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 On some combinatorial identities and harmonic sums

2017 ◽  
Vol 13 (07) ◽  
pp. 1695-1709 ◽  
Author(s):  
Necdet Batir
Keyword(s):  
Integral Representation ◽  
Closed Form ◽  
Generating Function ◽  
Combinatorial Identities ◽  
Harmonic Numbers ◽  
Generalized Harmonic Numbers

For any [Formula: see text] we first give new proofs for the following well-known combinatorial identities [Formula: see text] and [Formula: see text] and then we produce the generating function and an integral representation for [Formula: see text]. Using them we evaluate many interesting finite and infinite harmonic sums in closed form. For example, we show that [Formula: see text] and [Formula: see text] where [Formula: see text] are generalized harmonic numbers defined below.

scholarly journals Congruences involving alternating sums related to harmonic numbers and binomial coefficients

2020 ◽  
Vol 26 (4) ◽  
pp. 39-51
Author(s):  
Laid Elkhiri ◽  
◽  
Miloud Mihoubi ◽  
Abdellah Derbal ◽  
◽  
...  
Keyword(s):  
Prime Number ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Arithmetic Properties ◽  
Alternating Sums ◽  
Generalized Harmonic Numbers

In 2017, Bing He investigated arithmetic properties to obtain various basic congruences modulo a prime for several alternating sums involving harmonic numbers and binomial coefficients. In this paper we study how we can obtain more congruences modulo a power of a prime number p (super congruences) in the ring of p-integer \mathbb{Z}_{p} involving binomial coefficients and generalized harmonic numbers.

scholarly journals Some Properties and Generating Functions of Generalized Harmonic Numbers

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 577 ◽  
Author(s):  
Giuseppe Dattoli ◽  
Silvia Licciardi ◽  
Elio Sabia ◽  
Hari M. Srivastava
Keyword(s):  
Generating Functions ◽  
Integral Transforms ◽  
Higher Order ◽  
Harmonic Numbers ◽  
Efficient Tool ◽  
Type Method ◽  
The Subject ◽  
Generalized Harmonic Numbers

In this paper, we introduce higher-order harmonic numbers and derive their relevant properties and generating functions by using an umbral-type method. We discuss the link with recent works on the subject, and show that the combinations of umbral and other techniques (such as the Laplace and other types of integral transforms) yield a very efficient tool to explore the properties of these numbers.

Binomial sums involving harmonic numbers

2011 ◽  
Vol 61 (2) ◽  
Author(s):  
Marian Genčev
Keyword(s):  
Infinite Series ◽  
Rational Functions ◽  
Harmonic Numbers ◽  
Binomial Type ◽  
Hypergeometric Type ◽  
Number Π ◽  
Binomial Sums ◽  
Generalized Harmonic Numbers

AbstractThis paper develops the approach to the evaluation of a class of infinite series that involve special products of binomial type, generalized harmonic numbers of order 1 and rational functions. We give new summation results for certain infinite series of non-hypergeometric type. New formulas for the number π are included.

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