Irrationality proof of certain Lambert series using little q-Jacobi polynomials

Abstract

We apply the Padé technique to find rational approximations toA separate section is dedicated to the special case q i = q ri , r i ∈ N, q = 1/p, p ∈ N \ {1}. In this construction we make use of little q-Jacobi polynomials. Our rational approximations are good enough to prove the irrationality of h ± (q 1 , q 2 ) and give an upper bound for the irrationality measure.

Mellin transforms for multiple Jacobi–Piñeiro polynomials and a q-analogue

“…A simple Padé approximation problem used in [5] proves the irrationality of a family of Lambert series: h ± (q 1 , q 2 ) = ∞ k=1 q k 1 1 ± q k 2 where q 2 = 1/ p 2 , p 2 ∈ N \ {1}, q 1 ∈ Q, 0 < q 1 < 1. The function which has to be approximated is…”