Evaluation of sums involving products of Gaussian q-binomial coefficients with applications

2019 ◽  
Vol 69 (2) ◽  
pp. 327-338 ◽  
Author(s):  
Emrah Kiliç ◽  
Helmut Prodinger
Keyword(s):  
Weight Function ◽  
Binomial Coefficients ◽  
Partial Fraction ◽  
Partial Fraction Decomposition ◽  
Sums Of Products ◽  
Binomial Identities

Abstract Sums of products of two Gaussian q-binomial coefficients, are investigated, one of which includes two additional parameters, with a parametric rational weight function. By means of partial fraction decomposition, first the main theorems are proved and then some corollaries of them are derived. Then these q-binomial identities will be transformed into Fibonomial sums as consequences.

Evaluation of sums involving Gaussian q-binomial coefficients with rational weight functions

2016 ◽  
Vol 12 (02) ◽  
pp. 495-504 ◽  
Author(s):  
Emrah Kiliç ◽  
Helmut Prodinger
Keyword(s):  
Weight Function ◽  
Binomial Coefficients ◽  
Weight Functions ◽  
Partial Fraction ◽  
Decomposition Technique ◽  
Lucas Numbers ◽  
Finite Products ◽  
Partial Fraction Decomposition ◽  
Gaussian Formula

We consider sums of the Gaussian [Formula: see text]-binomial coefficients with a parametric rational weight function. We use the partial fraction decomposition technique to prove the claimed results. We also give some interesting applications of our results to certain generalized Fibonomial sums weighted with finite products of reciprocal Fibonacci or Lucas 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 A new identity via partial fraction decomposition

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1957-1967
Author(s):  
Ji-Ke Ge ◽  
Tao-Tao Liu ◽  
Qiu-Ming Luo
Keyword(s):  
Partial Fraction ◽  
Partial Fraction Decomposition ◽  
Binomial Identities

In this paper, we obtain a new identity using the partial fraction decomposition. As applications, some interesting binomial identities are also derived.

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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