Evaluation of some q-integrals in terms of the Dedekind eta function
A q-integral is a definite integral of a function of q having an expansion in non-negative powers of q for |q| < 1 (q-series). In his book on hypergeometric series, N. J. Fine [N. J. Fine, Basic Hypergeometric Series and Applications, Math. Surveys Monogr. 27, American Mathematical Society, Providence, 1988] explicitly evaluated three q-integrals. For example, he showed that e-π ∫0 ∞πn=1 (1 - q2n)20/(1 - qn)16 dq = 1/16. In this paper, we prove a general theorem which allows us to determine a wide class of integrals of this type. This class includes the three q-integrals evaluated by Fine as well as some of those evaluated by L.-C. Zhang [L.-C. Zhang, Some q-integrals associated with modular forms, J. Math Anal. Appl. 150 (1990), 264-273]. It also includes many new evaluations of q-integrals.
|Keywords||Dedekind eta function|
Doyle, G. (Greg), & Williams, K.S. (2018). Evaluation of some q-integrals in terms of the Dedekind eta function. Analysis (Germany). doi:10.1515/anly-2018-0002