A Bivariate Asymptotic Expansion of Coefficients of Powers of Generating Functions

✍ Scribed by Michael Drmota

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 329 KB
• Volume: 15
• Category: Article
• ISSN: 0195-6698
• DOI: 10.1006/eujc.1994.1016

✦ Synopsis

The aim of this paper is to give a bivariate asymptotic expansion of the coefficient (y_{n k}=\left[x^{n}\right] y(x)^{k}), where (y(x)=\sum y_{n} x^{n}) has a power series expansion with non-negative coefficients (y_{n} \geqslant 0). Such expansions are known for (k / n \in[a, b]) with (a>0). In the first part we provide two versions of full asymptotic series expansions for (y_{n k}) and in the second part we show how to generalize these expansions to the case (k / n \in[0, b]) if (y(x)) has an algebraic singularity of the kind (y(x)=g(x)-h(x) \sqrt{1-x / x_{0}}). 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.