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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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