Abstract
In this paper, firstly, we introduce the several definitions of classic gamma function and beta function, and the several definitions of k-gamma function and k-beta function. Secondly, we discuss the relations between the several definitions and properties of gamma function, beta function, k-gamma function and k-beta function. Finally, we prove these properties by mathematical induction and integral transformation method.
Highlights
The gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers
Karl Weierstrass further established the role of the gamma function in complex analysis, starting from yet another product representation
The gamma function can be defined by an infinite product form (Weierstrass form) (Krantz 1999, p. 157; Havil 2003, p. 57)
Summary
The gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers. The problem of extending the factorial to non-integer arguments was apparently first considered by Daniel Bernoulli and Christian Goldbach in the 1720s, and was solved at the end of the same decade by Leonhard Euler (see [1,2,3,4]). Euler gave two different definitions: the first was not his integral but an infinite product,. Karl Weierstrass further established the role of the gamma function in complex analysis, starting from yet another product representation. The gamma function can be defined by an infinite product form (Weierstrass form) The definitions of gamma function given by Euler, Carl Friedrich Gauss, Karl Weierstrass and Egan are equivalent to each other
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