Euler Sums and Contour Integral Representations

This paper develops an approach to the evaluation of Euler sums involving harmonic numbers either linearly or nonlinearly. We give explicit formul{æ} for certain classes of Euler sums in terms of values of the Riemann zeta function at positive integers. The approach is based on simple contour integral representations and residue computations

On a discrete composition of the fractional integral and Caputo derivative

We prove a discrete analogue for the composition of the fractional integral and Caputo derivative. This result is relevant in numerical analysis of fractional PDEs when one discretizes the Caputo derivative with the so-called L1 scheme. The proof is based on asymptotic evaluation of the discrete sums with the use of the Euler-Maclaurin summation formula.Comment: This is an accepted version of the manuscript published in Communications in Nonlinear Science and Numerical Simulations. The changes with the previous versions included some language corrections, additional numerical simulations, and new reference

On one dimensional digamma and polygamma series related to the evaluation of Feynman diagrams

We consider summations over digamma and polygamma functions, often with summands of the form (\pm 1)^n\psi(n+p/q)/n^r and (\pm 1)^n\psi^{(m)} (n+p/q)/n^r, where m, p, q, and r are positive integers. We develop novel general integral representations and present explicit examples. Special cases of the sums reduce to known linear Euler sums. The sums of interest find application in quantum field theory, including evaluation of Feynman amplitudes.Comment: to appear in J. Comput. Appl. Math.; corrected proof available online with this journal; no figure