Evaluation of some q-integrals in terms of the Dedekind eta function
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Bibliographic record
Abstract
Abstract 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 \int_{0}^{e^{-\pi}}\prod_{n=1}^{\infty}\frac{(1-q^{2n})^{20}}{(1-q^{n})^{16}}% dq=\frac{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.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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