Related papers: A closed formula for the generating function of $p…
We use analytic combinatorics to give a direct proof of the closed formula for the generating function of $p$-Bernoulli numbers.
In this paper, using geometric polynomials, we obtain a generating function of p-Bernoulli numbers. As a consequences this generating function, we derive closed formulas for the finite summation of Bernoulli and harmonic numbers involving…
In this note, we give an alternative proof of the generating function of $p$-Bernoulli numbers. Our argument is based on the Euler's integral representation.
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.
In the paper, 2 explicit formulas for the Euler numbers of the second kind are obtained. Based on those formulas a exponential generating function is deduced. Using the generating function some well-known and new identities for the Euler…
The main purpose of this paper is to introduce and investigate a class of generalized Bernoulli polynomials and Euler polynomials based on the generating function. we unify all forms of q-exponential functions by one more parameter. we…
We consider the class $\mathcal{E}_t(Y)$ of Appell polynomials whose generating function is given by means of a real power $t$ of the moment generating function of a certain random variable $Y$. For such polynomials, we obtain explicit…
We introduce the degenerate Bernoulli numbers of the second kind as a degenerate version of the Bernoulli numbers of the second kind. We derive a family of nonlinear differential equations satisfied by a function closely related to the…
We provide direct elementary proofs of several explicit expressions for Bernoulli numbers and Bernoulli polynomials. As a byproduct of our method of proof, we provide natural definitions for generalized Bernoulli numbers and polynomials of…
We derive an expression for the generalized Bernoulli numbers in terms of the Bernoulli numbers involving the (exponential) complete Bell polynomials.
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 $(0,\infty)$.
Main purpose of this paper is to reconstruct generating function of the Bernstein type polynomials. Some properties this generating functions are given. By applying this generating function, not only derivative of these polynomials but also…
We conjecture that the structure of Bernoulli numbers can be explicitly given in the closed form $$ B_n = (-1)^{\frac{n}{2}-1} \prod_{p-1 \nmid n} |n|_p^{-1} \prod\limits_{(p,l)\in\Psi^{\rm irr}_1 \atop n \equiv l \mods{p-1}} |p…
We prove a duality formula for certain sums of values of poly-Bernoulli polynomials which generalizes dualities for poly-Bernoulli numbers. We first compute two types of generating functions for these sums, from which the duality formula is…
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…
By using definition of Golden derivative, corresponding Golden exponential function and Fibonomial coefficients, we introduce generating functions for Bernoulli-Fibonacci polynomials and related numbers. Properties of these polynomials and…
New methods for derivation of Bell polynomials of the second kind are presented. The methods are based on an ordinary generating function and its composita. The relation between a composita and a Bell polynomial is demonstrated. Main…
This paper introduces and investigates degenerate versions of the A-algorithm and B-algorithm by incorporating a parameter lambda into their respective recurrence relations. We derive explicit formulas for the final sequences of these…
In this paper, we introduce a new class of polynomials, called probabilistic q-Bernstein polynomials, alongside their generating function. Assuming Y is a random variable satisfying moment conditions, we use the generating function of these…
We consider Problem 6.94 posed in the book Concrete Mathematics by Graham, Knuth, and Patashnik, and solve it by using bivariate exponential generating functions. The family of recurrence relations considered in the problem contains many…