A Generalized Closed Form of Ramanujan-Type Fourier Cosine Transform via Meijer's G-Function
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2026
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| author | Dar, S. A. Paris, R. P. |
| author_facet | Dar, S. A. Paris, R. P. |
| contents | In this paper, we obtain analytical evaluations of the Ramanujan integral \[\textbf{R}_{C}(m,n)= \int_{0}^{\infty}\frac{x^m\,\cos(πnx)}{\exp{(2π\sqrt{x})-1}}dx\] subject to suitable convergence conditions in terms of an infinite series of Meijer $G$-functions of one variable, by using Mellin-Barnes-type contour integral representations of the cosine function. %and Laplace transform method. We also consider some generalizations of the integral $\textbf{R}_{C}(m,n)$ given as the integrals $I_{C}^{*}(\upsilon,b,c,λ,y)$ ,$Ξ_{C}(\upsilon,b,c,λ,y)$, $\nabla_{C}(\upsilon,b,c,λ,y)$ and $I_{C}(\upsilon,b,λ,y)$. These integrals are also expressed in terms of infinite series of Meijer $G$-functions. Moreover, as an application of a Ramanujan's integral $\textbf{R}_C(m,n)$, the closed-form evaluations of nine infinite series of Meijer $G$-functions are obtained. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_13882 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Generalized Closed Form of Ramanujan-Type Fourier Cosine Transform via Meijer's G-Function Dar, S. A. Paris, R. P. Number Theory 33C60, 33C65, 33C70, 33B15, 33C05, 33C45, 33C10 In this paper, we obtain analytical evaluations of the Ramanujan integral \[\textbf{R}_{C}(m,n)= \int_{0}^{\infty}\frac{x^m\,\cos(πnx)}{\exp{(2π\sqrt{x})-1}}dx\] subject to suitable convergence conditions in terms of an infinite series of Meijer $G$-functions of one variable, by using Mellin-Barnes-type contour integral representations of the cosine function. %and Laplace transform method. We also consider some generalizations of the integral $\textbf{R}_{C}(m,n)$ given as the integrals $I_{C}^{*}(\upsilon,b,c,λ,y)$ ,$Ξ_{C}(\upsilon,b,c,λ,y)$, $\nabla_{C}(\upsilon,b,c,λ,y)$ and $I_{C}(\upsilon,b,λ,y)$. These integrals are also expressed in terms of infinite series of Meijer $G$-functions. Moreover, as an application of a Ramanujan's integral $\textbf{R}_C(m,n)$, the closed-form evaluations of nine infinite series of Meijer $G$-functions are obtained. |
| title | A Generalized Closed Form of Ramanujan-Type Fourier Cosine Transform via Meijer's G-Function |
| topic | Number Theory 33C60, 33C65, 33C70, 33B15, 33C05, 33C45, 33C10 |
| url | https://arxiv.org/abs/2605.13882 |