Sums of products of power sums

الأصل · EN

For any two arithmetic functions f,g let be the commutative and associative arithmetic convolution (f g)(k):=∑ₘ₌₀ᵏ (arrayc k m array)f(m)g(k-m) and for any n, fⁿ=f f be n-fold product of f∈ S. For any x, let S₀=e be the multiplicative identity of the ring (S,,+) and Sₓ(k):=Bₓ₊₁(k+1)-B₁(k+1)k+1, x≠ 0 denote the power sum defined by Bernoulli polynomials Bₓ(k)=Bₖ(x). We consider the sums of products Sₓⁿ(k), N₀. A closed form expression for Sⁿₓ(k)(x) generalizing the classical Faulhaber formula, is derived. Furthermore, some properties of α-Euler numbers JS9(a variant of Apostol Bernoulli numbers) and their sums of products, are considered using which a closed form expression for the sums of products of infinite series of the form ηα(k):=∑ₙ₌₀∞αⁿ nᵏ, 0<|α|<1, k₀ and the related Abel sums, is obtained which in particular, gives a closed form expression for well known Bernoulli numbers. A generalization of the sums of products of power sums to the sums of products of alternating power sums is also obtained. These considerations generalize in a unified way to define sums of products of power sums for all k hence connecting them with zeta functions.