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

Abstract

We show how one can use Hermite-Padé approximation and little q-Jacobi polynomials to construct rational approximants for ζ q (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 ζ q (2) and they allow us to calculate an upper bound for its measure of irrationality: µ (ζ q (2)) ≤ 10π 2 /(5π 2 − 24) ≈ 3.8936. This is sharper than the upper bound given by Zudilin (On the irrationality measure for a q-analogue of ζ(2), Mat. Sb. 193 (2002), no. 8, 49-70).

Multiple Askey-Wilson polynomials and related basic hypergeometric multiple orthogonal polynomials

“…Often one can take the orthogonality measures for classical orthogonal polynomials and by allowing r different parameters one gets r measures with respect to which one can look for the corresponding multiple orthogonal polynomials, see, e.g., [2,7,30]. Some of these 'classical' multiple orthogonal polynomials play an important role in applications, e.g., multiple Hermite polynomials and multiple Laguerre polynomials are used in the analysis of random matrices [10,11,18] or special determinantal processes [19], multiple Jacobi polynomials and multiple little q-Jacobi polynomials are used in irrationality proofs [26,27,28], multiple Charlier and multiple Meixner polynomials are used to describe non-Hermitian oscillator Hamiltonians [21,22,23], and in general multiple orthogonal polynomials they are useful in the analysis of multidimensional Schrödinger equations and the multidimensional Toda lattice [3,4].…”

On the irrationality of generalized q-logarithm

“…The advantage of Borwein's method [7,8], in which he uses the Padé approximations to (2), is that it allows one to "measure" the irrationality of the numbers in question; this quantitative counterpart is absent in Erdős' method [18] and it was also absent in Bézivin's original method [3,4] until the recent work [30] of I. Rochev (see [31] for a further development). The Padé approximation technique as originated in [7,8] was significantly generalized and extended in later works (for example, [1,[9][10][11][12][13]17,22,24,25,32,34,36,37] to list a few) to sharpen the irrationality measures of the values of q-harmonic and q-logarithm series as well as to prove the irrationality and linear independence results for some close relatives of the series. Bézivin's method from [3,4] is capable of dealing with a class of generalized qhypergeometric functions that are quite different from (2).…”

“…As already pointed out, there are no general methods at present to prove the irrationality of the values of q-hypergeometric series like (4), when q-Pochhammer products appear in the numerators of the terms, for a sufficiently generic set of parameters. A different from (4) three-parameter generalization of p (x, z) can be considered, which is suggested by rational Padé-type approximations in [10,11,32,34,37] that generalize the approximations to the q-logarithm function. Namely, we introduce the function…”

Arithmetic hypergeometric series

“…the prime number theorem Mertens' formula Quantitative estimates of type (59) for ζ q (2) (which show that the number is the non-Liuovillian for q −1 ∈ Z \ {0, ±1}) were not known before, although as mentioned earlier the irrationality [49] and even the transcendence of ζ q (2) for any algebraic q satisfying 0 < |q| < 1 follows from Nesterenko's theorem [102]. A different interpretation of the rational approximations to ζ q (2) in [131] allowed the authors to simplify the arithmetic part and to sharpen the estimate (59):…”