Transcendence of multi-indexed infinite series

We consider the transcendence of the multi-indexed series∑n1,…,nk=1∞f(n1,…,nk)n1⋯nk and then extend our results to series of the form∑n1,…,nk=0∞f(n1,…,nk)A1(n1)⋯Ak(nk)B1(n1)⋯Bk(nk) where f is a k  -periodic function and each Ai(x),Bi(x)Ai(x),Bi(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.