scholarly journals Binomial Coefficient – Harmonic Sum Identities Associated to Supercongruences

Integers ◽  
2011 ◽  
Vol 11 (6) ◽  
Author(s):  
Dermot McCarthy
Keyword(s):  
Decomposition Method ◽  
Binomial Coefficient ◽  
Partial Fraction ◽  
Partial Fraction Decomposition

AbstractWe establish two binomial coefficient-generalized harmonic sum identities using the partial fraction decomposition method. These identities are a key ingredient in the proofs of numerous supercongruences. In particular, in other works of the author, they are used to establish modulo

scholarly journals A Binomial Coefficient Identity Associated with Beukers' Conjecture on Apéry numbers

10.37236/1856 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Wenchang Chu
Keyword(s):  
Rational Function ◽  
Binomial Coefficient ◽  
Harmonic Number ◽  
Partial Fraction ◽  
Algebraic Identity ◽  
Apéry Numbers ◽  
Partial Fraction Decomposition ◽  
Limiting Case ◽  
Binomial Coefficient Identity

By means of partial fraction decomposition, an algebraic identity on rational function is established. Its limiting case leads us to a harmonic number identity, which in turn has been shown to imply Beukers' conjecture on the congruence of Apéry numbers.

scholarly journals Binomial symmetries inspired by Bruckman's problem

Filomat ◽  
2010 ◽  
Vol 24 (1) ◽  
pp. 41-46 ◽  
Author(s):  
Wenchang Chu ◽  
Ying You
Keyword(s):  
Decomposition Method ◽  
Partial Fraction ◽  
Subject Classifications ◽  
Special Cases ◽  
Secondary 05A30 ◽  
Primary 05A10 ◽  
Partial Fraction Decomposition ◽  
Mathematics Subject Classifications ◽  
Binomial Identities

The partial fraction decomposition method is employed to establish two general algebraic identities, which contain consequently several binomial identities and their q-analogues as special cases. 2010 Mathematics Subject Classifications. Primary 05A10; Secondary 05A30. .

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.

scholarly journals Rational Solutions of Riccati Differential Equation with Coefficients Rational

2011 ◽  
Vol 2011 ◽  
pp. 1-44
Author(s):  
Nadhem Echi
Keyword(s):  
Differential Equation ◽  
Galois Group ◽  
Efficient Method ◽  
Rational Solution ◽  
Partial Fraction ◽  
Rational Solutions ◽  
Riccati Differential Equation ◽  
Differential Galois Group ◽  
Partial Fraction Decomposition ◽  
Decomposition Of Solution

This paper presents a simple and efficient method for determining the rational solution of Riccati differential equation with coefficients rational. In case the differential Galois group of the differential equation , is reducible, we look for the rational solutions of Riccati differential equation , by reducing the number of checks to be made and by accelerating the search for the partial fraction decomposition of the solution reserved for the poles of which are false poles of . This partial fraction decomposition of solution can be used to code . The examples demonstrate the effectiveness of the method.

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