

Abstract:	Although general order multivariate Padé approximants were introduced some decades ago, very few explicit formulas for special functions have been given. We explicitly construct some general order multivariate Padé approximants to the class of so-called pseudo-multivariate functions, using the Padé approximants to their univariate versions. We also prove that the constructed approximants inherit the normality and consistency properties of their univariate relatives, which do not hold in general for multivariate Padé approximants. Examples include the multivariate forms of the exponential and the -exponential functions $\displaystyle E\left( x,y\right) =\sum_{i,j=0}^\infty \frac{x^iy^j}{\left( i+j\right) !}$ and $\displaystyle E_q\left( x,y\right) =\sum_{i,j=0}^\infty \frac{x^iy^j}{i+j]_q!},$ as well as the Appell function $\displaystyle F_1\left( a,1,1;c;x,y\right) =\sum_{i,j=0}^\infty \frac{\left( a\right) _{i+j}x^iy^j}{\left( c\right) _{i+j}}$ and the multivariate form of the partial theta function $\displaystyle F\left( x,y\right) =\sum_{i,j=0}^\infty q^{-\left( i+j\right) ^2/2}x^iy^j.$

Keywords:	Multivariate Pad\'{e} approximants pseudo-multivariate functions

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