Re: [Maxima-discuss] Simplifying gamma functions?

[Barton W. <wi...@un...>]

First waffle:

(%i122) load("opsubst")$

(%i123) gamma_product_simp(e):=block([a : map('first, gatherargs(e,'gamma))],
                for ak in a do (
                    if not featurep(ak,'integer) then e : ratsubst(%pi/sin(%pi*ak),gamma(ak)*gamma(1-ak),e)),

                a : map('first, gatherargs(e,'gamma)),
                 for ak in a do (
                    e : ratsubst(gamma(2*ak), gamma(x)*gamma((2*x+1)/2),e)),
                e)$

(%i124)        gamma_product_simp(gamma(2/3)*gamma(1/3));
(%o124)       (2*%pi)/sqrt(3)

(%i125)        gamma_product_simp(gamma(x)*gamma(x+1/2));
(%o125)       gamma(2*x)


Many things to ponder—the order of the substitutions, ratsubst or fullratsubst, featurep vs askinteger vs … the Legendre triplication formula, and what else?


From: Oscar Benjamin<mailto:osc...@gm...>
Sent: Wednesday, July 20, 2022 4:29 PM
To: Raymond Toy<mailto:toy...@gm...>
Cc: <max...@li...><mailto:max...@li...>
Subject: Re: [Maxima-discuss] Simplifying gamma functions?

Non-NU Email

On Wed, 20 Jul 2022 at 06:02, Raymond Toy <toy...@gm...> wrote:
>
> Is there some builtin function to simplify a product of gamma functions?
>
> I can't simplify gamma(1/4)*gamma(3/4).
>
> For example gamma(1-z)*gamma(z) = %pi/sin(%pi*z).  In this case if z = 1/4, we can see that the product is %pi/sin(%pi/4) = sqrt(2)*%pi.
>
> Or maybe note that beta(1/4,3/4) = gamma(1/4)*gamma(3/4)/gamma(1), so gamma(1/4)*gamma(3/4) = beta(1/4,3/4)*gamma(1).  And maxima knows that beta(1/4,3/4) = sqrt(2)*%pi.
>
> The beta form can be useful for gamma(x)*gamma(y) where x+y is an integer or half integer.
>
> The reflection formula is useful if y = 1-x.
>
> The duplication formula gamma(z)*gamma(z+1/2) = 2^(1-2*z)*sqrt(%pi)*gamma(2*z) is also useful here when z = 1/4.

Not sure if this is useful but there is a function gammasimp in SymPy
that handles the cases described above:

  >>> gammasimp(gamma(z)*gamma(1-z))
  pi/sin(pi*z)
  >>> gammasimp(gamma(S(1)/4)*gamma(S(3)/4))
  sqrt(2)*pi
  >>> gammasimp(gamma(z)*gamma(z + S(1)/2))
  2*sqrt(2)*2**(-2*z - 1/2)*sqrt(pi)*gamma(2*z)

The code for it is here:
https://urldefense.com/v3/__https://github.com/sympy/sympy/blob/master/sympy/simplify/gammasimp.py__;!!PvXuogZ4sRB2p-tU!FBC3ZGVwsQg5hM3uLxEVQOApRjziHgp4j8F52mOELMyuw_tDI7pTUZq_PMtPfE-Pbo7zbW_yZbt7gQc5NJZYbVUgDA$<https://urldefense.com/v3/__https:/github.com/sympy/sympy/blob/master/sympy/simplify/gammasimp.py__;!!PvXuogZ4sRB2p-tU!FBC3ZGVwsQg5hM3uLxEVQOApRjziHgp4j8F52mOELMyuw_tDI7pTUZq_PMtPfE-Pbo7zbW_yZbt7gQc5NJZYbVUgDA$>

I haven't studied it in detail.

--
Oscar


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