Combinatorial aspects of polylogarithms and Euler-Zagier sums. (Aspects combinatoires des polylogarithmes et des sommes d'Euler-Zagier.)
Let with , and , and let where We establish the asymptotic expansion where stands for the Bernoulli polynomials. Further, we prove that the functions and are completely monotonic in on for every if and only if and , respectively. This not only unifies the two known results but also yields some new results.
It is proved that if the increasing sequence kn n=0..∞ n=0 of nonnegative integers has density greater than 1/2 and D is an arbitrary simply connected subregion of CRthen the system of Hermite associated functions Gkn(z) n=0..∞ is complete in the space H(D) of complex functions holomorphic in D.