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Re: What is QGamma[q,z]

• To: mathgroup at smc.vnet.net
• Subject: [mg95554] Re: [mg95503] What is QGamma[q,z]
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Wed, 21 Jan 2009 06:47:53 -0500 (EST)
• References: <200901201047.FAA16820@smc.vnet.net>

On 20 Jan 2009, at 11:47, Ted Ersek wrote:

> Q function are new in Mathematica 7.
> What applications are there for these things?
> Also I can't understand the definition of these things because the
> Mathematica documentation uses notation I never herd of.
> Could sombody explain the definition of  QGamma[q,z]  in terms of
> functions I am likely to know about.
>
> Thanks,
>   Ted Ersek
>
>


No, they cannot be explained "in terms of" you are likely to know  
about but they are analogues of the functions you surely know about  
(the ones without the q). The classical ones correspond to q=1. The  
starting point is the q-binomial theorem and the q-Pascal Triangle,  
which is pretty well explained here:

http://demonstrations.wolfram.com/QPascalTriangle/

You might also look at this:
http://demonstrations.wolfram.com/QTrigonometricFunctions/

These things are quite useful, both in pure mathematics (e.g. the so  
called Ramanujan's Summation formula) and in physics in connection  
with quantum theory. Also, the q-binomial coefficients have a nice  
geometric interpretation  in terms of areas under lattice paths due to  
Polya, but all of this would take too long to describe here. If I can  
find the time for it I will make a Mathematica Demonstration of  
Polya's interpretation of the q-Binomial Theorem some day, but I it  
won't happen very soon (perhaps someone else might want to do it  
earlier). Anyway, all of this is described in the classic  
text:Andrews,Askey,Roy "Special Functions", Chapter 10.

Andrzej Kozlowski

• References:
  • What is QGamma[q,z]
    • From: "Ted Ersek" <ersekt@md.metrocast.net>