×

Identities of nonterminating series by Zeilberger’s algorithm. (English) Zbl 0945.65015

The Gosper algorithm and in particular the subsequent Zeilberger algorithm and the related WZ method have been extremely successful for an approach by computer algebra to identities involving hypergeometric functions [cf. M. Petovšek, H. S. Wilfe and D. Zeilberger, \(A=B\). Wellesley, MA: A. K. Peters (1996; Zbl 0848.05002)]. But this method remains restricted to the case of identities for the terminating hypergeometric series.
The author shows that automated proofs of the identities for non-terminating hypergeometric series are feasible by a combination of Zeilberger’s algorithm and asymptotic estimates. It is shown that the method extends to the non-terminating generalization of Saalschütz’ summation formula for a terminating Saalschützian \({_3F_2}\) hypergeometric series of argumentation \(1\).

MSC:

33C20 Generalized hypergeometric series, \({}_pF_q\)
33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
68W40 Analysis of algorithms

Citations:

Zbl 0848.05002

Software:

EKHAD; hsum.mpl
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Erdélyi, A., (Higher Transcendental Functions, vol. I (1953), McGraw-Hill), (reprinted in 1981 by R.E. Krieger)
[2] Gasper, G.; Rahman, M., Basic Hypergeometric Series (1990), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0695.33001
[3] Gessel, I. M., Finding identities with the WZ method, J. Symbolic Comput., 20, 537-566 (1995) · Zbl 0908.33004
[4] Koepf, W., Maple package hsum.mpl (February 1, 1998)
[5] Koornwinder, T. H., Jacobi functions as limit cases of \(q\)-ultraspherical polynomials, J. Math. Anal. Appl., 148, 44-54 (1990) · Zbl 0713.33010
[6] Olver, F. W.J., Asymptotics and Special Functions (1974), Academic Press: Academic Press New York, (reprinted in 1997 by A.K. Peters) · Zbl 0303.41035
[7] Petkovšek, M.; Wilf, H. S.; Zeilberger, D., \(A = B (1996)\), A.K. Peters: A.K. Peters Wellesley, MA
[8] Zeilberger, D., Maple package EKHAD (27 March 1997), obtainable from URL
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.