scholarly journals Applications of derivative and difference operators on some sequences

2020 ◽  
pp. 30-30
Author(s):  
Ayhan Dil ◽  
Erkan Muniroğlu
Keyword(s):  
Recurrence Relations ◽  
Binomial Coefficients ◽  
Difference Operators ◽  
Harmonic Numbers ◽  
Hyperharmonic Numbers ◽  
Derivatives Of ◽  
Generalized Harmonic Numbers

In this study, depending on the upper and the lower indices of the hyperharmonic number h(r), nonlinear recurrence relations are obtained. It is shown that generalized harmonic numbers and hyperharmonic numbers can be obtained from derivatives of the binomial coefficients. Taking into account of difference and derivative operators, several identities of the harmonic and hyperharmonic numbers are given. Negative-ordered hyperharmonic numbers are defined and their alternative representations are given.

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 Determinant Expressions for $q$-Harmonic Congruences and Degenerate Bernoulli Numbers

10.37236/787 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Karl Dilcher
Keyword(s):  
Bernoulli Numbers ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Generalized Harmonic Numbers

The generalized harmonic numbers $H_n^{(k)}=\sum_{j=1}^n j^{-k}$ satisfy the well-known congruence $H_{p-1}^{(k)}\equiv 0\pmod{p}$ for all primes $p\geq 3$ and integers $k\geq 1$. We derive $q$-analogs of this congruence for two different $q$-analogs of the sum $H_n^{(k)}$. The results can be written in terms of certain determinants of binomial coefficients which have interesting properties in their own right. Furthermore, it is shown that one of the classes of determinants is closely related to degenerate Bernoulli numbers, and new properties of these numbers are obtained as a consequence.

scholarly journals The Abel-Zeilberger Algorithm

10.37236/2013 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
William Y.C. Chen ◽  
Qing-Hu Hou ◽  
Hai-Tao Jin
Keyword(s):  
Recurrence Relations ◽  
Difference Operators ◽  
Harmonic Numbers ◽  
Summation By Parts ◽  
Polynomial Coefficients ◽  
Zeilberger’S Algorithm ◽  
Zeilberger's Algorithm

By combining Abel's lemma on summation by parts with Zeilberger's algorithm, we give an algorithm, called the Abel-Zeilberger algorithm, to find recurrence relations for definite summations. The role of Abel's lemma can be extended to the case of linear difference operators with polynomial coefficients. This approach can be used to verify and discover identities involving harmonic numbers and derangement numbers. As examples, we use the Abel-Zeilberger algorithm to prove the Paule-Schneider identities, an identity of Andrews and Paule, and an identity of Calkin.

scholarly journals Identities for Harmonic Numbers and Binomial Relations Via Legendre Polynomials

2018 ◽  
Vol 13 (2) ◽  
pp. 92-97
Author(s):  
B. M. Tuladhar ◽  
J. López-Bonilla ◽  
R. López-Vázquez
Keyword(s):  
Legendre Polynomials ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Derivatives Of ◽  
Binomial Identities

We employ the orthonormality of the Legendre polynomials to deduce binomial identities. The harmonic numbers Hn are connected with the derivatives of binomial coefficients, this fact allows to deduce identities involving the Hn.Kathmandu University Journal of Science, Engineering and TechnologyVol. 13, No. 2, 2017, page: 92-97

HANKEL DETERMINANTS OF FACTORIAL FRACTIONS

2021 ◽  
pp. 1-12
Author(s):  
WENCHANG CHU
Keyword(s):  
Integral Transforms ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Hankel Determinant ◽  
Harmonic Numbers ◽  
Expansion Formula ◽  
Hankel Determinants ◽  
Laplace Expansion ◽  
Generalized Harmonic Numbers

Abstract By making use of the Cauchy double alternant and the Laplace expansion formula, we establish two closed formulae for the determinants of factorial fractions that are then utilised to evaluate several determinants of binomial coefficients and Catalan numbers, including those obtained recently by Chammam [‘Generalized harmonic numbers, Jacobi numbers and a Hankel determinant evaluation’, Integral Transforms Spec. Funct.30(7) (2019), 581–593].

scholarly journals Summation Formulas Involving Binomial Coefficients, Harmonic Numbers, and Generalized Harmonic Numbers

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Junesang Choi
Keyword(s):  
Particle Physics ◽  
Analysis Of Algorithms ◽  
Summation Formula ◽  
Theoretical Physics ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Elementary Particle Physics ◽  
Summation Formulas ◽  
Wide Range ◽  
Generalized Harmonic Numbers

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).

scholarly journals Fractional Diffusion Equations for Lattice and Continuum: Grünwald-Letnikov Differences and Derivatives Approach

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Fractional Derivatives ◽  
Three Dimensional ◽  
Lattice Models ◽  
Fractional Diffusion ◽  
Lattice Diffusion ◽  
Diffusion Equations ◽  
Difference Operators ◽  
Fractional Diffusion Equations ◽  
Nonlocal Media ◽  
Derivatives Of

Fractional diffusion equations for three-dimensional lattice models based on fractional-order differences of the Grünwald-Letnikov type are suggested. These lattice fractional diffusion equations contain difference operators that describe long-range jumps from one lattice site to another. In continuum limit, the suggested lattice diffusion equations with noninteger order differences give the diffusion equations with the Grünwald-Letnikov fractional derivatives for continuum. We propose a consistent derivation of the fractional diffusion equation with the fractional derivatives of Grünwald-Letnikov type. The suggested lattice diffusion equations can be considered as a new microstructural basis of space-fractional diffusion in nonlocal media.

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

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