Abstract
Three proofs are given for a reciprocity theorem for a certain q-series found in Ramanujan’s lost notebook. The first proof uses Ramanujan’s 1ψ1 summation theorem, the second employs an identity of N. J. Fine, and the third is combinatorial. Next, we show that the reciprocity theorem leads to a two variable generalization of the quintuple product identity. The paper concludes with an application to sums of three squares.
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Adiga, C., Anitha, N.: On a reciprocity theorem of Ramanujan. Tamsui Oxford J. Math. Sci. 22, 9–15 (2006)
Andrews, G.E.: Ramanujan’s lost notebook. I. Partial θ-functions. Adv. Math. 41, 137–172 (1981)
Andrews, G.E.: Ramanujan’s lost notebook V: Euler’s partition identity. Adv. Math. 61, 156–164 (1986)
Andrews, G.E.: The fifth and seventh order mock theta functions. Trans. Amer. Math. Soc. 293, 113–134 (1986)
Berndt, B.C.: Ramanujan’s Notebooks, Part III. Springer-Verlag, New York (1991)
Bhargava, S., Adiga, C.: A basic bilateral series summation formula and its applications. Integral Transforms Special Functions 2, 165–184 (1994)
Bhargava, S., Adiga, C., Mahadeva Naika, M.S.: Ramanujan’s remarkable summation formula as a 2-parameter generalization of the quintuple product identity. Tamkang J. Math. 33, 285–288 (2002)
Bhargava, S., Adiga, C., Somashekara, D.D.: Three-square theorem as an application of Andrews’ identity. Fibonacci Quart. 31, 129–133 (1993)
Bhargava, S., Somashekara, D.D., Fathima, S.N.: Some q-Gamma and q-Beta function identities deducible from the reciprocity theorem of Ramanujan. Adv. Stud. Contemp. Math. (Kyungshang) 11, 227–234 (2005)
Fine, N.J.: Basic Hypergeometric Series and Applications. American Mathematical Society, Providence, RI (1988)
Kang, S.–Y.: Generalizations of Ramanujan’s reciprocity theorem and their applications. J. London Math. Soc. 74, (2006), to appear.
Kim, T., Somashekara, D.D., Fathima, S.N.: On a generalization of Jacobi’s triple product identity and its applications. Adv. Stud. Contemp. Math. (Kyungshang) 9, 165–174 (2004)
Liu, Z.–G.: Some operator identities and q-series transformation formulas. Discrete Math. 265, 119–139 (2003)
Rogers, L.J.: On a three-fold symmetry in the elements of Heine’s series. Proc. London Math. Soc. 24, 171–179 (1893)
Ramanujan, S.: The Lost Notebook and Other Unpublished Papers. Narosa, New Delhi (1988)
Somashekara, D.D., Fathima, S.N.: An interesting generalization of Jacobi’s triple product identity. Far East J. Math. Sci. 9, 255–259 (2003)
Yee, A.J.: Combinatorial proofs of Ramanujan’s 1 ψ1 summation and the q-Gauss summation. J. Combin. Thy. Ser. A. 105, 63–77 (2004)
Zhang, Z.: A generalization of an identity of Ramanujan, to appear.
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Dedicated to Richard Askey on the occasion of his 70th birthday.
2000 Mathematics Subject Classification Primary—33D15
B. C. Berndt: Research partially supported by grant MDA904-00-1-0015 from the National Security Agency.
A. J. Yee: Research partially supported by a grant from The Number Theory Foundation.
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Berndt, B.C., Chan, S.H., Yeap, B.P. et al. A reciprocity theorem for certain q-series found in Ramanujan’s lost notebook. Ramanujan J 13, 27–37 (2007). https://doi.org/10.1007/s11139-006-0241-5
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DOI: https://doi.org/10.1007/s11139-006-0241-5