A Note on a Formula of Riordan Involving Harmonic Numbers

CENTRAL EUROPEAN SYMPOSIUM ON THERMOPHYSICS 2019 (CEST)

In "Construction of some new families of Apostol-type numbers and polynomials via Dirichlet character and p-adic q-integrals Turk J Math (2018) 42: 557-577", we defined some new families of numbers and polynomials related to the Dirichlet character of a finite abelian group. In this paper, we investigate some properties and relations for these numbers and polynomials with their series representations and generating functions. By using functional equations of generating functions with their differential equations and series representations of these numbers and polynomials, we derive some formulas including recurrence relations, combinatorial numbers, combinatorial sums and special numbers and polynomials. By using p-adic integral method, we also derive p-adic series representations of these series representations of these numbers and polynomials.

Applicable Analysis and Discrete Mathematics, 2021

By means of the generating function approach, we derive several summation formulae involving multiple harmonic numbers Hn,? (?), as well as other combinatorial numbers named after Bernoulli, Euler, Bell, Genocchi and Stirling.

Journal Human Research in Rehabilitation, 2013

In this paper we give a representation of Stirling numbers of the second kind, we obtain explicit formulas for some cases of Stirling numbers of the second kind and illustrate a method for founding other such formulas. 2010 Mathematics Subject Classification. 11B73, 05A10.

Abstract and Applied Analysis, 2014

A variety of identities involving harmonic numbers and generalized harmonic numbers have been investigated since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics, and theoretical physics. Here we show how one can obtain further interesting and (almost) serendipitous identities about certain finite or infinite series involving binomial coefficients, harmonic numbers, and generalized harmonic numbers by simply applying the usual differential operator to well-known Gauss’s summation formula for2F1(1).

Journal of the Korean Mathematical Society, 2007

In this paper, we obtain important combinatorial identities of generalized harmonic numbers using symmetric polynomials. We also obtain the matrix representation for the generalized harmonic numbers whose inverse matrix can be computed recursively.

Far East Journal of Mathematical Sciences (FJMS), 2018

Stirling numbers play an important role in various research subjects, in particular, in the combinatorial analysis. A number of identities involving Stirling numbers, harmonic numbers and generalized harmonic numbers have been established in various ways. Here we aim to present certain presumably new identities involving Stirling numbers of the first kind, harmonic and generalized harmonic numbers by analyzing the Beta integral function.

2004

In this paper we prove that some results concerned the generalized Stirling numbers are the consequence of the results of Toscano and Chak. The new explicit expressions for generalized Stirling numbers are also given.

2022

The aim of this paper is to give some new classes of finite sums involving the numbers y (m, λ), the generalized harmonic functions, special numbers and polynomials, the Dedekind sums, and other combinatorial sum. Reciprocity laws for these sums are proven. Some applications of these reciprocity laws are presented. With aid of the reciprocity law of the Dedekind sums, formulas for many new finite sums are obtained. Relations among these new classes of finite sums, partial sum of the generalized harmonic functions, the Riemann zeta function, the Hurwitz zeta function, hypergeometric series, polylogarithms, digamma functions, polygmamma functions, and special numbers and polynomials and other combinatorial sums are given. Moreover, some formulas for the partial sum of the generalized harmonic functions and special numbers and polynomials are given. Finally, coments and observations on the results of this paper are given.

Discrete Mathematics, 2008

We consider an identity relating Fibonacci numbers to Pascal's triangle discovered by G. E. Andrews. Several authors provided proofs of this identity, most of them rather involved or else relying on sophisticated number theoretical arguments. We present a new proof, quite simple and based on a Riordan array argument. The main point of the proof is the construction of a new Riordan array from a given Riordan array, by the elimination of elements. We extend the method and as an application we obtain other identities, some of which are new. An important feature of our construction is that it establishes a nice connection between the generating function of the A−sequence of a certain class of Riordan arrays and hypergeometric functions.

Miskolc Mathematical Notes, 2017

In this paper, we study the r-Jacobi-Stirling numbers of the second kind introduced by Gelineau in his Phd thesis. We give, upon using combinatorial and analytic arguments, the ordinary generating function of these numbers, two recurrence relations, their exact expressions and the log-concavity.