A generalized modular relation of the form \(F(z, w, \alpha )=F(z, iw,\beta )\), where \(\alpha \beta =1\) and \(i=\sqrt{-1}\), is obtained in the course of evaluating an integral involving the Riemann \(\Xi \)-function. This modular relation involves a surprising new generalization of the Hurwitz zeta function \(\zeta (s, a)\), which we denote by \(\zeta _w(s, a)\). We show that \(\zeta _w(s, a)\) satisfies a beautiful theory generalizing that of \(\zeta (s, a)\). In particular, it is shown that for \(0<a<1\) and \(w\in \mathbb {C}\), \(\zeta _w(s, a)\) can be analytically continued to Re\((s)>-1\) except for a simple pole at \(s=1\). The theories of functions reciprocal in a kernel involving a combination of Bessel functions and of a new generalized modified Bessel function \({}_1K_{z,w}(x)\), which are also essential to obtain the generalized modular relation, are developed.

形式为\(F(z, w, \alpha )=F(z, iw,\beta )\)的广义模关系，其中\(\alpha \beta =1\)和\(i=\sqrt{ -1}\)，是在计算涉及黎曼\(\Xi \)函数的积分过程中获得的。这种模关系涉及 Hurwitz zeta 函数\(\zeta (s, a)\)的令人惊讶的新泛化，我们将其表示为\(\zeta _w(s, a)\)。我们证明\(\zeta _w(s, a)\)满足一个美丽的理论，概括了\(\zeta (s, a)\) 的理论。特别是，对于\(0<a<1\)和\(w\in \mathbb {C}\)，\(\zeta _w(s, a)\)除了\(s=1\)处的一个简单极点之外，可以解析继续到 Re \((s)>- 1\)。包含贝塞尔函数和新广义修正贝塞尔函数\({}_1K_{z,w}(x)\)的核中的函数倒数理论，对于获得广义模关系也是必不可少的，是发达。