## Abstract and Applied Analysis

### 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 ${D}_{\mathrm{2}k}(n)=(\mathrm{1}/\mathrm{4}){\sigma }_{\mathrm{2}k+\mathrm{1,0}}(n;\mathrm{2})-\mathrm{2}·{\mathrm{4}}^{\mathrm{2}k}{\sigma }_{\mathrm{2}k+\mathrm{1}}(n/\mathrm{4})\mathrm{ }-(\mathrm{1}/\mathrm{2})[\sum \begin{smallmatrix}d|n,d\equiv \mathrm{1}\mathrm{ }(\mathrm{4})\end{smallmatrix}\{{E}_{\mathrm{2}k}(d)+{E}_{\mathrm{2}k}(d-\mathrm{1})\}+{\mathrm{2}}^{\mathrm{2}k}$ ${\sum }_{d|n,d\equiv \mathrm{1}\mathrm{ }(\mathrm{2})}^{\begin{smallmatrix}\end{smallmatrix}}{E}_{\mathrm{2}k}((d+(-\mathrm{1}{)}^{(d-\mathrm{1})/\mathrm{2}})/\mathrm{2})]$, ${U}_{\mathrm{2}k}(p,q)={\mathrm{2}}^{\mathrm{2}k-\mathrm{2}}[-((p+q)/\mathrm{2}){E}_{\mathrm{2}k}((p+q)/\mathrm{2}+\mathrm{1})+((q-p)/\mathrm{2}){E}_{\mathrm{2}k}$ $((q-p)/\mathrm{2})-{E}_{\mathrm{2}k}((p+\mathrm{1})/\mathrm{2})-{E}_{\mathrm{2}k}((q+\mathrm{1})/\mathrm{2})+{E}_{\mathrm{2}k+\mathrm{1}}((p+q)/\mathrm{2}+\mathrm{1})-{E}_{\mathrm{2}k+\mathrm{1}}((q-p)/\mathrm{2})]$, 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 289187, 6 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412277174

Digital Object Identifier
doi:10.1155/2014/289187

Mathematical Reviews number (MathSciNet)
MR3256242

Zentralblatt MATH identifier
07022098

#### Citation

Kim, Daeyeoul; Bayad, Abdelmejid; Park, Joongsoo. Euler Polynomials and Combinatoric Convolution Sums of Divisor Functions with Even Indices. Abstr. Appl. Anal. 2014 (2014), Article ID 289187, 6 pages. doi:10.1155/2014/289187. https://projecteuclid.org/euclid.aaa/1412277174