Multiple Gamma Function and Its Application to Computation of Series

The multiple gamma function $\Gamma_n$, defined by a recurrence-functional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the multiple gamma function has been revived. This paper discusses some theoretical aspects of the $\Gamma_n$ function and their applications to summation of series and infinite products.Comment: 20 pages, Latex, uses kluwer.cls, will appear in The Ramanujan Journa

A generalization of the "probléme des rencontres"

In this paper, we study a generalization of the classical \emph{probl\'eme des rencontres} (\emph{problem of coincidences}), consisting in the enumeration of all permutations \pi \in \SS_n with $k$ fixed points, and, in particular, in the enumeration of all permutations \pi \in \SS_n with no fixed points (derangements). Specifically, we study this problem for the permutations of the $n+m$ symbols $1$, $2$, \ldots, $n$, $v_1$, $v_2$, \ldots, $v_m$, where $v_i \not\in\{1,2,\ldots,n\}$ for every $i=1,2,\ldots,m$. In this way, we obtain a generalization of the derangement numbers, the rencontres numbers and the rencontres polynomials. For these numbers and polynomials, we obtain the exponential generating series, some recurrences and representations, and several combinatorial identities. Moreover, we obtain the expectation and the variance of the number of fixed points in a random permutation of the considered kind. Finally, we obtain some asymptotic formulas for the generalized rencontres numbers and the generalized derangement numbers