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)σ21,0k+(n;224)-·42kσ21k+(n/) -(1/2)[∑d|n,d≡14 () {E2k(d)+E2k(d-1)}+22k∑d|n,d≡12 ()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){σ21k+†(n)-σ2k†(n)}. As applications of these identities, we give several concrete interpretations in terms of the procedural modelling method.