Some evaluations of infinite series involving Dirichlet type parametric harmonic numbers
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In this paper, we formally introduce the notion of a general parametric digamma function $\Psi (-s; A, a)$ and we find the Laurent expansion of $\Psi (-s; A, a)$ at the integers and poles. Considering the contour integrations involving $\Psi (-s; A, a)$, we present some new identities for infinite series involving Dirichlet type parametric harmonic numbers by using the method of residue computation. Then applying these formulas obtained, we establish some explicit relations of parametric linear Euler sums and some special functions (e.g.~trigonometric functions, digamma functions, Hurwitz zeta functions etc.). Moreover, some illustrative special cases as well as immediate consequences of the main results are also considered.
Abstract
In this paper, we formally introduce the notion of a general parametric digamma function $\Psi (-s; A, a)$ and we find the Laurent expansion of $\Psi (-s; A, a)$ at the integers and poles. Considering the contour integrations involving $\Psi (-s; A, a)$, we present some new identities for infinite series involving Dirichlet type parametric harmonic numbers by using the method of residue computation. Then applying these formulas obtained, we establish some explicit relations of parametric linear Euler sums and some special functions (e.g.~trigonometric functions, digamma functions, Hurwitz zeta functions etc.). Moreover, some illustrative special cases as well as immediate consequences of the main results are also considered.
참고문헌 (24)
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