Series transformations and integrals involving the Riemann \(\Xi \)-function (Q972489)

Let \(\psi(x):=\frac{\Gamma'(x)}{\Gamma(x)}=-\gamma-\sum_{m=0}^{\infty}\big(\frac{1}{m+x}-\frac{1}{m+1}\big)\) and \(\xi(s):=(s-1)\pi^{-1/2s}\Gamma\big(1+\frac{1}{2}s\big)\zeta(s)\) be the logarithmic derivative of the gamma-function \(\Gamma(x)\) and the Riemann \(\xi\)-function, respectively (here \(\gamma\) is Euler's constant, and \(\zeta(s)\) is the Riemann zeta-function). Then the Riemann \(\Xi\)-function is defined by the formula \(\Xi(t):=\xi\bigl(\tfrac12+it\bigr). \) The paper is devoted to the unification the transformation formulas of Ramanujan, Hardy, Koshliakov and Ferrar in the sense that all theses formulas come back from the general formula involving an integral of the \(\Xi\)-function. In the proof of these formulas, the theory of Mellin transforms and the residue theorem are used. Also, the new extensions for the formulas of Koshliakov and Ferrar via their connection with integrals involving the Riemann \(\Xi\)-function are obtained.