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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

General order multivariate Padé approximants for pseudo-multivariate functions


Authors: Annie Cuyt, Jieqing Tan and Ping Zhou
Journal: Math. Comp. 75 (2006), 727-741
MSC (2000): Primary 41A21
Published electronically: February 1, 2006
MathSciNet review: 2196989
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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 $ q$-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. $


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Additional Information

Annie Cuyt
Affiliation: Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, B-2020 Antwerpen, Belgium
Email: annie.cuyt@ua.ac.be

Jieqing Tan
Affiliation: Institute of Applied Mathematics, Hefei University of Technology, 193 Tunxi Road, 230009 Hefei, People’s Republic of China
Email: jqtan@mail.hf.ah.cn

Ping Zhou
Affiliation: Mathematics, Statistics and Computer Science Department, St. Francis Xavier University, Antigonish, Nova Scotia, Canada B2G 2W5
Email: pzhou@stfx.ca

DOI: http://dx.doi.org/10.1090/S0025-5718-06-01789-3
PII: S 0025-5718(06)01789-3
Keywords: Multivariate Pad\'{e} approximants, pseudo-multivariate functions
Received by editor(s): July 5, 2004
Received by editor(s) in revised form: January 3, 2005
Published electronically: February 1, 2006
Additional Notes: The first author is the Research Director of FWO-Vlaanderen
The second author’s research was supported by the National Natural Science Foundation of China under Grant No.60473114
The third author’s research was supported by NSERC of Canada
Article copyright: © Copyright 2006 American Mathematical Society