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A reciprocity theorem for certain $$q$$-series found in Ramanujan’s lost notebook. (English) Zbl 1123.33012
Let $$(a)_n:=(1-a)(1-aq)\cdots (1-aq^{n-1})$$ and $\rho(a,b) = \left(1+\frac{1}{b}\right)\sum_{n=0}^\infty (-1)^nq^{n(n+1)/2}\frac{a^nb^{-n}}{(-aq)_n},$ where $$a$$ and $$b$$ are any complex numbers, except that $$a\neq -q^{-n}$$ for any positive integer $$n$$. In this article, the authors offer three different proofs of Ramanujan’s identity for $$\rho(a,b)$$, namely, for $$a,b\neq q^{-n}$$,
$\rho(a,b)-\rho(b,a) =\left(\frac{1}{b}-\frac{1}{a}\right) {\prod_{k=1}^\infty} \frac{\left(1-\dfrac{a}{b}q^k\right)\left(1-\dfrac{b}{a}q^k\right)\left(1-q^k\right)}{\left(1+aq^k\right)\left(1+bq^k\right)}.$
The first proof uses Ramanujan’s $$_1\psi_1$$ summation formula, the second proof uses identities due to N.J. Fine and Jacobi’s triple product identity and the third proof uses purely combinatorial arguments. The paper ends with a generalization of the quintuple product identity, as well as a new representation for the function $\left(\sum_{n=-\infty}^\infty q^{n^2}\right)^3.$

##### MSC:
 33D15 Basic hypergeometric functions in one variable, $${}_r\phi_s$$
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##### References:
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