Little q - Legendre polynomials and irrationality of

Little q-Legendre polynomials and irrationality of certain Lambert series

Certain q-analogs hp(1) of the harmonic series, with p = 1/q an integer greater than one, were shown to be irrational by Erdős [9]. In 1991–1992 Peter Borwein [4] [5] used Padé approximation and complex analysis to prove the irrationality of these q-harmonic series and of q-analogs lnp(2) of the natural logarithm of 2. Recently Amdeberhan and Zeilberger [1] used the qEKHAD symbolic package to f...

Legendre polynomials for numerical solution of linear fuzzy Fredholm integral equations

Irrationality proof of a q-extension of ζ(2) using little q-Jacobi polynomials

We show how one can use Hermite-Padé approximation and little q-Jacobi polynomials to construct rational approximants for ζq(2). These numbers are qanalogues of the well known ζ(2). Here q = 1 p , with p an integer greater than one. These approximants are good enough to show the irrationality of ζq(2) and they allow us to calculate an upper bound for its measure of irrationality: μ (ζq(2)) ≤ 10...

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...

Irrationality of ζ q ( 1 ) and ζ q ( 2 ) ?

In this paper we show how one can obtain simultaneous rational approximants for ζq(1) and ζq(2) with a common denominator by means of Hermite-Padé approximation using multiple little q-Jacobi polynomials and we show that properties of these rational approximants prove that 1, ζq(1), ζq(2) are linearly independent over Q. In particular this implies that ζq(1) and ζq(2) are irrational. Furthermor...