Related papers: Motzkin numbers of higher rank: Generating functio…
We have formulated a generating function for the Hermite polynomials by comparing two expressions of the same coherent states attached to planar Landau levels. A first expression is obtained by generalizing the canonical coherent states…
We evaluate four families of determinants of matrices, where the entries are sums or differences of generating functions for paths consisting of up-steps, down-steps and level steps. By specialisation, these determinant evaluations have…
Traditional mathematical notation can lead to confusion. Expressions that appear to define composite functions sometimes do not. A particular example with engineering applications is studied in detail.
The number of ordered factorizations and the number of recursive divisors are two related arithmetic functions that are recursively defined. But it is hard to construct explicit representations of these functions. Taking advantage of their…
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 study generating functions in the context of Rota-Baxter algebras. We show that exponential generating functions can be naturally viewed in a very special case of complete free commutative Rota-Baxter algebras. This allows us to use free…
We summarize some combinatoric problems solved by the higher Catalan numbers. These problems are generalizations of the combinatoric problems solved by the Catalan numbers. The generating function of the higher Catalan numbers appeared…
Motzkin numbers are derived from a special case of Random Domino Automaton - recently proposed toy model of earthquakes. An exact solution of the set of equations describing stationary state of Random Domino Automaton in "inverse-power"…
In this paper we study substitutions and some of their associated generating functions. This association takes aperiodicity to transcendence, and vice-versa. These generating functions have a recursive structure arising from the…
In a recent article a generalization of the binomial distribution associated with a sequence of positive numbers was examined. The analysis of the nonnegativeness of the formal expressions was a key-point to allow to give them a statistical…
In this manuscript, the authors derive closed formula for definite integrals of combinations of powers and logarithmic functions of complicated arguments and express these integrals in terms of the Hurwitz zeta. These derivations are then…
An averaged generating function for coloured hard-dimers is being investigated by proving estimates for the latter. Furthermore, two different enumerating problems and their distributions are studied numerically.
In this paper we use computational method based on operational point of view to prove a new generating function of exponential polynomials. We give its applications involving geometric polynomials, Bernoulli and Euler numbers.
Obtained a new property of superposition of the generating functions ln(1/(1-F(x))), where F(x) - generating function with integer coefficients, which allows the construction a primality tests. The theorem which is based on compositions of…
New identities and congruences involving the ranks and cranks of partitions are proved. The proof depends on a new partial differential equation connecting their generating functions.
We derive generating functions for the ranks of pre-modular categories associated to quantum groups at roots of unity.
We present a higher genus generalization of $bc$-Motzkin numbers, which are themselves a generalization of Catalan numbers, and we derive a recursive formula which can be used to calculate them. Further, we show that this leads to a…
We find a generating function for interval-closed sets of the product of two chains poset by constructing a bijection to certain bicolored Motzkin paths. We also find a functional equation for the generating function of interval-closed sets…
In this paper, we perform an arithmetization of well-formed parenthesis strings with zeros (Motzkin words) and of corresponding Motzkin paths. The transformations used are reminiscent of G\"odel numbering for mathematical objects of some…
In this paper, we present a new approach to the convolved Fibonacci numbers arising from the generating function of them and give some new and explicit identities for the convolved Fibonacci numbers.