(maxima.info)Functions and Variables for zeilberger

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89.2 Functions and Variables for zeilberger
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 -- Function: AntiDifference (F_k, <k>)

     Returns the hypergeometric anti-difference of F_k, if it exists.
     Otherwise 'AntiDifference' returns 'no_hyp_antidifference'.

 -- Function: Gosper (F_k, <k>)
     Returns the rational certificate R(k) for F_k, that is, a rational
     function such that F_k = R(k+1) F_(k+1) - R(k) F_k, if it exists.
     Otherwise, 'Gosper' returns 'no_hyp_sol'.

 -- Function: GosperSum (F_k, <k>, <a>, <b>)

     Returns the summmation of F_k from <k> = <a> to <k> = <b> if F_k
     has a hypergeometric anti-difference.  Otherwise, 'GosperSum'
     returns 'nongosper_summable'.

     Examples:

          (%i1) load ("zeilberger")$
          (%i2) GosperSum ((-1)^k*k / (4*k^2 - 1), k, 1, n);
          Dependent equations eliminated:  (1)
                                     3       n + 1
                                (n + -) (- 1)
                                     2               1
          (%o2)               - ------------------ - -
                                            2        4
                                2 (4 (n + 1)  - 1)
          (%i3) GosperSum (1 / (4*k^2 - 1), k, 1, n);
                                          3
                                    - n - -
                                          2       1
          (%o3)                  -------------- + -
                                          2       2
                                 4 (n + 1)  - 1
          (%i4) GosperSum (x^k, k, 1, n);
                                    n + 1
                                   x          x
          (%o4)                    ------ - -----
                                   x - 1    x - 1
          (%i5) GosperSum ((-1)^k*a! / (k!*(a - k)!), k, 1, n);
                                          n + 1
                          a! (n + 1) (- 1)              a!
          (%o5)       - ------------------------- - ----------
                        a (- n + a - 1)! (n + 1)!   a (a - 1)!
          (%i6) GosperSum (k*k!, k, 1, n);
          Dependent equations eliminated:  (1)
          (%o6)                     (n + 1)! - 1
          (%i7) GosperSum ((k + 1)*k! / (k + 1)!, k, 1, n);
                            (n + 1) (n + 2) (n + 1)!
          (%o7)             ------------------------ - 1
                                    (n + 2)!
          (%i8) GosperSum (1 / ((a - k)!*k!), k, 1, n);
          (%o8)                  NON_GOSPER_SUMMABLE

 -- Function: parGosper (F_(n,k), <k>, <n>, <d>)

     Attempts to find a <d>-th order recurrence for F_(n,k).

     The algorithm yields a sequence [s_1, s_2, ..., s_m] of solutions.
     Each solution has the form

     [R(n, k), [a_0, a_1, ..., a_d]].

     'parGosper' returns '[]' if it fails to find a recurrence.

 -- Function: Zeilberger (F_(n,k), <k>, <n>)

     Attempts to compute the indefinite hypergeometric summation of
     F_(n,k).

     'Zeilberger' first invokes 'Gosper', and if that fails to find a
     solution, then invokes 'parGosper' with order 1, 2, 3, ..., up to
     'MAX_ORD'.  If Zeilberger finds a solution before reaching
     'MAX_ORD', it stops and returns the solution.

     The algorithms yields a sequence [s_1, s_2, ..., s_m] of solutions.
     Each solution has the form

     [R(n,k), [a_0, a_1, ..., a_d]].

     'Zeilberger' returns '[]' if it fails to find a solution.

     'Zeilberger' invokes 'Gosper' only if 'Gosper_in_Zeilberger' is
     'true'.

89.3 General global variables
=============================

 -- Global variable: MAX_ORD
     Default value: 5

     'MAX_ORD' is the maximum recurrence order attempted by
     'Zeilberger'.

 -- Global variable: simplified_output
     Default value: 'false'

     When 'simplified_output' is 'true', functions in the 'zeilberger'
     package attempt further simplification of the solution.

 -- Global variable: linear_solver
     Default value: 'linsolve'

     'linear_solver' names the solver which is used to solve the system
     of equations in Zeilberger's algorithm.

 -- Global variable: warnings
     Default value: 'true'

     When 'warnings' is 'true', functions in the 'zeilberger' package
     print warning messages during execution.

 -- Global variable: Gosper_in_Zeilberger
     Default value: 'true'

     When 'Gosper_in_Zeilberger' is 'true', the 'Zeilberger' function
     calls 'Gosper' before calling 'parGosper'.  Otherwise, 'Zeilberger'
     goes immediately to 'parGosper'.

 -- Global variable: trivial_solutions
     Default value: 'true'

     When 'trivial_solutions' is 'true', 'Zeilberger' returns solutions
     which have certificate equal to zero, or all coefficients equal to
     zero.

89.4 Variables related to the modular test
==========================================

 -- Global variable: mod_test
     Default value: 'false'

     When 'mod_test' is 'true', 'parGosper' executes a modular test for
     discarding systems with no solutions.

 -- Global variable: modular_linear_solver
     Default value: 'linsolve'

     'modular_linear_solver' names the linear solver used by the modular
     test in 'parGosper'.

 -- Global variable: ev_point
     Default value: 'big_primes[10]'

     'ev_point' is the value at which the variable <n> is evaluated when
     executing the modular test in 'parGosper'.

 -- Global variable: mod_big_prime
     Default value: 'big_primes[1]'

     'mod_big_prime' is the modulus used by the modular test in
     'parGosper'.

 -- Global variable: mod_threshold
     Default value: 4

     'mod_threshold' is the greatest order for which the modular test in
     'parGosper' is attempted.