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 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 Walter Greiner: In Memoriam

2017 ◽  
Vol 45 ◽  
pp. 1760001
Author(s):  
César Zen Vasconcellos ◽  
Helio T. Coelho ◽  
Peter Otto Hess
Keyword(s):  
Elementary Particle ◽  
Particle Physics ◽  
National Laboratory ◽  
Theoretical Physics ◽  
Relativistic Astrophysics ◽  
Full Professor ◽  
Heavy Ion Physics ◽  
Oak Ridge ◽  
Elementary Particle Physics ◽  
The University

Walter Greiner (29 October 1935 - 6 October 2016) was a German theoretical physicist. His scientific research interests include the thematic areas of atomic physics, heavy ion physics, nuclear physics, elementary particle physics (particularly quantum electrodynamics and quantum chromodynamics). He is most known in Germany for his series of books in theoretical physics, but he is also well known around the world. Greiner was born on October 29, 1935, in Neuenbau, Sonnenberg, Germany. He studied physics at the University of Frankfurt (Goethe University in Frankfurt Am Main), receiving in this institution a BSci in physics and a Master’s degree in 1960 with a thesis on plasma-reactors, and a PhD in 1961 at the University of Freiburg under Hans Marshal, with a thesis on the nuclear polarization in [Formula: see text]-mesic atoms. During the period of 1962 to 1964 he was assistant professor at the University of Maryland, followed by a position as research associate at the University of Freiburg, in 1964. Starting in 1965, he became a full professor at the Institute for Theoretical Physics at Goethe University until 2003. Greiner has been a visiting professor to many universities and laboratories, including Florida State University, the University of Virginia, the University of California, the University of Melbourne, Vanderbilt University, Yale University, Oak Ridge National Laboratory and Los Alamos National Laboratory. In 2003, with Wolf Singer, he was the founding Director of the Frankfurt Institute for Advanced Studies (FIAS), and gave lectures and seminars in elementary particle physics. He died on October 6, 2016 at the age of 80. Walter Greiner was an excellent teacher, researcher, friend. And he was a great supporter of the series of events known by the acronyms IWARA - International Workshop on Astronomy and Relativistic Astrophysics, STARS - Caribbean Symposium on Cosmology, Gravitation, Nuclear and Astroparticle Physics, and SMFNS - International Symposium on Strong Electromagnetic Fields and Neutron Stars. Walter Greiner left us. But his memory will remain always alive among us who have had the privilege of knowing him and enjoy his wisdom and joy of living.

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 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.

Sign in / Sign up

Export Citation Format

Share Document