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 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.