Linear recurrent sequences and irrationality measures

Abstract

Inspired by the proof of the irrationality of ξ(2) and ξ(3), Alladi and Robinson used Legendre polynomials to obtain some irrationality measures for numbers of the form log (1+ z ). We extend these results by using Gegenbauer polynomials. They satisfy the following three-term recurrent relation, ( n + 1 ) u n + 1 − ( 2 n + 1 + α ) u n + x ( n + α ) u n − 1 = 0 the case α =0 corresponding to Legendre polynomials. We deduce from this recurrence relation the asymptotic behavior of Gegenbauer polynomials; we also give explicit formulas for these polynomials, that are used to describe their arithmetic properties. This yields some irrationality measures for F 1 2 ( 1 , 1/ 2 ( a + 3 ) / 2 ; x ) under some suitable conditions on α and x . We also present a possible generalization of the proof to numbers of the form Γ ( 1 + β ) Γ ( ( 1 + α ) / 2 ) Γ ( 1 + β + ( 1 + α ) / 2 ) F 1 2 ( ( 1 + β ) / 2 , 1 − β / 2 1 + β + ( 1 + α ) / 2 ; x )