Problems using hand-made transformation functions

Question

I experience difficulties when using custom transformation rules. Below I give two concrete problems. I am interested in fixing each of them and in understanding what is wrong with my attitude to the issue in general.

1) Suppose I try to make symbolic computations involving gamma function $\Gamma(z)$ more effective. Particularly, I want $\it{Mathematica}$ to make use of the property $\frac{\Gamma(z+1)}{\Gamma(z)}=z$ in simplification, which it doesn't do on its own. Evaluation of

Simplify[Gamma[z + 1]/Gamma[z]]

returns

Gamma[1 + z]/Gamma[z]

In order to handle this task let me define the first transformation function

tf1[e_] := e /. Gamma[x_]/Gamma[y_] /; Simplify[x - y] == 1 :> y;

Now evaluation of

 Simplify[Gamma[z + 1]/Gamma[z], 
 TransformationFunctions -> {Automatic, tf1}]

indeed gives

z

However, execution of

Simplify[Gamma[z + 1]^2/Gamma[z]^2, 
TransformationFunctions -> {Automatic, tf1}]

results in no real simplification and gives

Gamma[1 + z]^2/Gamma[z]^2

Well, one unwitty way to deal with this problem is to define a separate transformation function for the ratio of squares. But the ratio of the third powers will then require a third transformation function and so on. I would like to handle all these possibilities universally.

I've tried something like

tfUniversal[e_] := 
e /. (Gamma[x_]/Gamma[y_])^n_ /; Simplify[x - y] == 1 :> y^n;

but this didn't work. Is there a way to solve this problem?

2) The second chapter of my adventures includes teaching "Mathematica" identity $\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin{\pi z}}$. Here problem comes right at the start. I define transformation function

tf2[e_] := 
e /. Gamma[x_] Gamma[y_] /; Simplify[x + y] == 1 :> Pi/Sin[Pi x];

and then evaluate

Simplify[{Gamma[-z] Gamma[1 + z], Gamma[z] Gamma[1 - z]}, 
TransformationFunctions -> {Automatic, tf2}]

which gives

{-\[Pi] Csc[\[Pi] z], Gamma[1 - z] Gamma[z]}

so that simplification occurs in the first case but not in the second. I wonder what is the problem here.

Overall, I am very often troubled with such issues. Is it possible to find some references in the literature on using custom transformation rules?

Any help is highly appreciated, thanks :)

Answer 1

Your rule for tfUniversal is almost correct, and I fixed it up by inspecting objects via FullForm.

FullForm[Gamma[z]^2/Gamma[z + 1]^2]
(* Times[Power[Gamma[z],2], Power[Gamma[Plus[1,z]],-2] *)

while the pattern in your rule has FullForm

FullForm[(Gamma[x_]/Gamma[y_])^n_]
(* Power[Times[Gamma[Pattern[x,Blank[]]],
               Power[Gamma[Pattern[y,Blank[]]],-1]], Pattern[n,Blank[]]]]*)

The important difference is your rule has Power on the outside, exponentiating the ratio, while the quantity you had tried to simplify was a Times of two Powers.

I made a working tfUniversalNew by following the expression you're interested in:

tfUniversalNew[e_] := e /. (
        (Gamma[x_]^p_. Gamma[y_]^q_.) :> Gamma[x]^(p + q) y^(-q) /; And[Simplify[x - y] == 1, p q < 0]
    )

The p q < 0 ensures that you have a ratio. Now

Simplify[Gamma[z + 1]^7/Gamma[z]^2, TransformationFunctions -> {Automatic, tfUniversalNew}]

gives

z^2 Gamma[1 + z]^5

I think similar FullForm inspection can help you create other rules for cases of interest...

Answer 2

How about

Simplify[Gamma[z+1]^2/Gamma[z]^2, 
  TransformationFunctions -> {Simplify`SimplifyGamma, Automatic}]
(* z^2 *)

or

FullSimplify[Gamma[z+1]^2/Gamma[z]^2]
(* z^2 *)

Answer 3

Immediately after posting I've found the answer to the second question and decided to put it here while the first question still remains unresolved.

The thing is that I expected $\it{Mathematica}$ to give

-\[Pi] Csc[\[Pi] (1 - z)]

when evaluating

Simplify[Gamma[z] Gamma[1 - z], 
TransformationFunctions -> {Automatic, tf2}]

but the LeafCount of the latter expression is actually less the that of the former. So $\it{Mathematica}$ does not render it as a simplification and therefore does not apply transformation.

A way to avoid it is to use some expression with a low LeafCount in the transformation function, e.g.

tf2simpler[e_] := 

e /. Gamma[x_] Gamma[y_] /; Simplify[x + y] == 1 :> simple[y];

and making a replacement rule

replacement = simple[y_] :> Pi/Sin[Pi y];

then, evaluation of

Simplify[{Gamma[-z] Gamma[1 + z], Gamma[z] Gamma[1 - z]}, 
TransformationFunctions -> {Automatic, tf2simpler}] /. replacement

Gives the desirable answer.

{\[Pi] Csc[\[Pi] (1 + z)], \[Pi] Csc[\[Pi] z]}