gamma function with 2 parameters

Helen — Fri, 07 Dec 2007 01:41:43 -0500

Hi everyone.

I am trying to find the summation of the following symbolic
expression

(z^k)/(n+k)!

with respect to its symbolic variable k from 0 to 2m-1.

I have done this as follows

>> syms n real
>> syms k real
>> syms z real
>> syms m real
>> simplify(symsum((z^k)/maple('factorial', n+k), k, 0, 2*m-1))

(For some reason symsum((z^k)/factorial(n+k), k, 0, 2*m-1)
does not work.)

and I obtain

ans =

z^(-n)*exp(z)*(-gamma(n,z)*gamma(2*m+n)+gamma(2*m+n,z)*gamma(n))/gamma(n)/gamma(2*m+n)

Why does gamma have 2 parameters? The gamma function can
only have one input argument

Please help.
Helen

Re: gamma function with 2 parameters

Roger Stafford — Fri, 07 Dec 2007 06:47:40 -0500

"Helen " <helenlegakis@hotmail.com> wrote in message <fja8cm$fuk
$1@fred.mathworks.com>...
> Hi everyone.
>
> I am trying to find the summation of the following symbolic
> expression
>
> (z^k)/(n+k)!
>
> with respect to its symbolic variable k from 0 to 2m-1.
>
> I have done this as follows
>
> >> syms n real
> >> syms k real
> >> syms z real
> >> syms m real
> >> simplify(symsum((z^k)/maple('factorial', n+k), k, 0, 2*m-1))
>
> (For some reason symsum((z^k)/factorial(n+k), k, 0, 2*m-1)
> does not work.)
>
>
> and I obtain
>
> ans =
>
> z^(-n)*exp(z)*(-gamma(n,z)*gamma(2*m+n)+gamma(2*m+n,z)*gamma
(n))/gamma(n)/gamma(2*m+n)
>
> Why does gamma have 2 parameters? The gamma function can
> only have one input argument
>
> Please help.
> Helen
---------
  The two-argument 'gamma' function undoubtedly refers to the incomplete
gamma function. Unfortunately it is defined in more than one way.
Mathworks' numerical 'gammainc' function is defined as a normalized entity,
but others define it differently. For example, MathWorld defines an 'upper'
and a 'lower' incomplete gamma function. See

http://mathworld.wolfram.com/IncompleteGammaFunction.html

  You can tell which of the definitions your symbolic toolbox is using in your
expression above by writing

int('exp(-t)*t^(n-1)','t',0,'m') or
int('exp(-t)*t^(n-1)','t','m',inf)

  In any case, writing

int('exp(-t)*t^(n-1)','t',0,inf)

should always give you the single argument function, gamma(n).

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gamma function with 2 parameters

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