Relations between R_α, R_β and R_m functions related to Jacobi’s triple-product identity and the family of theta-function identities

M. P. Chaudhary
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 2, Pages 1–11
DOI: 10.7546/nntdm.2021.27.2.1-11
Full paper (PDF, 207 Kb)

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Authors and affiliations

M. P. Chaudhary
Department of Mathematics, Netaji Subhas University of Technology
Sector 3, Dwarka, New Delhi 110078, India

Abstract

In this paper, the author establishes a set of three new theta-function identities involving R_α, R_β and R_m functions which are based upon a number of q-product identities and Jacobi’s celebrated triple-product identity. These theta-function identities depict the inter-relationships that exist among theta-function identities and combinatorial partition-theoretic identities. Here, in this paper we answer a open question of Srivastava et al [33], and established relations in terms of R_α, R_β and R_m (for m = 1, 2, 3), and q-products identities. Finally, we choose to further emphasize upon some close connections with combinatorial partition-theoretic identities.

Keywords

• Theta-function identities
• Multivariable R-functions
• Jacobi’s triple-product identity
• Ramanujan’s theta functions
• q-Product identities
• Euler’s Pentagonal Number Theorem
• Rogers–Ramanujan continued fraction
• Rogers–Ramanujan identities
• Combinatorial partition-theoretic identities

2020 Mathematics Subject Classification

• 05A17
• 05A30
• 11F27
• 11P83

References

1. Adiga, C., Kim, T., Mahadeva Naika, M. S., & Madhusudhan, H. S. (2004). On Ramanujan’s cubic continued fraction and explicit evaluations of theta-functions. Indian Journal of Pure and Applied Mathematics, 35(9), 1047–1062.
2. Andrews, G. E. (1998). The Theory of Partitions, Cambridge University Press, Cambridge.
3. Andrews, G. E., Bringman K., & Mahlburg, K. (2015). Double series representations for Schur’s partition function and related identities, Journal of Combinatorial Theory, Series A, 132, 102–119.
4. Apostol, T. M. (1976). Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer-Verlag New York.
5. Berndt, B. C. (1991). Ramanujan’s Notebooks, Part III, Springer-Verlag New York.
6. Cao, J., Srivastava H. M., & Luo, Z.-G. (2018). Some iterated fractional q-integrals and their applications. Fractional Calculus and Applied Analysis, 21, 672–695.
7. Chaudhary, M. P. (2020). A family of theta-function identities based upon R_α, R_β and R_m-functions related to Jacobi’s triple-product identity, Publications de l’Institut Mathematique (Beograd), 108(122), 23–32.
8. Chaudhary, M. P. (2012). Generalization of Ramanujan’s identities in terms of q-products and continued fractions. Global Journal of Science Frontier Research: (F) Mathematics and Decision, 12, 53–60.
9. Chaudhary, M. P. (2015). Some relationships between q-product identities, combinatorial partition identities and continued-fractions identities, III. Pacific Journal of Applied Mathematics, 7, 87–95.
10. Chaudhary, M. P., & Chaudhary, S. (2020). On relationships between q-product identities, R_α, R_β and R_m-functions related to Jacobi’s triple-product identity. Mathematica Moravica, 24(2), 133–144.
11. Chaudhary, M. P., & Chaudhary, S. (2017). Note on Ramanujan’s modular equations of degrees three and nine, Pacific Journal of Applied Mathematics, 8, 143–148.
12. Chaudhary, M. P., Chaudhary, S., & Salilew, G. A. (2020). A family of theta-function identities in the light of Jacobi’s triple-product identity. Applied Analysis and Optimization, 4(3), 283–289.
13. Chaudhary, M. P., Chaudhary, S., & Choi, J. (2016). Certain identities associated
    with 3-dissection property, continued fractions and combinatorial partition. Applied Mathematical Sciences, 10, 37–44.
14. Chaudhary, M. P., Chaudhary, S., & Choi, J. (2016). Note on Ramanujan’s modular equation of degree seven. International Journal of Mathematical Analysis, 10, 661–667.
15. Chaudhary, M. P., & Luis Cimadevilla Villacorta, J. (2017). Representations of certain theta function identities in terms of combinatorial partition identities. Far East Journal of Mathematical Sciences, 102, 1605–1611.
16. Chaudhary, M. P., Chaudhary, S., & Minz, S. (2020). On relationships between q-products identities and combinatorial partition identities. Mathematica Moravica, 24, 83–91.
17. Chaudhary, M. P., & Choi, J. (2015). Note on modular relations for Roger–Ramanujan type identities and representations for Jacobi identities. East Asian Mathematical Journal, 31, 659–665.
18. Chaudhary, M. P., & Choi, J. (2016). Certain identities associated with Eisenstein series, Ramanujan–Göllnitz–Gordon continued fraction and combinatorial partition identities. International Journal of Mathematical Analysis, 10, 237–244.
19. Chaudhary, M. P., & Choi, J. (2016). Certain identities associated with character formulas, continued fraction and combinatorial partition identities. East Asian Mathematical Journal, 32, 609–619.
20. Chaudhary, M. P., Uddin, S., & Choi, J. (2017). Certain relationships between q-product identities, combinatorial partition identities and continued-fraction identities. Far East Journal of Mathematical Sciences, 101, 973–982.
21. Chaudhary, M. P., Salilew, G. A., & Choi, J. (2017). Five relationships between continued fraction identities, q-product identities and combinatorial partition identities. Far East Journal of Mathematical Sciences, 102, 855–863.
22. Chaudhary, M. P., & Kaba Wakene, F. (2019). On interrelationships between q-product identities and combinatorial partition identities II. Pacific Journal of Applied Mathematics, 9, 255–261.
23. Chaudhary, M. P., & Kaba Wakene, F. (2019). On interrelationships between q-product identities and combinatorial partition identities V. Uzbek Mathematical Journal, 63, 42–49.
24. Hahn, H.-Y., Huh, J.-S., Lim, E.-S., & Sohn, J.-B. (2018). From partition identities to a combinatorial approach to explicit Satake inversion, Annals of Combinatorics, 22, 543–562.
25. Jacobi, C. G. J. (1829). Fundamenta Nova Theoriae Functionum Ellipticarum. Regiomonti, Sumtibus Fratrum Borntrager, Konigsberg, Germany; Reprinted in  Gesammelte Mathematische Werke, 1 (1829), 497–538, American Mathematical Society, Providence, Rhode Island (1969), 97–239.
26. Munagi, A. O. (2016). Combinatorial identities for restricted set partitions. Discrete Mathematics, 339, 1306–1314.
27. Ramanujan, S. (1957). Notebooks, Vols. 1 and 2, Tata Institute of Fundamental Research, Bombay.
28. Ramanujan, S. (1988). The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi.
29. Srivastava, H. M. (2020). Operators of basic (or q-) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis. Iranian Journal of Science and Technology. Transaction A, Science, 44, 327–344.
30. Srivastava, H. M., & Chaudhary, M. P. (2015). Some relationships between q-product identities, combinatorial partition identities and continued-fraction identities. Advanced Studies in Contemporary Mathematics, 25, 265–272.
31. Srivastava, H. M., Chaudhary, M. P., & Chaudhary, S. (2018). Some theta-function identities related to Jacobi’s triple-product identity. European Journal of Pure and Applied Mathematics, 11(1), 1–9.
32. Srivastava, H. M., Chaudhary, M. P., & Chaudhary, S. (2020). A family of theta-function identities related to Jacobi’s triple-product identity. Russian Journal of Mathematical Physics, 27, 139–144.
33. Srivastava, H. M., Srivastava, R., Chaudhary, M. P., & Uddin, S. (2020). 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.
34. Srivastava, H. M., Chaudhary, M. P., & Kaba Wakene, F. (2020). A family of theta-function identities based upon q-binomial theorem and Heine’s transformations. Montes Taurus Journal of Pure and Applied Mathematics, 2(2), 1–6.
35. Srivastava, H. M., & Choi, J. (2012). Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier Science Publishers.
36. Srivastava, H. M., & Karlsson, P. W. (1985). Multiple Gaussian Hypergeometric Series. Ellis Horwood, Chichester; Halsted Press, NY.
37. Srivastava, H. M., & Zhang, C.-H. (2009). A certain class of identities of the
    Rogers–Ramanujan type. Panamerican Mathematical Journal, 19, 89–102.
38. Yee, A.-J. (2003). Combinatorial proofs of generating function identities for F-partitions. Journal of Combinatorial Theory, Series A, 102, 217–228.
39. Yi, J.-H. (2004). Theta-function identities and the explicit formulas for theta-function and their applications. Journal of Mathematical Analysis and Applications, 292, 381–400.