Riordan Array Approach to the Coefficients of Ramanujan’s Harmonic Number Expansion

2016 ◽  
Vol 71 (3-4) ◽  
pp. 1413-1419 ◽  
Author(s):  
Lei Feng ◽  
Weiping Wang
Keyword(s):  
Harmonic Number ◽  
Riordan Array

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.

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.

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 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.

On the Ramanujan–Lodge harmonic number expansion

2015 ◽  
Vol 251 ◽  
pp. 423-430 ◽  
Author(s):  
Cristinel Mortici ◽  
Mark B. Villarino
Keyword(s):  
Harmonic Number

A congruence for some generalized harmonic type sums

2018 ◽  
Vol 14 (04) ◽  
pp. 1033-1046 ◽  
Author(s):  
Haydar Göral ◽  
Doğa Can Sertbaş
Keyword(s):  
Harmonic Number ◽  
Congruence Modulo

In 1862, Wolstenholme proved that the numerator of the [Formula: see text]th harmonic number is divisible by [Formula: see text] for any prime [Formula: see text]. A variation of this theorem was shown by Alkan and Leudesdorf. Motivated by these results, we prove a congruence modulo some odd primes for some generalized harmonic type sums.

Harmonic number identities via the Newton–Andrews method

2013 ◽  
Vol 35 (2) ◽  
pp. 263-285 ◽  
Author(s):  
Weiping Wang ◽  
Cangzhi Jia
Keyword(s):  
Harmonic Number
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