The Abel lemma and the q-Gosper algorithm

Keywords

the Abel lemma / Abel pairs / basic hypergeometric series / the q-Gosper algorithm / Gosper pairs / 33D15 / 33F10

References

[1] Chu W. C. Bailey’s very well-poised 6ω6-series identity. J Combin Theory, Ser A, 2006, 113: 966-979.

[2] Bailey W. N. Series of hypergeometric type which are infinite in both directions. Quart J Math (Oxford), 1936, 7: 105-115.

[3] Gasper G., Rahman M. Basic Hypergeometric Series, 2004 2nd ed. Cambridge: Cambridge University Press.

[4] Abramov S. A., Paule P., Petkovšek M. q-Hypergeometric solutions of q-difference equations. Discrete Math, 1998, 180: 3-22.

[5] Böing H., Koepf W. Algorithm for q-hypergeometric summation in computer algebra. J Symbolic Comput, 1999, 28: 777-799.

[6] Koornwinder T. H. On Zeilberger’s algorithm and its q-analogue. J Comput Appl Math, 1993, 48: 91-111.

[7] Mohammed M., Zeilberger D. Sharp upper bounds for the orders of the recurrences outputted by the Zeilberger and q-Zeilberger Algorithms. J Symbolic Comput, 2005, 39: 201-207.

[8] Riese A. A generalization of Gosper’s algorithm to bibasic hypergeometric summation. Electron J Combin, 1996, 3: R19

[9] Wilf H. S., Zeilberger D. An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities. Invent Math, 1992, 108: 575-633.

[10] Andrews G. E. On Ramanujan’s summation of 1ω1 (a, b, z). Proc Amer Math Soc, 1969, 22: 552-553.

[11] Andrews G. E., Askey R. A simple proof of Ramanujan’s summation of the 1ω1. Aequationes Math, 1978, 18: 333-337.

[12] Berndt B. C. Murty M. R. Ramanujan’s theory of theta-functions. Theta Functions from the Classical to the Modern, 1993, Providence: Amer Math Soc, 1-63.

[13] Ismail M. E. H. A simple proof of Ramanujan’ 1ω1 sum. Proc Amer Math Soc, 1977, 63: 185-186.

[14] Schlosser M. Abel-Rothe type generalizations of Jacobi’s triple product identity. In: Ismail M E H, Koelink E, eds. Theory and Applications of Special Functions. A Volume Dedicated to Mizan Rahman. Dev Math, 2005, 13: 383–400

[15] Shukla H. S. A note on the sums of certain bilateral hypergeometric series. Proc Cambridge Phil Soc, 1959, 55: 262-266.

[16] Schlosser M. Inversion of bilateral basic hypergeometric series. Electron J Combin, 2003, 10: R10