Polynomial series expansions for confluent and Gaussian hypergeometric functions

Luh, W.; Müller, J.; Ponnusamy, S.; Vasundhra, P.

doi:10.1090/S0025-5718-05-01734-5

Polynomial series expansions for confluent and Gaussian hypergeometric functions
HTML articles powered by AMS MathViewer

by W. Luh, J. Müller, S. Ponnusamy and P. Vasundhra PDF

Math. Comp. 74 (2005), 1937-1952 Request permission

Abstract:

Based on the Hadamard product of power series, polynomial series expansions for confluent hypergeometric functions $M(a,c;\cdot )$ and for Gaussian hypergeometric functions $F(a,b;c;\cdot )$ are introduced and studied. It turns out that the partial sums provide an interesting alternative for the numerical evaluation of the functions $M(a,c;\cdot )$ and $F(a,b;c;\cdot )$, in particular, if the parameters are also viewed as variables.

References

Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions, with formulas, graphs and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 20402, 1966. Fifth printing, with corrections; National Bureau of Standards, Washington, D.C., (for sale by the Superintendent of Documents). MR 0208798
R. Balasubramanian, S. Ponnusamy, and M. Vuorinen, On hypergeometric functions and function spaces, J. Comput. Appl. Math. 139 (2002), no. 2, 299–322. MR 1876641, DOI 10.1016/S0377-0427(01)00417-4
Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. I, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman; With a preface by Mina Rees; With a foreword by E. C. Watson; Reprint of the 1953 original. MR 698779
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, DOI 10.1080/17476939308814650
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
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
Jürgen Müller, The Hadamard multiplication theorem and applications in summability theory, Complex Variables Theory Appl. 18 (1992), no. 3-4, 155–166. MR 1157924, DOI 10.1080/17476939208814542
J. Müller, Convergence acceleration of Taylor sections by convolution, Constr. Approx. 15 (1999), no. 4, 523–536. MR 1702803, DOI 10.1007/s003659900120
Ch. Schwarz, Computation of confluent hypergeometric functions and application to parabolic boundary control problems, Ph.D Thesis, University of Trier, 2001.
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
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, DOI 10.1002/9781118032572
J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I., 1960. MR 0218587