Fine’s function and partial theta function
Authors:
Heng Huat Chan, Song Heng Chan, Kuo-Jye Chen and Sen-Shan Huang
Journal:
Proc. Amer. Math. Soc. 149 (2021), 2561-2572
MSC (2020):
Primary 33D15
DOI:
https://doi.org/10.1090/proc/15380
Published electronically:
March 29, 2021
MathSciNet review:
4246806
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Abstract | References | Similar Articles | Additional Information
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.
- 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 https://doi.org/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 https://doi.org/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 https://doi.org/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
- 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
- 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 https://doi.org/10.1112/S002461150201403X
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Additional Information
Heng Huat Chan
Affiliation:
Department of Mathematics, National University of Singapore, Singapore 119076, Singapore;
Department of Mathematics, National Changhua University of Education, Jin-De Campus, No. 1, Jin-De Road, Changhua City, Taiwan
MR Author ID:
365568
Email:
matchh@nus.edu.sg
Song Heng Chan
Affiliation:
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
MR Author ID:
730779
Email:
chansh@ntu.edu.sg
Kuo-Jye Chen
Affiliation:
Department of Mathematics, National Changhua University of Education, Jin-De Campus, No. 1, Jin-De Road, Changhua City, Taiwan
MR Author ID:
352175
Email:
kjchen@cc.ncue.edu.tw
Sen-Shan Huang
Affiliation:
Department of Mathematics, National Changhua University of Education, Jin-De Campus, No. 1, Jin-De Road, Changhua City, Taiwan
MR Author ID:
620036
Email:
sshuang@cc.ncue.edu.tw
Received by editor(s):
October 1, 2020
Received by editor(s) in revised form:
October 8, 2020
Published electronically:
March 29, 2021
Dedicated:
Dedicated to the memory of Professor Richard Askey
Communicated by:
Mourad Ismail
Article copyright:
© Copyright 2021
American Mathematical Society
