A NEW CONVOLUTION IDENTITY DEDUCIBLE FROM THE REMARKABLE FORMULA OF RAMANUJAN

Abstract

In this paper we obtain a convolution identity for the coefficients $B_n(\alpha,\theta,q)$ defined by \[ \sum_{n=-\infty}^{\infty} B_{n}(\alpha,\theta,q) x^{n} = \frac{\prod\limits_{n=1}^{\infty} (1 + 2x q^n \cos \theta + x^2 q^{2n})}{\prod\limits_{n=1}^{\infty} (1 + \alpha q^n xe^{i\theta})}, \] using the well-known Ramanujan’s $_{1}\psi_1$-summation formula. The work presented here complements the works of K.-W. Yang, S. Bhargava, C. Adiga and D. D. Somashekara and of H. M. Srivastava.