Connection formulas for the confluent hypergeometric functions and the functional relation for the Hurwitz zeta function

Zeros of combinations of the Riemann $Ξ$-function and the confluent hypergeometric function on bounded vertical shifts

In 1914, Hardy proved that infinitely many non-trivial zeros of the Riemann zeta function lie on the critical line using the transformation formula of the Jacobi theta function. Recently the first author obtained an integral representation involving the Riemann $Ξ$-function and the confluent hypergeometric function linked to the general theta transformation. Using this result, we show that a series consisting of bounded vertical shifts of a product of the Riemann $Ξ$-function and the real part of a confluent hypergeometric function has infinitely many zeros on the critical line, thereby genera…

A relation between the Mordell-Tornheim multiple Dirichlet series and the confluent hypergeometric function

We investigate a relation between the Mordell-Tornheim type of multiple Dirichlet series and a confluent hypergeometric function. We prove it by applying the Mellin-Barnes integral formula. Also, main results in this paper contain two kinds of functional equations for double $L$-functions as a special case.

利用有限域上的合流超几何函数表示有限域上 Appell-Lauricella 函数的求和及其应用

Sum representations of Appell-Lauricella functions over finite fields using confluent hypergeometric functions and their applications

本文利用有限域上的合流超几何函数证明了 Appell-Lauricella 函数的求和表示，并以此推导出其在有限域上的变换、求和与化简公式。

多重zeta函数与广义合流超几何函数的一个函数方程

A functional equation for multiple zeta functions and generalized confluent hypergeometric functions

本文引入多重合流超几何函数，并利用其为$r$变量Euler--Zagier多重zeta函数建立函数方程，该方程在$r=2$时包含了Matsumoto得到的著名二重zeta函数方程。

Generalized hypergeometric expansion related to the Hurwitz zeta function

We study the incomplete Mellin transformation of the fractional part and the related log-sine function when composed by an affine complex map. We evaluate the corresponding integral in two different ways which yields equalities with series in hypergeometric functions on each side. These equalities capture basic analytic properties of the Hurwitz zeta function like meromorphic extension and the functional equation. We give rates of convergence for the involved series. As a special case we find that the exponential rate of convergence for the deformation of the log-sine to the fractional part in…

An explicit formula for Bell numbers in terms of Stirling numbers and hypergeometric functions

In the paper, the author finds an explicit formula for computing Bell numbers in terms of Kummer confluent hypergeometric functions and Stirling numbers of the second kind.