Polynomial series expansions for confluent and Gaussian hypergeometric functions

Authors:
W. Luh, J. Müller, S. Ponnusamy and P. Vasundhra

Journal:
Math. Comp. **74** (2005), 1937-1952

MSC (2000):
Primary 33C05, 33C15, 33F05, 65D20

DOI:
https://doi.org/10.1090/S0025-5718-05-01734-5

Published electronically:
March 15, 2005

MathSciNet review:
2164104

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Abstract | References | Similar Articles | Additional Information

Abstract: Based on the Hadamard product of power series, polynomial series expansions for confluent hypergeometric functions and for Gaussian hypergeometric functions are introduced and studied. It turns out that the partial sums provide an interesting alternative for the numerical evaluation of the functions and , in particular, if the parameters are also viewed as variables.

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Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi,*Higher transcendental functions. Vol. III*, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman; Reprint of the 1955 original. MR**698781****4.**Karl-Goswin Grosse-Erdmann,*On the Borel-Okada theorem and the Hadamard multiplication theorem*, Complex Variables Theory Appl.**22**(1993), no. 1-2, 101–112. MR**1277015****5.**Yudell L. Luke,*The special functions and their approximations, Vol. I*, Mathematics in Science and Engineering, Vol. 53, Academic Press, New York-London, 1969. MR**0241700****6.**Yudell L. Luke,*The special functions and their approximations. Vol. II*, Mathematics in Science and Engineering, Vol. 53, Academic Press, New York-London, 1969. MR**0249668****7.**Jürgen Müller,*The Hadamard multiplication theorem and applications in summability theory*, Complex Variables Theory Appl.**18**(1992), no. 3-4, 155–166. MR**1157924****8.**J. Müller,*Convergence acceleration of Taylor sections by convolution*, Constr. Approx.**15**(1999), no. 4, 523–536. MR**1702803**, https://doi.org/10.1007/s003659900120**9.**CH. SCHWARZ, Computation of confluent hypergeometric functions and application to parabolic boundary control problems, Ph.D Thesis, University of Trier, 2001.**10.**Ch. Schwarz,*Computation and confluent hypergeometric functions on compact intervals*, Functions, series, operators (Budapest, 1999) János Bolyai Math. Soc., Budapest, 2002, pp. 357–366. MR**1981575****11.**Nico M. Temme,*Special functions*, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996. An introduction to the classical functions of mathematical physics. MR**1376370****12.**J. L. Walsh,*Interpolation and approximation by rational functions in the complex domain*, Third edition. American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I., 1960. MR**0218587**

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Additional Information

**W. Luh**

Affiliation:
University of Trier, FB IV, Mathematics, D-54286 Trier, Germany

Email:
luh@uni-trier.de

**J. Müller**

Affiliation:
University of Trier, FB IV, Mathematics, D-54286 Trier, Germany

Email:
jmueller@uni-trier.de

**S. Ponnusamy**

Affiliation:
Department of Mathematics, Indian Institute of Technology, IIT-Madras, Chennai- 600 036, India

Email:
samy@iitm.ac.in

**P. Vasundhra**

Affiliation:
Department of Mathematics, Indian Institute of Technology, IIT-Madras, Chennai- 600 036, India

Email:
vasu2kk@yahoo.com

DOI:
https://doi.org/10.1090/S0025-5718-05-01734-5

Keywords:
Hypergeometric series,
Hadamard product,
polynomial expansions

Received by editor(s):
December 3, 2003

Received by editor(s) in revised form:
May 18, 2004

Published electronically:
March 15, 2005

Additional Notes:
The work of the authors was supported by DST-DAAD under Project Based Personal Exchange Programme (Sanction No. INT/DAAD/P-64/2002).

Article copyright:
© Copyright 2005
American Mathematical Society