scholarly journals Some Combinatorial Identities Concerning Harmonic Numbers and Binomial Coefficients

2021 ◽  
Vol 8 ◽  
pp. 41-48
Keyword(s):  
Combinatorial Identities ◽  
Binomial Coefficients ◽  
Harmonic Numbers

scholarly journals Harmonic Numbers and Cubed Binomial Coefficients

2011 ◽  
Vol 2011 ◽  
pp. 1-14
Author(s):  
Anthony Sofo
Keyword(s):  
Closed Form ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Polygamma Functions ◽  
Form Representation ◽  
The Given

Euler related results on the sum of the ratio of harmonic numbers and cubed binomial coefficients are investigated in this paper. Integral and closed-form representation of sums are developed in terms of zeta and polygamma functions. The given representations are new.

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 The multiparameter r-Whitney numbers

Filomat ◽  
2019 ◽  
Vol 33 (3) ◽  
pp. 931-943 ◽  
Author(s):  
B. El-Desouky ◽  
F.A. Shiha ◽  
Ethar Shokr
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Matrix Representation ◽  
Stirling Numbers ◽  
Combinatorial Identities ◽  
Harmonic Numbers ◽  
Explicit Formulas ◽  
Whitney Numbers ◽  
Special Cases ◽  
Generalized Stirling Numbers

In this paper, we define the multiparameter r-Whitney numbers of the first and second kind. The recurrence relations, generating functions , explicit formulas of these numbers and some combinatorial identities are derived. Some relations between these numbers and generalized Stirling numbers of the first and second kind, Lah numbers, C-numbers and harmonic numbers are deduced. Furthermore, some interesting special cases are given. Finally matrix representation for these relations are given.

scholarly journals Two congruences involving harmonic numbers with applications

2016 ◽  
Vol 12 (02) ◽  
pp. 527-539 ◽  
Author(s):  
Guo-Shuai Mao ◽  
Zhi-Wei Sun
Keyword(s):  
Bernoulli Polynomial ◽  
Combinatorial Identities ◽  
Harmonic Numbers

The harmonic numbers [Formula: see text] play important roles in mathematics. Let [Formula: see text] be a prime. With the help of some combinatorial identities, we establish the following two new congruences: [Formula: see text] and [Formula: see text] where [Formula: see text] denotes the Bernoulli polynomial of degree [Formula: see text]. As an application, we determine [Formula: see text] and [Formula: see text] modulo [Formula: see text], where [Formula: see text] with [Formula: see text].

scholarly journals Supercongruences related to 3F2(1) involving harmonic numbers

2018 ◽  
Vol 14 (04) ◽  
pp. 1093-1109 ◽  
Author(s):  
Roberto Tauraso
Keyword(s):  
Infinite Series ◽  
Binomial Coefficients ◽  
Harmonic Numbers

We provide various supercongruences for truncated series which involve central binomial coefficients and harmonic numbers. The corresponding infinite series are also evaluated.

Identities for the harmonic numbers and binomial coefficients

2010 ◽  
Vol 25 (1) ◽  
pp. 93-113 ◽  
Author(s):  
Anthony Sofo ◽  
H. M. Srivastava
Keyword(s):  
Binomial Coefficients ◽  
Harmonic Numbers
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