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

We apply the Padé technique to find rational approximations to h±(q1, q2) = ∞ ∑ k=1 q 1 1± q 2 , 0 < q1, q2 < 1, q1 ∈ Q, q2 = 1/p2, p2 ∈ N \ {1}. A separate section is dedicated to the special case qi = q ri , ri ∈ 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(q1, q2) and give an upper bound for the irrationality measure.