Problems with the Gamma function

I have further added some code to simpgamma to expand expressions like
gamma(n*z) and gamma(z+n) with n an integer. The expansion depends on a flag
$gammaexpand. Especially specint gives with this code not in all cases but often
a much more simple result.

Here are examples:

(%i25) gamma(2*z);
(%o25) 2^(2*z-1)*gamma(z)*gamma(z+1/2)/sqrt(%pi)

(%i26) gamma(z+1);
(%o26) z*gamma(z)

(%i27) gamma(z-1);
(%o27) -gamma(z)/(1-z)

(%i28) gamma(z+1)/gamma(z);
(%o28) z

An example with three Gamma functions which we get in results of specint. In
this case the Gamma functions vanish completely:

(%i29) factor(gamma(2*k)/gamma(k)/gamma((2*k+1)/2));
(%o29) 2^(2*k)/(2*sqrt(%pi))

Here is the code added to simpgamma:

          ((and $gammaexpand
                (mtimesp j) 
                (integerp (cadr j))
                (> (cadr j) 0))
           ;; Expand for multiple arguments:
           ;; gamma(n*z) and n a positive integer
           (let ((n (cadr j))
                 (z (cons '(mtimes) (cddr j))))
             (mul
               (power n (add (mul n z) (div -1 2)))
               (power (mul 2 '$%pi) (div (sub 1 n) 2))
               (let ((index (gensumindex)))
                 (dosum
                   ($gamma (add z (div index n)))
                   index 0 (sub n 1) nil)))))

          ((and $gammaexpand 
                (mplusp j) 
                (integerp (cadr j)))
           ;; gamma(z+n) or gamma(z-n) and n a positive integer
           (let ((n (cadr j))
                 (z (cons '(mplus) (cddr j))))
             (cond 
               ((> n 0)
                (mul ($pochhammer z n) ($gamma z)))
               ((< n 0)
                (setq n (- n))
                (div
                  (mul (power -1 n) ($gamma z))
                  ($pochhammer (sub 1 z) n))))))

Dieter Kaiser