scholarly journals New Multiple Harmonic Sum Identities

10.37236/3946 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Roberto Tauraso ◽  
Helmut Prodinger
Keyword(s):  
Infinite Series ◽  
Special Class ◽  
Partial Fraction ◽  
Harmonic Numbers ◽  
Elementary Method ◽  
Partial Fraction Decomposition ◽  
Binomial Sums

We consider a special class of binomial sums involving harmonic numbers and we prove three identities by using the elementary method of the partial fraction decomposition. Some applications to infinite series  and congruences are given.

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.

scholarly journals A note on harmonic number identities, Stirling series and multiple zeta values

2019 ◽  
Vol 15 (07) ◽  
pp. 1323-1348
Author(s):  
Markus Kuba ◽  
Alois Panholzer
Keyword(s):  
Infinite Series ◽  
General Type ◽  
Multiple Zeta Values ◽  
Harmonic Numbers ◽  
Special Values ◽  
Multiple Zeta ◽  
Special Cases ◽  
Zeta Values ◽  
Finite Sums ◽  
Binomial Sums

We study a general type of series and relate special cases of it to Stirling series, infinite series discussed by Choi and Hoffman, and also to special values of the Arakawa–Kaneko zeta function, studied before amongst others by Candelpergher and Coppo, and also by Young. We complement and generalize earlier results. Moreover, we survey properties of certain truncated multiple zeta and zeta star values, pointing out their relation to finite sums of harmonic numbers. We also discuss the duality result of Hoffman, relating binomial sums and truncated multiple zeta star values.

scholarly journals A Fast Algorithm for MacMahon's Partition Analysis

10.37236/1811 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Guoce Xin
Keyword(s):  
Fast Algorithm ◽  
Special Class ◽  
Laurent Series ◽  
Rational Functions ◽  
Constant Term ◽  
Partial Fraction ◽  
Running Time ◽  
Partition Analysis ◽  
Several Variables ◽  
Partial Fraction Decomposition

This paper deals with evaluating constant terms of a special class of rational functions, the Elliott-rational functions. The constant term of such a function can be read off immediately from its partial fraction decomposition. We combine the theory of iterated Laurent series and a new algorithm for partial fraction decompositions to obtain a fast algorithm for MacMahon's Omega calculus, which (partially) avoids the "run-time explosion" problem when eliminating several variables. We discuss the efficiency of our algorithm by investigating problems studied by Andrews and his coauthors; our running time is much less than that of their Omega package.

scholarly journals Some extensions for the several combinatorial identities

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Gao-Wen Xi ◽  
Qiu-Ming Luo
Keyword(s):  
Combinatorial Identities ◽  
Partial Fraction ◽  
Harmonic Numbers ◽  
Bell Polynomial ◽  
In Series ◽  
Partial Fraction Decomposition

AbstractIn this paper, we give some extensions for Mortenson’s identities in series with the Bell polynomial using the partial fraction decomposition. As applications, we obtain some combinatorial identities involving the harmonic numbers.

scholarly journals Two-loop leading colour QCD corrections to $$ q\overline{q} $$ → γγg and qg → γγq

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Bakul Agarwal ◽  
Federico Buccioni ◽  
Andreas von Manteuffel ◽  
Lorenzo Tancredi
Keyword(s):  
Numerical Code ◽  
Tree Level ◽  
Partial Fraction ◽  
Integration By Parts ◽  
Analytic Expressions ◽  
Partial Fraction Decomposition ◽  
Loop Amplitudes

Abstract We present the leading colour and light fermionic planar two-loop corrections for the production of two photons and a jet in the quark-antiquark and quark-gluon channels. In particular, we compute the interference of the two-loop amplitudes with the corresponding tree level ones, summed over colours and polarisations. Our calculation uses the latest advancements in the algorithms for integration-by-parts reduction and multivariate partial fraction decomposition to produce compact and easy-to-use results. We have implemented our results in an efficient C++ numerical code. We also provide their analytic expressions in Mathematica format.

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