Introduction to celine
======================

Maxima implementation of Sister Celine's method. Barton Willis wrote this code. It is released under the Creative Commons CC0 license (https://creativecommons.org/about/cc0).

Celine's method is described in Sections 4.1--4.4 of the book "A=B", by Marko Petkovsek, Herbert S. Wilf, and Doron Zeilberger. This book is available at http://www.math.rutgers.edu/~zeilberg/AeqB.pdf

Let f = F(n,k). The function celine returns a set of recursion relations for F of the form

p_0(n) * fff(n,k) + p_1(n) * fff(n+1,k) + ... +  p_p(n) * fff(n+p,k+q),

where p_0 through p_p are polynomials. If Maxima is unable to determine that sum(sum(a(i,j) * F(n+i,k+j),i,0,p),j,0,q) / F(n,k) is a rational function of n and k, celine returns the empty set. When f involves parameters (variables other than n or k), celine might make assumptions about these parameters. Using 'put' with a key of 'proviso,' Maxima saves these assumptions on the input label.

To use this function, first load the package integer_sequence, opsubst, and to_poly_solve.

Examples:

     (%i1) load("integer_sequence")$
     (%i2) load("opsubst")$
     (%i3) load("to_poly_solve")$
     (%i4) load("celine")$

     (%i5) celine(n!,n,k,1,0);
     (%o5)       {fff(n + 1, k) - n fff(n, k) - fff(n, k)}

Verification that this result is correct:

     (%i1) load("integer_sequence")$
     (%i2) load("opsubst")$
     (%i3) load("to_poly_solve")$
     (%i4) load("celine")$

     (%i5) g1:{fff(n+1,k)-n*fff(n,k)-fff(n,k)};
     (%o5)       {fff(n + 1, k) - n fff(n, k) - fff(n, k)}

     (%i6) ratsimp(minfactorial(first(g1))),fff(n,k) := n!;
     (%o6)                           0

An example with parameters including the test that the result of the example is correct:

     (%i1) load("integer_sequence")$
     (%i2) load("opsubst")$
     (%i3) load("to_poly_solve")$
     (%i4) load("celine")$

     (%i5) e : pochhammer(a,k) * pochhammer(-k,n) / (pochhammer(b,k));
                                (a)  (- k)
                                   k      n
     (%o5)                      -----------
                                   (b)
                                      k

     (%i6) recur : celine(e,n,k,2,1);
     (%o6) {fff(n + 2, k + 1) - fff(n + 2, k) - b fff(n + 1, k + 1)
      + n ((- fff(n + 1, k + 1)) + 2 fff(n + 1, k) - a fff(n, k)
      - fff(n, k)) + a (fff(n + 1, k) - fff(n, k)) + 2 fff(n + 1, k)
         2
      - n  fff(n, k)}

     (%i7) /* Test this result for correctness */
     (%i8) first(%), fff(n,k) := ''(e)$

     (%i9) makefact(makegamma(%))$
     (%o9)                           0

     (%i10) minfactorial(factor(minfactorial(factor(%))));

The proviso data suggests that setting a = b may result in a lower order recursion which is shown by the following example:

     (%i1) load("integer_sequence")$
     (%i2) load("opsubst")$
     (%i3) load("to_poly_solve")$
     (%i4) load("celine")$

     (%i5) e : pochhammer(a,k) * pochhammer(-k,n) / (pochhammer(b,k));
                                (a)  (- k)
                                   k      n
     (%o5)                      -----------
                                   (b)
                                      k

     (%i6) recur : celine(e,n,k,2,1);
     (%o6) {fff(n + 2, k + 1) - fff(n + 2, k) - b fff(n + 1, k + 1)
      + n ((- fff(n + 1, k + 1)) + 2 fff(n + 1, k) - a fff(n, k)
      - fff(n, k)) + a (fff(n + 1, k) - fff(n, k)) + 2 fff(n + 1, k)
         2
      - n  fff(n, k)}

     (%i7) get('%,'proviso);
     (%o7)                         false

     (%i8) celine(subst(b=a,e),n,k,1,1);
     (%o8) {fff(n + 1, k + 1) - fff(n + 1, k) + n fff(n, k)
                                                          + fff(n, k)}