Related papers: A Note on the generating function of p-Bernoulli n…
In this note we define a generalization of Hall-Littlewood symmetric functions using formal group law and give an elementary proof of the generating function formula for the generalized Hall-Littlewood symmetric functions. We also give some…
We consider here a particular quadratic equation linking two elements of a C-Algebra. By analysing powers of the unknowns, it appears a double sequence of polynomials related to classical Bernoulli polynomials. We get the generating…
The main object of this paper is to construct a new generating function of the (q-) Bernstein type polynomials. We establish elementary properties of this function. By using this generating function, we derive recurrence relation and…
We derive a closed form for the generalized Bernoulli polynomial of order $n$ in terms of Bell polynomials and Stirling numbers of the second kind using the Fa\`a di Bruno's formula.
We focus on writing closed forms of generating functions for the number of partitions with gap conditions as double sums starting from a combinatorial construction. Some examples of the sets of partitions with gap conditions to be discussed…
A procedure is described that makes use of the generating function of characters to obtain a new generating function $H$ giving the multiplicities of each weight in all the representations of a simple Lie algebra. The way to extract from…
The rapid development of q-calculus has led to the discovery of new generalizations of Bernstein polynomials and Genocchi polynomials involving q-integers. The present paper deals with weighted q-Bernstein polynomials and q-Genocchi numbers…
The purpose of this paper is to construct q-Euler numbers and polynomials by using p-adic q-integral equations on Zp. Finally, we will give some interesting formulae related to these q-Euler numbers and polynomials.
In this paper, we present a probabilistic extension of the Fubini polynomials and numbers associated with a random variable satisfying some appropriate moment conditions. We obtain the exponential generating function and an integral…
We prove a closed formula for the derivative, of any order, of a implicit function, in terms of some binomial building blocks, and explain the combinatorics behind the coefficients appearing in the formula.
In this note, we present a simple summation formula for $k$-bonacci numbers. The derivation consists in obtaining the generating function of such numbers, and noting that its evaluation at a particular value yields a formula generalizing a…
In this paper, we consider degenerate poly-Bernoulli numbers and polynomials associated with polylogarithmic function and p-adic invariant integral on Zp. By using umbral calculus, we derive some identities of those numbers and polynomials
We give yet another proof for Fa\`{a} di Bruno's formula for higher derivatives of composite functions. Our proof technique relies on reinterpreting the composition of two power series as the generating function for weighted integer…
Charlier configurations provide a combinatorial model for Charlier polynomials. We use this model to give a combinatorial proof of a multilinear generating function for Charlier polynomials. As special cases of the multilinear generating…
In this note we describe a new method of counting the number of unordered factorizations of a natural number by means of a generating function and a recurrence relation arising from it, which improves an earlier result in this direction.
In the paper, the author establishes an integral representation and properties of Bernoulli numbers of the second kind and reveals that the generating function of Bernoulli numbers of the second kind is a Bernstein function on $(0,\infty)$.
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 analyze the combinatorics behind the operation of taking the logarithm of the generating function $G_k$ for $k^\text{th}$ generalized Catalan numbers. We provide combinatorial interpretations in terms of lattice paths and in terms of…
We consider a class of generalized binomials emerging in fractional calculus. After establishing some general properties, we focus on a particular yet relevant case, for which we provide several ready-for-use combinatorial identities,…
Sequences are often conveniently encoded in the form of a generating function depending on a formal variable. This note presents two observations that allow one to draw conclusions about the generated sequence from the generating function.…