General order multivariate Padé approximants for pseudo-multivariate functions
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- by Annie Cuyt, Jieqing Tan and Ping Zhou PDF
- Math. Comp. 75 (2006), 727-741 Request permission
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 \[ E\left ( x,y\right ) =\sum _{i,j=0}^\infty \frac {x^iy^j}{\left ( i+j\right ) !} \] and \[ E_q\left ( x,y\right ) =\sum _{i,j=0}^\infty \frac {x^iy^j}{[i+j]_q!}, \] as well as the Appell function \[ F_1\left ( a,1,1;c;x,y\right ) =\sum _{i,j=0}^\infty \frac {\left ( a\right ) _{i+j}x^iy^j}{\left ( c\right ) _{i+j}} \] and the multivariate form of the partial theta function \[ F\left ( x,y\right ) =\sum _{i,j=0}^\infty q^{-\left ( i+j\right ) ^2/2}x^iy^j. \]References
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Additional Information
- Annie Cuyt
- Affiliation: Department of Mathematics and Computer Science, University of Antwerp, Middelheimlaan 1, B-2020 Antwerpen, Belgium
- MR Author ID: 53570
- 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
- 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 - © Copyright 2006 American Mathematical Society
- Journal: Math. Comp. 75 (2006), 727-741
- MSC (2000): Primary 41A21
- DOI: https://doi.org/10.1090/S0025-5718-06-01789-3
- MathSciNet review: 2196989