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General order multivariate Padé approximants for Pseudo-multivariate functions. II

Authors: Ping Zhou, Annie Cuyt and Jieqing Tan
Journal: Math. Comp. 78 (2009), 2137-2155
MSC (2000): Primary 41A21
Published electronically: February 2, 2009
MathSciNet review: 2521282
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Abstract: Explicit formulas for general order multivariate Padé approximants of pseudo-multivariate functions are constructed on specific index sets. Examples include the multivariate forms of the exponential function

$\displaystyle E\left(\underline{x}\right) =\sum_{j_{1},j_{2},\ldots ,j_{m}=0}^{... ...x_{2}^{j_{2}}\cdots x_{m}^{j_{m}}}{\left( j_{1}+j_{2}+\cdots +j_{m}\right) !}, $

the logarithm function

$\displaystyle L(\underline{x})=\sum_{j_{1}+j_{2}+\cdots +j_{m}\geq 1}\frac{ x_{1}^{j_{1}}x_{2}^{j_{2}}\cdots x_{m}^{j_{m}}}{j_{1}+j_{2}+\cdots +j_{m}}, $

the Lauricella function

$\displaystyle F_{D}^{\left( m\right) }\left( a,1,\ldots ,1;c;x_{1},\ldots ,x_{m... ...}}}{\left( c\right) _{j_{1}+\cdots +j_{m}}} x_{1}^{j_{1}}\cdots x_{m}^{j_{m}}, $

and many more. We prove that the constructed approximants inherit the normality and consistency properties of their univariate relatives. These properties do not hold in general for multivariate Padé approximants. A truncation error upperbound is also given.

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Additional Information

Ping Zhou
Affiliation: Department of Mathematics, Statistics and Computer Science, St. Francis Xavier University, Antigonish, NS, Canada, B2G 2W5

Annie Cuyt
Affiliation: Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, B-2020 Antwerpen, Belgium

Jieqing Tan
Affiliation: Institute of Applied Mathematics, Hefei University of Technology, 193 Tunxi Road, 230009 Hefei, People’s Republic of China

Keywords: Multivariate Pad\'{e} approximants, pseudo-multivariate functions
Received by editor(s): August 10, 2007
Received by editor(s) in revised form: September 5, 2008
Published electronically: February 2, 2009
Additional Notes: The first author’s research is supported by NSERC of Canada
The second author is Research Director of FWO-Vlaanderen
The third author’s research is supported by the National Natural Science Foundation of China under Grant No. 60473114
Article copyright: © Copyright 2009 American Mathematical Society