Asymptotic expansion related to the generalized Somos recurrence constant

Abstract

The generalized Somos recurrence is defined by [Formula: see text] In the present paper, for fixed [Formula: see text], we study the following asymptotic series [Formula: see text] as [Formula: see text]. By using the Bell polynomials and the cycle indicator polynomials of the symmetric group, recurrence relations and a general explicit expression for the coefficients [Formula: see text] are developed. It is also shown that these coefficients are closely related to the Apostol–Bernoulli numbers. Furthermore, an asymptotic estimation for the coefficients [Formula: see text] is given as [Formula: see text].