Pochhammer k-symbol

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, 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 Γk, with k > 0, is defined as

Gamma_k(x) = lim_{ntoinfty} 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.


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