Transcendence of multi-indexed infinite series

摘要

We consider the transcendence of the multi-indexed series. ∑_(n1,...,nk=1)~∞ f(n_1,...,n_k)_1...n_k and then extend our results to series of the form. ∑_(n1,...,nk=0)~∞ f(n_1,...,n_k)A_1(n_1)...A_k(n_k)/B_1(n_1)...B_k(n_k) where f is a k-periodic function and each A_i(x),B_i(x) is a polynomial with algebraic coefficients. We relate these series to special values of the digamma function as well as polygamma functions. We also show some conditional implications on the transcendence of the series via Schanuel's Conjecture.