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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,<ref>
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| {{cite arxiv
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| |date=2005
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| |class=math.CA
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| |eprint=math/0405596
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| |first=Rafael |last=Díaz
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| |coauthors=Eddy Pariguan
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| |title=On hypergeometric functions and k-Pochhammer symbol
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| }}</ref> 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 [[integer]]s.
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| The Pochhammer ''k''-symbol (''x'')<sub>''n,k''</sub> is defined as
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| : <math> (x)_{n,k} = x(x + k)(x + 2k) \cdots (x + (n-1)k),\, </math> | |
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| and the ''k''-gamma function Γ<sub>''k''</sub>, with ''k'' > 0, is defined as
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| : <math>\Gamma_k(x) = \lim_{n\to\infty} \frac{n!k^n (nk)^{x/k - 1}}{(x)_{n,k}}. </math>
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| When ''k'' = 1 the standard Pochhammer symbol and gamma function are obtained.
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| 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 [[Exponentiation|power]] ''x<sup>n</sup>''.
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| 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<sup>n</sup>'' continue to hold when ''x<sup>n</sup>'' is replaced by (''x'')<sub>''n,k''</sub>.
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| ==References==
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| <references />
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| [[Category:Gamma and related functions]]
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| [[Category:Factorial and binomial topics]]
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Andrew Berryhill is what his wife loves to contact him and he completely digs that name. My spouse and I live in Mississippi but now I'm considering other options. I am really fond of handwriting but I can't make it my profession truly. Credit authorising is how he tends to make cash.
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