Closed-form evaluations of certain integrals of \(J_{0}(\xi )\), the Bessel function of the first kind, have been crucial in the studies on the electromagnetic field of alternating current in a circuit with two groundings, as can be seen from the works of Fock and Bursian, Schermann, etc. Koshliakov’s generalization of one such integral, which contains \(J_s(\xi )\) in the integrand, encompasses several important integrals in the literature including Sonine’s integral. Here, we derive an analogous integral identity where \(J_{s}(\xi )\) is replaced by a kernel consisting of a combination of \(J_{s}(\xi )\), \(K_{s}(\xi )\) and \(Y_{s}(\xi )\). This kernel is important in number theory because of its role in the Voronoï summation formula for the sum-of-divisors function \(\sigma _s(n)\). Using this identity and the Voronoï summation formula, we derive a general transformation relating infinite series of products of Bessel functions \(I_{\lambda }(\xi )\) and \(K_{\lambda }(\xi )\) with those involving the Gaussian hypergeometric function. As applications of this transformation, several important results are derived, including what we believe to be a corrected version of the first identity found on page 336 of Ramanujan’s Lost Notebook.

中文翻译：

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电磁引起的福克型积分的类比及其在数论中的应用

对第一类贝塞尔函数\（J_ {0}（\ xi）\）的某些积分的闭式求值，对于研究具有两个接地的电路中的交流电的电磁场至关重要，因为可以从Fock和Bursian，Schermann等的著作中看到。Koshliakov对这样一个积分的推广，在被积数中包含\（J_s（\ xi）\），涵盖了文学中的几个重要积分，包括Sonine积分。在这里，我们得到一个类似的积分恒等式，其中\（J_ {s}（\ xi）\）被由\（J_ {s}（\ xi）\），\（K_ {s} （\ xi）\）和\（Y_ {s}（\ xi）\）。该核在数论中很重要，因为它在除数函数\（\ sigma _s（n）\）的Voronoï求和公式中起作用。使用该恒等式和Voronoï求和公式，我们推导了将Bessel函数\（I _ {\ lambda}（\ xi）\）和\（K _ {\ lambda}（\ xi）\）的乘积的无穷级数与那些涉及高斯超几何函数的函数。作为这种转换的应用，得出了一些重要的结果，包括我们认为是Ramanujan的“失落笔记本”第336页上发现的第一个身份的更正版本。