On relationships between \(q\)-products identities, \(R_\alpha\), \(R_\beta\) and \(R_m\) functions related to Jacobi’s triple-product identity. (English) Zbl 1474.11112
Summary: The authors establish a set of two new relationships involving \(q\)-product identities, \(R_\alpha\), \(R_\beta\) and \(R_m\) \((m = 1, 2, 3, \dots)\) functions; and answer a open question of Srivastava et al. H. M. Srivastava et al. [“A family of theta-function identities based upon combinatorial partition identities related to Jacobi’s triple-product identity”, Mathematics 8, No. 6, Paper No. 918, 14 p. (2020; doi:10.3390/math8060918)]. The present work is motivated and based upon recent findings of the authors et al. [Math. Morav. 24, No. 1, 83–91 (2020; Zbl 1474.05023)].
MSC:
| 11F27 | Theta series; Weil representation; theta correspondences |
| 11P83 | Partitions; congruences and congruential restrictions |
| 05A17 | Combinatorial aspects of partitions of integers |
| 05A30 | \(q\)-calculus and related topics |
Keywords:
theta-function identities; \(R_\alpha\), \(R_\beta\) and \(R_m\) functions; Jacobi’s triple-product identity; \(q\)-product identities; Rogers-Ramanujan continued fraction; Rogers-Ramanujan identitiesCitations:
Zbl 1474.05023References:
| [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, Journal of Number Theory, 158 (2016), 281-297. · Zbl 1400.11134 |
| [2] | C. Adiga, N. A. S. Bulkhali, Y. Simsek and H. M. Srivastava,A continued fraction of Ramanujan and some Ramanujan-Weber class invariants, Filomat, 31 (2017), 3975-3997. · Zbl 1499.11029 |
| [3] | G. E. Andrews,The Theory of Partitions, Cambridge University Press, Cambridge, London and New York, 1998. · Zbl 0996.11002 |
| [4] | G. E. Andrews, K. Bringman and K. Mahlburg,Double series representations for Schur’s partition function and related identities, Journal of Combinatorial Theory, Series A, 132 (2015), 102-119. · Zbl 1307.05224 |
| [5] | T. M. Apostol,Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer-Verlag, Berlin, New York and Heidelberg, 1976. · Zbl 0335.10001 |
| [6] | B. C. Berndt,Ramanujan’s Notebooks, Part III, Springer-Verlag, Berlin, Heidelberg and New York, 1991. · Zbl 0733.11001 |
| [7] | J. Cao, H. M. Srivastava and Z.-G. Luo,Some iterated fractionalq-integrals and their applications, Fractional Calculus and Applied Analysis, 21 (2018), 672-695. · Zbl 1403.05012 |
| [8] | M. P. Chaudhary, Sangeeta Chaudhary and Sonajharia Minz,On relationships betweenq-products identities and combinatorial partition identities, Mathematica Moravica, 24 (2020), 83-91. · Zbl 1474.05023 |
| [9] | H.-Y. Hahn, J.-S. Huh, E.-S. Lim and J.-B. Sohn,From partition identities to a combinatorial approach to explicit Satake inversion, Annals of Combinatorics, 22 (2018), 543-562. · Zbl 1454.11189 |
| [10] | C. G. J. Jacobi,Fundamenta Nova Theoriae Functionum Ellipticarum, Regiomonti, Sumtibus Fratrum Bornträger, Königsberg, Germany, 1829; Reprinted inGesammelte Mathematische Werke, 1 (1829), 497-538; American Mathematical Society, Providence, Rhode Island, 1969, 97-239. |
| [11] | M. S. M. Naika, K. S. Bairy and N. P Suman,Modular relations for remarkable product of theta-functions and evaluations, Recent Advances in Mathematics, RMSLecture Notes Series, 21 (2015), 133-145. · Zbl 1350.11056 |
| [12] | S. Ramanujan,Notebooks, Vols. 1 and 2, Tata Institute of Fundamental Research, Bombay, 1957. · Zbl 0138.24201 |
| [13] | S. Ramanujan,The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988. · Zbl 0639.01023 |
| [14] | H. M. Srivastava,Operators of basic (orq-) calculus and fractionalq-calculus and their applications in geometric function theory of complex analysis, Iranian Journal of Science and Technology, Transaction A: Science, 44 (2020), 327-344. |
| [15] | H. M. Srivastava and M. P. Chaudhary,Some relationships betweenq-product identities, combinatorial partition identities and continued-fraction identities, Advanced Studies in Contemporary Mathematics, 25 (2015), 265-272. · Zbl 1326.05017 |
| [16] | H. M. Srivastava, M. P. Chaudhary and S. Chaudhary,Some theta-function identities related to Jacobi’s triple-product identity, European Journal of Pure and Applied Mathematics, 11 (1) (2018), 1-9. · Zbl 1382.05008 |
| [17] | H. M. Srivastava, M. P. Chaudhary and S. Chaudhary,A family of theta-function identities related to Jacobi’s triple-product identity, Russian Journal of Mathematical Physics, 27 (2020), 139-144. · Zbl 1435.11077 |
| [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) (2020), Article ID: 918, 1-14. |
| [19] | H. M. Srivastava and J. Choi,Zeta andq-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012. · Zbl 1239.33002 |
| [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. · Zbl 0552.33001 |
| [21] | H. M. Srivastava and C.-H. Zhang,A certain class of identities of the RogersRamanujan type, Pan-American Mathematical Journal, 19 (2009), 89-102. · Zbl 1180.05017 |
| [22] | A.-J. Yee,Combinatorial proofs of generating function identities for F-partitions, Journal of Combinatorial Theory, Series A, 102 (2003), 217-228. · Zbl 1017.05019 |
| [23] | J.-H. Yi,Theta-function identities and the explicit formulas for theta-function and their applications, Journal of Mathematical Analysis and Applications, 292 (2004), 381-400 · Zbl 1060.11027 |
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