On integral representations of q-gamma and q–beta functions Alberto De Sole , Victor G. Kac Department of Mathematics, MIT 77 Massachusetts Avenue, Cambridge, MA 02139, USA E–mails: [email protected] [email protected] Abstract We study q–integral representations of the q–gamma and the q–beta functions. This

Duplication formula for Gamma function, Physics 2400 - Mathematical methods for the physical sciences, Spring semester 2017 Author: Michael Rozman Keywords: Mathematical methods, Euler, Gamma function, Beta function Created Date: 5/5/2017 6:54:54 PM

A key property of the beta function is its relationship to the gamma function; proof is given below in the section on relationship between gamma function and beta function B ( x , y ) = Γ ( x ) Γ ( y ) Γ ( x + y ) . {\displaystyle \mathrm {B} (x,y)={\frac {\Gamma (x)\,\Gamma (y)}{\Gamma (x+y)}}.}

Properties of the Gamma function The purpose of this paper is to become familiar with the gamma function, a very important function in mathematics and statistics. The gamma function is a continuous extension to the factorial function, which is only de ned for the nonnegative integers. While there are other continuous extensions to the 1.5. The Beta Function and the Gamma Function 14 2. Stirling’s Formula 17 2.1. Stirling’s Formula and Probabilities 18 2.2. Stirling’s Formula and Convergence of Series 20 2.3. From Stirling to the Central Limit Theorem 21 2.4. Integral Test and the Poor Man’s Stirling 24 2.5. Elementary Approaches towards Stirling’s Formula 25 2.6. This MATLAB function returns the beta function evaluated at the elements of Z and W. Toggle Main Navigation. Products; ... The Γ (z) term is the gamma function. Solved problems: gamma and beta functions, Legendre polynomials, Bessel functions. Responsibility ... Gamma functions.