Fine’s function and partial theta function

Chan, Heng Huat; Chan, Song Heng; Chen, Kuo-Jye; Huang, Sen-Shan

doi:10.1090/proc/15380

Fine’s function and partial theta function
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by Heng Huat Chan, Song Heng Chan, Kuo-Jye Chen and Sen-Shan Huang

Proc. Amer. Math. Soc. 149 (2021), 2561-2572

DOI: https://doi.org/10.1090/proc/15380

Published electronically: March 29, 2021

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Abstract:

In this article, we use identities found in N.J. Fine’s book Basic hypergeometric series and its applications to derive a one-parameter generalization of the product of two partial theta functions discovered by G.E. Andrews and S.O. Warnaar. We also give two different proofs of this generalization, one of which is motivated by the work of A. Berkovich and the other is given by M.E.H. Ismail.

References

George E. Andrews and Bruce C. Berndt, Ramanujan’s lost notebook. Part II, Springer, New York, 2009. MR 2474043
George E. Andrews and S. Ole Warnaar, The product of partial theta functions, Adv. in Appl. Math. 39 (2007), no. 1, 116–120. MR 2319567, DOI 10.1016/j.aam.2005.12.003
Alexander Berkovich, On the difference of partial theta functions, Bull. Malays. Math. Sci. Soc. 44 (2021), no. 2, 563–570. MR 4217074, DOI 10.1007/s40840-020-00961-4
H. van Haeringen and L. P. Kok, Table errata: Higher transcendental functions, Vol. I [McGraw-Hill, New York, 1953; MR 15, 419] by A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Math. Comp. 41 (1983), no. 164, 778. MR 717721, DOI 10.1090/S0025-5718-1983-0717721-1
Nathan J. Fine, Basic hypergeometric series and applications, Mathematical Surveys and Monographs, vol. 27, American Mathematical Society, Providence, RI, 1988. With a foreword by George E. Andrews. MR 956465, DOI 10.1090/surv/027
George Gasper and Mizan Rahman, Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 96, Cambridge University Press, Cambridge, 2004. With a foreword by Richard Askey. MR 2128719, DOI 10.1017/CBO9780511526251
Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2009. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey; Reprint of the 2005 original. MR 2542683
Mourad E. H. Ismail, private communication.
S. Ole Warnaar, Partial theta functions. I. Beyond the lost notebook, Proc. London Math. Soc. (3) 87 (2003), no. 2, 363–395. MR 1990932, DOI 10.1112/S002461150201403X