Pochhammer k-symbol

In the mathematical theory of special functions, the Pochhammer "k"-symbol and the "k"-gamma function, introduced by Rafael Díaz and Eddy Pariguan [cite web
date=2006-05-23
url=http://arxiv.org/abs/math/0405596v2
first=Rafael
last=Díaz
coauthors=Eddy Pariguan
title=On hypergeometric functions and k-Pochhammer symbol
accessdate=2007-07-05
] , are generalizations of the Pochhammer symbol and gamma function. They differ from the Pochhammer symbol and gamma function in that they can be related to a general arithmetic progression in the same manner as those are related to the sequence of consecutive integers.

The Pochhammer "k"-symbol ("x")_"n,k" is defined as

: $(x)\_\{n,k\}\; =\; x(x\; +\; k)(x\; +\; 2k)\; cdots\; (x\; +\; (n-1)k),,$

and the "k"-gamma function Γ_"k", with "k" > 0, is defined as

: $Gamma\_k(x)\; =\; lim\_\{n\; oinfty\}\; frac\{n!k^n\; (nk)^\{x/k\; -\; 1\{(x)\_\{n,k.$

When "k" = 1 the standard Pochhammer symbol and gamma function are obtained.

Díaz and Pariguan use these definitions to demonstrate a number of properties of the hypergeometric function. Although Díaz and Pariguan restrict these symbols to "k" > 0, the Pochhammer "k"-symbol as they define it is well-defined for all real "k," and for negative "k" gives the falling factorial, while for "k" = 0 it reduces to the power "xⁿ".

The Díaz and Pariguan paper does not address the many analogies between the Pochhammer "k"-symbol and the power function, such as the fact that the binomial theorem can be extended to Pochhammer "k"-symbols. It is true, however, that many equations involving the power function "xⁿ" continue to hold when "xⁿ" is replaced by ("x")_"n,k".

References