×

On a reciprocity theorem of Ramanujan. (English) Zbl 1149.33003

For a complex \(q\) with \(| q| <1\), let \((a)_n=(a;q)_n=\prod_{k=0}^{n-1}(1-aq^{k-1})\) denote the \(q\)-shifted factorial, \(n=0,1,\ldots\). Ramanujan’s reciprocity theorem states that, for \(a,b\notin-q^{\mathbb Z_{\leq0}}\), one has
\[ \rho(a,b)-\rho(b,a) =\biggl(\frac1b-\frac1a\biggr) \frac{(aq/b)_\infty(bq/a)_\infty(q)_\infty}{(-aq)_\infty(-bq)_\infty}, \]
where
\[ \rho(a,b)=\biggl(1+\frac1b\biggr) \sum_{n=0}^\infty\frac{(-1)^nq^{n(n+1)/2}(a/b)^n}{(-aq)_n}. \]
The authors give a short proof of the theorem using classical Heine’s transformation of the Heine (basic) hypergeometric series and show that Jacobi’s triple product identity and Euler’s formula
\[ \Gamma(s)=\int_0^\infty e^{-x}x^{s-1}dx, \qquad s>-1, \]
for the gamma function are special cases of this theorem. The last section is illustrated by four eta-function identities (\(\eta(\tau)=q^{1/24}(q;q)_\infty\) with \(q=e^{2\pi i\tau}\), \(\text{Im}\tau>0\)) deduced from Ramanujan’s reciprocity theorem; one example is
\[ \frac{\eta^2(\tau)\eta(6\tau)}{\eta(2\tau)\eta(3\tau)} =q^{1/8}\frac{1+q^3}{1+q}\sum_{n=0}^\infty\frac{(-1)^nq^{n(3n-1)/2}}{(-q^2;q^3)_n} +q^{9/8}\sum_{n=0}^\infty\frac{(-1)^nq^{n(3n+7)/2}}{(-q^4;q^3)_n}. \]

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
11F11 Holomorphic modular forms of integral weight
11F20 Dedekind eta function, Dedekind sums
33D05 \(q\)-gamma functions, \(q\)-beta functions and integrals
PDF BibTeX XML Cite