Abstract
A theorem of Meinardus provides asymptotics of the number of weighted partitions under certain assumptions on associated ordinary and Dirichlet generating functions. The ordinary generating functions are closely related to Euler's generating function $\prod_{k=1}^\infty S(z^k)$ for partitions, where $S(z)=(1-z)^{-1}$. By applying a method due to Khintchine, we extend Meinardus' theorem to find the asymptotics of the Taylor coefficients of generating functions of the form $\prod_{k=1}^\infty S(a_kz^k)^{b_k}$ for sequences $a_k$, $b_k$ and general $S(z)$. We also reformulate the hypotheses of the theorem in terms of the above generating functions. This allows novel applications of the method. In particular, we prove rigorously the asymptotics of Gentile statistics and derive the asymptotics of combinatorial objects with distinct components.
Highlights
Meinardus [11] proved a theorem about the asymptotics of weighted partitions with weights satisfying certain conditions
By applying a method due to Khintchine, we extend Meinardus’ theorem to find the asymptotics of the Taylor coefficients of generating functions of the form
We extend Meinardus’ theorem further to a general framework, which encompasses a variety of models in physics and combinatorics, including previous results
Summary
Meinardus [11] proved a theorem about the asymptotics of weighted partitions with weights satisfying certain conditions. 2. The Khintchine-Meinardus method used in this work covers a variety of models given by generating functions F (δ) = f (e−δ) exhibiting exponential asymptotics, as δ → 0+ (see Lemma 1 below) which are essentially implied by Meinardus Condition 1 of the main theorem. The Khintchine-Meinardus method used in this work covers a variety of models given by generating functions F (δ) = f (e−δ) exhibiting exponential asymptotics, as δ → 0+ (see Lemma 1 below) which are essentially implied by Meinardus Condition 1 of the main theorem In this connection note that there exists a rich literature Such models are studied by a quite different singularity analysis
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