Euler Polynomials and Combinatoric Convolution Sums of Divisor Functions with Even Indices

Abstract

We study combinatoric convolution sums of certain divisor functions involving even indices. We express them as a linear combination of divisor functions and Euler polynomials and obtain identities $D_{\mathrm{2}k}(n)=(\mathrm{1}/\mathrm{4}){\sigma }_{\mathrm{2}k+\mathrm{1,0}}(n;\mathrm{2})-\mathrm{2}·{\mathrm{4}}^{\mathrm{2}k}{\sigma }_{\mathrm{2}k+\mathrm{1}}(n/\mathrm{4})-(\mathrm{1}/\mathrm{2})[\sum \begin{array}{c}d|n,d\equiv \mathrm{1}(\mathrm{4})\end{array}\{E_{\mathrm{2}k}(d)+E_{\mathrm{2}k}(d-\mathrm{1})\}+{\mathrm{2}}^{\mathrm{2}k}$ ${\sum }_{d|n,d\equiv \mathrm{1}(\mathrm{2})}^{\begin{array}{}\end{array}}{E}_{\mathrm{2}k}((d+(-\mathrm{1}{)}^{(d-\mathrm{1})/\mathrm{2}})/\mathrm{2})]$, $U_{\mathrm{2}k}(p,q)={\mathrm{2}}^{\mathrm{2}k-\mathrm{2}}[-((p+q)/\mathrm{2})E_{\mathrm{2}k}((p+q)/\mathrm{2}+\mathrm{1})+((q-p)/\mathrm{2})E_{\mathrm{2}k}$ $((q-p)/\mathrm{2})-E_{\mathrm{2}k}((p+\mathrm{1})/\mathrm{2})-E_{\mathrm{2}k}((q+\mathrm{1})/\mathrm{2})+E_{\mathrm{2}k+\mathrm{1}}((p+q)/\mathrm{2}+\mathrm{1})-E_{\mathrm{2}k+\mathrm{1}}((q-p)/\mathrm{2})]$, and $F_{\mathrm{2}k}(n)=(\mathrm{1}/\mathrm{2})\{{\sigma }_{\mathrm{2}k+\mathrm{1}}^{†}(n)-{\sigma }_{\mathrm{2}k}^{†}(n)\}$. As applications of these identities, we give several concrete interpretations in terms of the procedural modelling method.