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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. ## References

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

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

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 x^{n}.

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 x^{n} continue to hold when x^{n} is replaced by (x)_{n,k}.

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Last updated on Monday June 23, 2008 at 10:47:28 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Monday June 23, 2008 at 10:47:28 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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