Some applications of the Gamma and polygamma functions involving convolutions of the Rayleigh functions, multiple Euler sums and log‐sine integrals

Abstract

The Gamma function and its nth logarithmic derivatives (that is, the polygamma or the psi-functions) have found many interesting and useful applications in a variety of subjects in pure and applied mathematics. Here we mainly apply these functions to treat convolutions of the Rayleigh functions by recalling a general identity expressing a certain class of series as psi-functions and to evaluate a class of log-sine integrals in an algorithmic way. We also evaluate some Euler sums and give much simpler psi-function expressions for some known parameterized multiple sums.

Log-Sine and Log-Cosine Integrals

“…Motivated essentially by their potential for applications in a wide range of mathematical and physical problems, the log-sine and log-cosine integrals have been evaluated, in the existing literature on the subject, in many different ways (see, for example, [2,3,6,8,9,13,15,17,18,19,20,21] and the references therein). By making use of the familiar Beta function B(α, β) (see, for example, [18,p.…”

Identities for the harmonic numbers and binomial coefficients

“…Further work in the summation of harmonic numbers and binomial coefficients has also been done by Flajolet and Salvy [11], Basu [2], Basu and Apostol [3], Choi and Srivastava ([7] and [8]), and Rassias and Srivastava [17]. The book by Srivastava and Choi [23] also has many results associated with zeta and related functions.…”

Euler Sums and Integral Connections

“…Some other Euler identities are given in [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37].…”