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

Abstract

We apply the Padé technique to find rational approximations to h ± ( q 1 , q 2 ) = ∑ k = 1 ∞ q 1 k 1 ± q 2 k , 0 < q 1 , q 2 < 1 , q 1 ∈ Q , q 2 = 1 / p 2 , p 2 ∈ N ∖ { 1 } . A separate section is dedicated to the special case q i = q r i , 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.