Distributions, continued fractions, and the Ehrenfest urn model
Let f(x) be a function of a real variable and \(Lf(x)=x^{- 1}\int^{\infty}_{0}f(t)e^{-tx^{-1}}dt\). There are many results of the form \[ Lf(x)= \text{ the continued fraction }\frac{1}{1-a_ 0x- }\frac{b_ 0x^ 2}{1-a_ 1x-}\frac{b_ 1x^ 2}{1-a_ 2x-}... \] subject to conditions on x. Results of this type are studied for the ring \(Qx\) of formal power series in x, where Q denotes the rationals. If \(f(x)=\sum_{i\geq 0}f_ ix^ i/i!\), then \(Lf(x)=\sum_{i\geq 0}f_ ix^ i\). A number of identities have been established by analytic and combinatorial methods. Examples are \[ L(Cosh^ m(x))=\frac{1}{1- }\frac{m\cdot 1x^ 2}{1-} \frac{(m-1)\cdot 2x^ 2}{1-}... \frac{1\cdot mx^ 2}{1} \] and \[ L(\sum_{k\geq 0}\{\frac{x^{3k}}{(3k)!}- \frac{x^{3k+1}}{(3k+1)!}\})^{-1}= \frac{1}{1-x-}\frac{1^ 2x^ 2}{1-2x-} \frac{2^ 2x^ 2}{1-3x-}...\quad. \] In this paper a bijection between lattice paths and distributions (ordered partitions) is obtained to give a combinatorial proof of the first example. Furthermore this yields a combinatorial derivation of transition probabilities P(n,m,i,k) in the Ehrenfest urn model proposed for resolving the discrepancy between irreversibility and recurrence in Boltzmann's theory of gases.
scientific article; zbMATH DE number 3931018