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 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.