Euler Polynomials and Combinatoric Convolution Sums of Divisor Functions with Even Indices
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 D2k(n)=(1/4)σ2k+1,0(n;2)-2·42kσ2k+1(n/4) -(1/2)[∑d|n,d≡1 (4){E2k(d)+E2k(d-1)}+22k∑d|n,d≡1 (2)E2k((d...
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2014-01-01
|
| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/289187 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1849401554681987072 |
|---|---|
| author | Daeyeoul Kim Abdelmejid Bayad Joongsoo Park |
| author_facet | Daeyeoul Kim Abdelmejid Bayad Joongsoo Park |
| author_sort | Daeyeoul Kim |
| collection | DOAJ |
| description | 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 D2k(n)=(1/4)σ2k+1,0(n;2)-2·42kσ2k+1(n/4) -(1/2)[∑d|n,d≡1 (4){E2k(d)+E2k(d-1)}+22k∑d|n,d≡1 (2)E2k((d+(-1)(d-1)/2)/2)], U2k(p,q)=22k-2[-((p+q)/2)E2k((p+q)/2+1)+((q-p)/2)E2k((q-p)/2)-E2k((p+1)/2)-E2k((q+1)/2)+E2k+1((p+q)/2 +1)-E2k+1((q-p)/2)], and F2k(n)=(1/2){σ2k+1†(n)-σ2k†(n)}. As applications of these identities, we give several concrete interpretations in terms of the procedural modelling method. |
| format | Article |
| id | doaj-art-8d1dcee0311f4d34aab9be8a88a5c2dc |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-8d1dcee0311f4d34aab9be8a88a5c2dc2025-08-20T03:37:44ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/289187289187Euler Polynomials and Combinatoric Convolution Sums of Divisor Functions with Even IndicesDaeyeoul Kim0Abdelmejid Bayad1Joongsoo Park2National Institute for Mathematical Sciences, Daejeon 305-811, Republic of KoreaDépartement de Mathématiques, Université d'Evry Val d'Essonne, FranceWoosuk University, Samlae, Wanju, Jeonbuk 565-701, Republic of KoreaWe 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 D2k(n)=(1/4)σ2k+1,0(n;2)-2·42kσ2k+1(n/4) -(1/2)[∑d|n,d≡1 (4){E2k(d)+E2k(d-1)}+22k∑d|n,d≡1 (2)E2k((d+(-1)(d-1)/2)/2)], U2k(p,q)=22k-2[-((p+q)/2)E2k((p+q)/2+1)+((q-p)/2)E2k((q-p)/2)-E2k((p+1)/2)-E2k((q+1)/2)+E2k+1((p+q)/2 +1)-E2k+1((q-p)/2)], and F2k(n)=(1/2){σ2k+1†(n)-σ2k†(n)}. As applications of these identities, we give several concrete interpretations in terms of the procedural modelling method.http://dx.doi.org/10.1155/2014/289187 |
| spellingShingle | Daeyeoul Kim Abdelmejid Bayad Joongsoo Park Euler Polynomials and Combinatoric Convolution Sums of Divisor Functions with Even Indices Abstract and Applied Analysis |
| title | Euler Polynomials and Combinatoric Convolution Sums of Divisor Functions with Even Indices |
| title_full | Euler Polynomials and Combinatoric Convolution Sums of Divisor Functions with Even Indices |
| title_fullStr | Euler Polynomials and Combinatoric Convolution Sums of Divisor Functions with Even Indices |
| title_full_unstemmed | Euler Polynomials and Combinatoric Convolution Sums of Divisor Functions with Even Indices |
| title_short | Euler Polynomials and Combinatoric Convolution Sums of Divisor Functions with Even Indices |
| title_sort | euler polynomials and combinatoric convolution sums of divisor functions with even indices |
| url | http://dx.doi.org/10.1155/2014/289187 |
| work_keys_str_mv | AT daeyeoulkim eulerpolynomialsandcombinatoricconvolutionsumsofdivisorfunctionswithevenindices AT abdelmejidbayad eulerpolynomialsandcombinatoricconvolutionsumsofdivisorfunctionswithevenindices AT joongsoopark eulerpolynomialsandcombinatoricconvolutionsumsofdivisorfunctionswithevenindices |