An analytic method for convergence acceleration of certain hypergeometric series

Authors:
Stanisław Lewanowicz and Stefan Paszkowski

Journal:
Math. Comp. **64** (1995), 691-713

MSC:
Primary 33C45; Secondary 65B10, 65D20

DOI:
https://doi.org/10.1090/S0025-5718-1995-1277769-6

MathSciNet review:
1277769

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A method is presented for convergence acceleration of the generalized hypergeometric series with the argument , using analytic properties of their terms. Iterated transformation of the series is performed analytically, which results in obtaining new fast converging expansions for some special functions and mathematical constants.

**[1]**M. Abramowitz and I. A. Stegun (eds.),*Handbook of mathematical functions*, Nat. Bureau of Standards, Washington, 1964.**[2]**W. N. Bailey,*Contiguous hypergeometric functions of the type*, Proc. Glasgow Math. Assoc.**2**(1954), 62-65. MR**0064918 (16:356e)****[3]**C. Brezinski and M. Redivo Zaglia,*Extrapolation methods. Theory and practice*, North-Holland, Amsterdam, 1991. MR**1140920 (93d:65001)****[4]**A. Erdélyi (ed.),*Higher transcendental functions*, McGraw-Hill, New York, 1953.**[5]**I. S. Gradshteyn and I. M. Ryzhik,*Tables of integrals, series and products*, Academic Press, New York, 1980.**[6]**S. L. Kalla, S. Conde, and Y. L. Luke,*Integrals of Jacobi functions*, Math. Comp.**38**(1982), 207-214. MR**637298 (83a:33005)****[7]**K. Knopp,*Theory and application of infinite series*, Hafner, New York, 1949.**[8]**Y. L. Luke,*The special functions and their approximations*, Academic Press, New York, 1969.**[9]**-,*Mathematical functions and their approximations*, Academic Press, New York, 1975. MR**0501762 (58:19039)****[10]**-,*Rational approximations for the logarithmic derivative of the gamma function*, Appl. Anal.**1**(1979), 65-73. MR**0279965 (43:5686)****[11]**A. Sidi,*Convergence properties of some nonlinear sequence transformations*, Math. Comp.**33**(1979), 315-326. MR**514827 (81h:65003)****[12]**-,*Analysis of convergence of the T-transformation for power series*, Math. Comp.**35**(1980), 833-850. MR**572860 (83d:41039)****[13]**E. J. Weniger,*Nonlinear sequence transformations for the acceleration of convergence and the summation of divergent series*, Comp. Phys. Reports**10**(1989), 189-371.**[14]**J. Wimp,*Irreducible recurrences and representation theorems for*, Comput. Math. Appl.**9**(1983), 663-678. MR**726815 (85b:33005)****[15]**-,*Computation with recurrence relations*, Pitman, Boston-London, 1984.**[16]**D. Zeilberger,*Gauss's**cannot be generalized to*, J. Comput. Appl. Math.**39**(1992), 379-382. MR**1164298 (93i:33002)**

Retrieve articles in *Mathematics of Computation*
with MSC:
33C45,
65B10,
65D20

Retrieve articles in all journals with MSC: 33C45, 65B10, 65D20

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1995-1277769-6

Keywords:
Analytic convergence acceleration,
generalized hypergeometric series,
polygamma functions,
Beta function

Article copyright:
© Copyright 1995
American Mathematical Society