On the denominator of generalized Bernoulli numbers (Q1090357)

The generalized Bernoulli numbers \(A_ k(x)\) are defined by \[ (t/(e^ t-1))^ x = \sum^{\infty}_{k=0}A_ k(t) t^ k/k!\;(| t| <2\pi). \] For \(x=1\) we get the ordinary Bernoulli numbers \(B_ k=A_ k(1)\). The function \(A_ k(x)\) is a polynomial of degree k with rational coefficients and indeterminate x. The main result is the computation of the least common multiples of the denominators of these coefficients given in the following Theorem: ''For \(k\geq 0\), \(A_ k(x)=(1/d_ k)P_ k(x)\), where \(P_ k(x)\) is a polynomial with relatively prime integer coefficients, and \(d_ k\) is the integer \[ d_ k = (2^{k- 2[k/2]}/k!)\prod^{[k/2]}_{j=1}denom(B_{2j}/4j)\quad.'' \] The author introduces some propositions which are used for the proof of the theorem.