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...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/289187 |
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| Summary: | 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. |
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| ISSN: | 1085-3375 1687-0409 |