Inverse Generalized Harmonic Numbers with Riordan Arrays

2013 ◽  
Vol 842 ◽  
pp. 750-753
Author(s):  
Gao Wen Xi ◽  
Lan Long ◽  
Xue Quan Tian ◽  
Zhao Hui Chen
Keyword(s):  
Stirling Numbers ◽  
Harmonic Number ◽  
Harmonic Numbers ◽  
Riordan Arrays ◽  
Combinatorial Sums ◽  
Generalized Harmonic Numbers ◽  
Riordan Array

In this paper, By observing that the infinite triangle obtained from some generalized harmonic numbers follows a Riordan array, we obtain connections between the Stirling numbers of both kinds and other inverse generalized harmonic numbers. Further, we proved some combinatorial sums and inverse generalized harmonic number identities.

Summations of Inverse Generalized Harmonic Numbers with Riordan Arrays

2014 ◽  
Vol 687-691 ◽  
pp. 1394-1398
Author(s):  
Gao Wen Xi ◽  
Zheng Ping Zhang
Keyword(s):  
Stirling Numbers ◽  
Harmonic Number ◽  
Harmonic Numbers ◽  
Riordan Arrays ◽  
Combinatorial Sums ◽  
Generalized Harmonic Numbers ◽  
Riordan Array

By observing that the infinite triangle obtained from some generalized harmonic numbers follows a Riordan array, we using connections between the Stirling numbers of both kinds and other inverse generalized harmonic numbers. we proved some combinatorial sums and inverse generalized harmonic number identities.

scholarly journals On identities involving generalized harmonic, hyperharmonic and special numbers with Riordan arrays

2021 ◽  
Vol 9 (1) ◽  
pp. 22-30
Author(s):  
Sibel Koparal ◽  
Neşe Ömür ◽  
Ömer Duran
Keyword(s):  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Riordan Arrays ◽  
Riordan Array

Abstract In this paper, by means of the summation property to the Riordan array, we derive some identities involving generalized harmonic, hyperharmonic and special numbers. For example, for n ≥ 0, ∑ k = 0 n B k k ! H ( n . k , α ) = α H ( n + 1 , 1 , α ) - H ( n , 1 , α ) , \sum\limits_{k = 0}^n {{{{B_k}} \over {k!}}H\left( {n.k,\alpha } \right) = \alpha H\left( {n + 1,1,\alpha } \right) - H\left( {n,1,\alpha } \right)} , and for n > r ≥ 0, ∑ k = r n - 1 ( - 1 ) k s ( k , r ) r ! α k k ! H n - k ( α ) = ( - 1 ) r H ( n , r , α ) , \sum\limits_{k = r}^{n - 1} {{{\left( { - 1} \right)}^k}{{s\left( {k,r} \right)r!} \over {{\alpha ^k}k!}}{H_{n - k}}\left( \alpha \right) = {{\left( { - 1} \right)}^r}H\left( {n,r,\alpha } \right)} , where Bernoulli numbers Bn and Stirling numbers of the first kind s (n, r).

scholarly journals Generalized harmonic numbers with Riordan arrays

2008 ◽  
Vol 128 (2) ◽  
pp. 413-425 ◽  
Author(s):  
Gi-Sang Cheon ◽  
M.E.A. El-Mikkawy
Keyword(s):  
Harmonic Numbers ◽  
Riordan Arrays ◽  
Generalized Harmonic Numbers

On 3-adic Valuations of Generalized Harmonic Numbers

Integers ◽  
2012 ◽  
Vol 12 (2) ◽  
Author(s):  
Ken Kamano
Keyword(s):  
Harmonic Numbers ◽  
Generalized Harmonic Numbers

Abstract.We investigate 3-adic valuations of generalized harmonic numbers

scholarly journals Some Properties of the(p,q)-Fibonacci and(p,q)-Lucas Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
GwangYeon Lee ◽  
Mustafa Asci
Keyword(s):  
Generating Functions ◽  
Combinatorial Identities ◽  
Fibonacci Polynomials ◽  
Lucas Polynomials ◽  
Pascal Matrix ◽  
Riordan Arrays ◽  
Combinatorial Sums

Riordan arrays are useful for solving the combinatorial sums by the help of generating functions. Many theorems can be easily proved by Riordan arrays. In this paper we consider the Pascal matrix and define a new generalization of Fibonacci polynomials called(p,q)-Fibonacci polynomials. We obtain combinatorial identities and by using Riordan method we get factorizations of Pascal matrix involving(p,q)-Fibonacci polynomials.

scholarly journals On a Class of Combinatorial Sums Involving Generalized Factorials

2007 ◽  
Vol 2007 ◽  
pp. 1-9 ◽  
Author(s):  
W.-S. Chou ◽  
L. C. Hsu ◽  
P. J.-S. Shiue
Keyword(s):  
Stirling Numbers ◽  
Combinatorial Sums ◽  
Generalized Stirling Numbers

The object of this paper is to show that generalized Stirling numbers can be effectively used to evaluate a class of combinatorial sums involving generalized factorials.

scholarly journals Sharp inequalities for the harmonic numbers

2012 ◽  
Vol 28 (2) ◽  
pp. 223-229
Author(s):  
CHAO-PING CHEN ◽  
Keyword(s):  
Harmonic Number ◽  
Harmonic Numbers ◽  
Double Inequality

Let Hn be the nth harmonic number, and let γ be the Euler-Mascheroni constant. We prove that for all integers n ≥ 1, the double-inequality ... holds with the best possible constants ... We also establish inequality for the Euler-Mascheroni constant.

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