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factorial_expand

Run Example
(%i1)f(k):=(2*n)!/(k!*(2*n-k)!)*a^(2*n-k)*b^k;
                                  (2 n)!      2 n - k  k
(%o1)                  f(k) := ------------- a        b
                               k! (2 n - k)!
(%i2) facexpand: true, factorial_expand: true;
(%o2)                                true
(%i3) f(k+1)/f(k);
                                b k! (2 n - k)!
(%o3)                      -------------------------
                           a (k + 1)! (2 n - k - 1)!
(%i4) minfactorial(f(k+1)/f(k));
                                  b (2 n - k)
(%o4)                             -----------
                                   a (k + 1)
(%i5) eq: sum(f(kk),kk,0,n);
                                 n
                                ====     2 n - kk  kk
                                \       a         b
(%o5)                    (2 n)!  >     ---------------
                                /      kk! (2 n - kk)!
                                ====
                                kk = 0
(%i6) solve_rec();
(%o6)                             solve_rec()
(%i7) 
Run Example
p(n,a,k):=1/(k!*(n-k)!)*(-1)^k*gamma(n+a+1)/gamma(k+a+1);
                                  1           k
                             ----------- (- 1)  gamma(n + a + 1)
                             k! (n - k)!
(%o1)          p(n, a, k) := -----------------------------------
                                      gamma(k + a + 1)
(%i2) gamma_expand: true;
(%o2)                                true
(%i3) factorial_expand: true;
(%o3)                                true
(%i4) f:p(n,a,k)/p(n,a,k+1);
                               (k + 1) (k + a + 1)
(%o4)                        - -------------------
                                      n - k
(%i5) collact(expand(denom(f)-num(f)),k);
                                  2
(%o5)                collact(n + k  + a k + k + a + 1, k)
(%i6) 
Run Example
p(n,a,k):=1/(k!*(n-k)!)*(-1)^k*gamma(n+a+1)/gamma(k+a+1);
                                  1           k
                             ----------- (- 1)  gamma(n + a + 1)
                             k! (n - k)!
(%o1)          p(n, a, k) := -----------------------------------
                                      gamma(k + a + 1)
(%i2) gamma_expand: true;
(%o2)                                true
(%i3) factorial_expand: true;
(%o3)                                true
(%i4) f:p(n,a,k)/p(n,a,k+1);
                               (k + 1) (k + a + 1)
(%o4)                        - -------------------
                                      n - k
(%i5) expand(num(f)-denom(f));
                                 2
(%o5)                     - n - k  - a k - k - a - 1
(%i6) 

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