# Article ID: MTJPAM-D-20-00024

## Title: A Family of Theta-Function Identities Based Upon q-Binomial Theorem and Heine’s Transformations

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-20-00024; Volume 2 / Issue 2 / Year 2020, Pages 1-6

Document Type: Research Paper

Author(s): Hari Mohan Srivastava a , Mahendra Pal Chaudhary b , Feyissa Kaba Wakene c

aDepartment of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada – Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan, Republic of China – Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007 Baku, Azerbaijan

bDepartment of Mathematics, Netaji Subhas University of Technology, Sector 3, Dwarka, New Delhi 110078, India

cDepartment of Mathematics, Madda Walabu University, Robe City, Bale Zone, Ethiopia

Received: 12 August 2020, Accepted: 21 August 2020, Available online: 23 September 2020.

Corresponding Author: Hari Mohan Srivastava (Email address: harimsri@math.uvic.ca)

Full Text: PDF

Abstract

The authors establish a set of two presumably new theta-function identities which are based upon q-binomial theorem and Heine’s transformations. Several closely-related identities such as (for example) q-product identities and Jacobi’s triple-product identity are also considered.

Keywords: Jacobi’s triple-product identity, q-Product identities, Euler’s Pentagonal Number Theorem, q-Binomial theorem, Heine’s transformations

References:
1. C. Adiga, N. A. S. Bulkhali, D. Ranganatha and H. M. Srivastava, Some new modular relations for the Rogers-Ramanujan type functions of order eleven with applications to partitions, J. Number Theory 158, 281–297, 2016.
2. G. E. Andrews, The Theory of Partitions, Cambridge University Press, Cambridge, London and New York, 1998.
3. T. M. Apostol, Introduction to Analytic Number Theory – Undergraduate Texts in Mathematics, Springer-Verlag, Berlin, New York and Heidelberg, 1976.
4. N. D. Baruah and J. Bora, Modular relations for the nonic analogues of the Rogers-Ramanujan functions with applications to partitions, J. Number Theory 128, 175–206, 2008.
5. B. C. Berndt, Ramanujan’s Notebooks – Part III, Springer-Verlag, Berlin, Heidelberg and New York, 1991.
6. J. Cao, H. M. Srivastava and Z.-G. Luo, Some iterated fractional q-integrals and their applications, Fract. Calc. Appl. Anal. 21, 672–695, 2018.
7. A. L. Cauchy, Memoire sur les fonctions dont plusieurs valeurs sont liées entree elles par une équation linéaire, et sur diverses transformations de produits composés d’un nombre indéfini de facteurs, C. R. Acad. Sci. Paris 17, 523, 1893.
8. G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers – (Revised by D. R. Heath-Brown and J. H. Silverman), Sixth edition (with Foreword by Andrew Wiles), Oxford University Press, Oxford, London and New York, 2008.
9. E. Heine, Untersuchungen über die Reihe

$1+\frac{(1-q^{\alpha})(1-q^{\beta})}{(1-q)(1-q^{\gamma})}\;x +\frac{(1-q^{\alpha})(1-q^{\alpha+1})(1-q^{\beta})(1-q^{\beta+1})} {(1-q)(1-q^{\gamma})(1-q^{\gamma+1})}\;x^{2}+\cdots,$

J. Reine Angrew. Math. 34, 285–328, 1847.
10. C. G. J. Jacobi, Fundamenta Nova Theoriae Functionum Ellipticarum (Regiomonti, Sumtibus Fratrum Bornträger, Königsberg, Germany, 1829; Reprinted in Gesammelte Mathematische Werke 1, 497–538, 1829), American Mathematical Society, Providence, Rhode Island, 97–239, 1969.
11. S. Ramanujan, Notebooks – Vols. 1 and 2, Tata Institute of Fundamental Research, Bombay, 1957.
12. S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988.
13. L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, London and New York, 1966.
14. H. M. Srivastava, Operators of basic (or q) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis, Iran. J. Sci. Technol. Trans. A: Sci. 44, 327–344, 2020.
15. H. M. Srivastava and M. P. Chaudhary, Some relationships between q-product identities, combinatorial partition identities and continued-fraction identities, Adv. Stud. Contemp. Math. 25, 265–272, 2015.
16. H. M. Srivastava, M. P. Chaudhary and S. Chaudhary, Some theta-function identities related to Jacobi’s triple-product identity, European J. Pure Appl. Math. 11 (1), 1–9, 2018.
17. H. M. Srivastava, M. P. Chaudhary and S. Chaudhary, A family of theta-function identities related to Jacobi’s triple-product identity, Russian J. Math. Phys. 27, 139–144, 2020.
18. H. M. Srivastava, R. Srivastava, M. P. Chaudhary and S. Uddin, A family of theta- function identities based upon combinatorial partition identities and related to Jacobi’s triple-product identity, Mathematics 8 (6), Article ID 918, 1–14, 2020.
19. H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.
20. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985.
21. H. M. Srivastava and N. Saikia, Some congruences for overpartitions with restriction, Math. Notes 107, 488–498, 2020.
22. J.-H. Yi, Theta-function identities and the explicit formulas for theta-function and their applications, J. Math. Anal. Appl. 292, 381–400, 2004.