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makefact-pochhammer

makefact(pochhammer(a...

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makefact-pochhammer

makefact(pochhammer(a...

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makefact

Run Example
(%i1)load(zeilberger);
(%o1)  /usr/share/maxima/5.21.1/share/contrib/Zeilberger/zeilberger.mac
(%i2) load(solve_rec);
(%o2)   /usr/share/maxima/5.21.1/share/contrib/solve_rec/solve_rec.mac
(%i3) F(N,i):=binomial(N+k-i-1,k-1)*(i+k)!/i!;
                         binomial(N + k - i - 1, k - 1) (i + k)!
(%o3)         F(N, i) := ---------------------------------------
                                           i!
(%i4) res: Zeilberger(F(N,i),i,N);
                    i (N + k - i)
(%o4)             [[-------------, [N + 2 k + 1, - (N + 1)]]]
                      N - i + 1
(%i5) res: res[1];
                    i (N + k - i)
(%o5)              [-------------, [N + 2 k + 1, - (N + 1)]]
                      N - i + 1
(%i6) eq: sum(res[ii]*G[N+ii-1],ii,1,length(res));
                              i (N + k - i) G
                                             N
(%o6) [(N + 2 k + 1) G      + ----------------, 
                      N + 1      N - i + 1
                                             i (N + k - i) G
                                                            N
                                             ---------------- - (N + 1) G     ]
                                                N - i + 1                N + 1
(%i7) solve_rec(eq,F[N],F[0]=k!);
(%o7)                                false
(%i8) nn:1;
(%o8)                                  1
(%i9) f(i):=binomial(N+k-i-1,k-1)*(i+k)!/i!;
                        binomial(N + k - i - 1, k - 1) (i + k)!
(%o9)           f(i) := ---------------------------------------
                                          i!
(%i10) a(i):=i+k+1;
(%o10)                         a(i) := i + k + 1
(%i11) b(i):=i+1;
(%o11)                           b(i) := i + 1
(%i12) c(i):=i-N-1;
(%o12)                         c(i) := i - N - 1
(%i13) x(i,p):=sum(X[ii]*i^ii,ii,0,p);
                                           ii
(%o13)                 x(i, p) := sum(X   i  , ii, 0, p)
                                       ii
(%i14) eq: rat(expand(a(i)*x(i+1,nn)-b(i-1)*x(i,nn)-c(i)),i);
(%o14)/R/     (X  k + 2 X  - 1) i + N + (X  + X ) k + X  + X  + 1
                1        1                1    0       1    0
(%i15) res: solve(makelist(coeff(eq,i,ii),ii,0,nn),makelist(X[ii],ii,0,nn));
                           (k + 2) N + 2 k + 3         1
(%o15)            [[X  = - -------------------, X  = -----]]
                     0         2                 1   k + 2
                              k  + 3 k + 2
(%i16) res: res[1];
                           (k + 2) N + 2 k + 3         1
(%o16)             [X  = - -------------------, X  = -----]
                     0         2                 1   k + 2
                              k  + 3 k + 2
(%i17) xx: rat(subst(res,x(i,nn)));
                         (k + 2) N + (- i + 2) k - i + 3
(%o17)/R/              - -------------------------------
                                   2
                                  k  + 3 k + 2
(%i18) factor(rat(makefact(b(N-1)*subst(N,i,xx)/c(N)*f(N))));
                           N (N + 2 k + 3) (N + k)!
(%o18)                     ------------------------
                              (k + 1) (k + 2) N!
(%i19) 
Run Example
f(n,k):=n!*n!/(n*k!*(n-k)!*(k-1)!*(n-k+1)!);
                                          n! n!
(%o1)           f(n, k) := -----------------------------------
                           n k! (n - k)! (k - 1)! (n - k + 1)!
(%i2) makefact:true;
(%o2)                                true
(%i3) facexpand: true, factorial_expand: true;
(%o3)                                true
(%i4) eq: num(rat(A*minfactorial(f(n-1,k)/f(n,k))+B*minfactorial(f(n-1,k-1)/f(n,k))-1));
               2            2                    2           2
(%o4)/R/     (k  - k) B + (n  + (- 2 k + 1) n + k  - k) A - n  + n
(%i5) eq: rat(expand(eq),n);
                  2                             2            2
(%o5)/R/ (A - 1) n  + ((- 2 k + 1) A + 1) n + (k  - k) B + (k  - k) A
(%i6) subst(a1*n+a2*k+a3,A,eq);
        2
(%o6) (k  - k) B + n ((1 - 2 k) (a1 n + a2 k + a3) + 1)
                          2                            2
                      + (k  - k) (a1 n + a2 k + a3) + n  (a1 n + a2 k + a3 - 1)
(%i7) 
Run Example
f(j):=binomial(k,j)*3^(k-j)*gamma(k/2+j+a)/gamma(k/2+j);
                                        k - j       k
                        binomial(k, j) 3      gamma(- + j + a)
                                                    2
(%o1)           f(j) := --------------------------------------
                                           k
                                     gamma(- + j)
                                           2
(%i2) minfactorial(makefact(f(j+1)/f(j)));
                               k
                              (- + j + a) (k - j)
                               2
(%o2)                         -------------------
                                          k
                               3 (j + 1) (- + j)
                                          2
(%i3) 

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