Abstract and Applied Analysis

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

Daeyeoul Kim, Abdelmejid Bayad, and Joongsoo Park

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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 2 k ( n ) = ( 1 / 4 ) σ 2 k + 1,0 ( n ; 2 ) - 2 · 4 2 k σ 2 k + 1 ( n / 4 )    - ( 1 / 2 ) [ d | n , d 1    ( 4 ) { E 2 k ( d ) + E 2 k ( d - 1 ) } + 2 2 k d | n , d 1    ( 2 ) E 2 k ( ( d + ( - 1 ) ( d - 1 ) / 2 ) / 2 ) ] , U 2 k ( p , q ) = 2 2 k - 2 [ - ( ( p + q ) / 2 ) E 2 k ( ( p + q ) / 2 + 1 ) + ( ( q - p ) / 2 ) E 2 k ( ( q - p ) / 2 ) - E 2 k ( ( p + 1 ) / 2 ) - E 2 k ( ( q + 1 ) / 2 ) + E 2 k + 1 ( ( p + q ) / 2 + 1 ) - E 2 k + 1 ( ( q - p ) / 2 ) ] , and F 2 k ( n ) = ( 1 / 2 ) { σ 2 k + 1 ( n ) - σ 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

Permanent link to this document
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


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