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Generalizations of Ramanujan’s reciprocity theorem and their applications. (English) Zbl 1120.33015

The author surveys some results on Ramanujan’s reciprocity theorem for a \(q\)-series related to partial theta functions and gives a new proof of this theorem. He actually gives a proof of a theorem of G. E. Andrews [Ramanujan’s “lost” notebook. I. Partial \(\theta\)- functions, Adv. Math. 41, 137–172 (1981; Zbl 0477.33001)] and derives four-variable generalizations of Jacobi’s triple product identity and the quintuple product identity from the generalization. The author presents several applications of the generalized reciprocity theorems and product identities, including new representations for the generating functions for sums of six squares and sums of four triangular numbers. He also gives as applications known and new results such as \(q\)-Gauss summation formula and \(q\)-series identities related to overpartition theory.

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
11B65 Binomial coefficients; factorials; \(q\)-identities
05A15 Exact enumeration problems, generating functions

Citations:

Zbl 0477.33001
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References:

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