A Bivariate Asymptotic Expansion of Coefficients of Powers of Generating Functions

The aim of this paper is to give a bivariate asymptotic expansion of the coefficient ynk = [xn]y(x)k, where y(x) = ∑ ynxn has a power series expansion with non-negative coefficients yn ⩾ 0. Such expansions are known for k/n ϵ [a, b] with a > 0. In the first part we provide two versions of full asymptotic series expansions for ynk and in the second part we show how to generalize these expansions to the case k/n ϵ [0, b] if y(x) has an algebraic singularity of the kind y(x) = g(x) - h (x)1 - x/x0. A concluding section provides extensions to multivariate asymptotic expansions and applications to multivariate generating functions. As a byproduct, we obtain a remarkable identity for Catalan numbers.