Related papers: Enumeration of multiplex juggling card sequences u…
In this paper, the formal derivative operator defined with respect to context-free grammars is used to prove some properties about binomial coefficients and multifactorial numbers. In addition, we extend the formal derivative operator to…
This is a compendium of generating functions involving single, double sums and definite integrals. These generating functions also involve special functions in both the summand function and closed form solution.
We consider the problem of enumerating periodic $\sigma$-juggling sequences of length $n$ for multiplex juggling, where $\sigma$ is the initial state (or {\em landing schedule}) of the balls. We first show that this problem is equivalent to…
Using the theoretical basis developed by Yao and Zeilberger, we consider certain graph families whose structure results in a rational generating function for sequences related to spanning tree enumeration. Said families are Powers of Cycles…
A solution is proposed for the problem of composition of ordinary generating functions. A new class of functions that provides a composition of ordinary generating functions is introduced; main theorems are presented; compositae are written…
We present the first combinatorial scheme for counting labelled 4-regular planar graphs through a complete recursive decomposition. More precisely, we show that the exponential generating function of labelled 4-regular planar graphs can be…
We investigate meandric systems with a large number of loops using tools inspired by free probability. For any fixed integer $r$, we express the generating function of meandric systems on $2n$ points with $n-r$ loops in terms of a finite…
Following our earlier work, where doubly indexed and irreducible over Q two-variable Laguerre polynomials were introduced, we prove for such polynomials some recurrence formulas and obtain a generating function. In addition, we show how…
We define a generalized vector partition function and derive an identity for generating series of such functions associated with solutions of basic recurrence relation of combinatorial analysis. As a consequence, we obtain the generating…
Generating functions, which are widely used in combinatorics and probability theory, encode function values into the coefficients of a polynomial. In this paper, we explore their use as a tractable probabilistic model, and propose…
In this paper, we count acyclic and strongly connected uniform directed labeled hypergraphs. For these combinatorial structures, we introduce a specific generating function allowing us to recover and generalize some results on the number of…
We give a generating function for the number of graphs with given numerical properties and prescribed weighted number of connected components. As an application, we give a generating function for the number of bipartite graphs of given…
We study generating functions for multiple zeta star values in general form. These generating functions provide a connection between multiple zeta star values and multiple Euler sums, which allows us to express each multiple zeta star value…
A Lambert series generating function is a special series summed over an arithmetic function $f$ defined by \[ L_f(q) := \sum_{n \geq 1} \frac{f(n) q^n}{1-q^n} = \sum_{m \geq 1} (f \ast 1)(m) q^m. \] Because of the way the left-hand-side…
In this paper we consider the q-extension of the generating function for the higher-order generalized Genocchi numbers and polynomials attached to Dirichlet's character.
A generating function for reciprocal binomial coefficients is written down, integral representations of this function are obtained, generating functions for sums of reciprocal binomial coefficients are derived, new identities are obtained,…
In this paper, we study the explicit expressions of multiple t-star values with an arbitrary number of blocks of twos of general level. We give an expression of a generating function of such values, which generalizes the results for…
We present an overview of the role of generating functions in quantum mechanical contexts, mainly in the modern theory of polarization and in the study of quantum phase transitions. Generating functions enable the derivation of moments and…
We study a generating function for the sum over fatgraphs with specified valences of vertices and faces, inversely weighted by the order of their symmetry group. A compact expression is found for general (i.e. non necessarily connected)…
We introduce a $q$-analog of the higher continued fractions introduced by the last three authors in a previous work (together with Gregg Musiker), which are simultaneously a generalization of the $q$-rational numbers of Morier-Genoud and…