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Returns the hypergeometric anti-difference of \(F_k\), if it exists.
Otherwise AntiDifference returns no_hyp_antidifference.
Returns the rational certificate \(R(k)\) for \(F_k\), that is,
a rational function such
that
\(F_k = R\left(k+1\right) \, F_{k+1} - R\left(k\right) \, F_k,\)
if it exists.
Otherwise, Gosper returns no_hyp_sol.
Returns the summation 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
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.
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.
Default value: 5
MAX_ORD is the maximum recurrence order attempted by Zeilberger.
Default value: false
When simplified_output is true,
functions in the zeilberger package attempt
further simplification of the solution.
Default value: linsolve
linear_solver names the solver which is used to solve the system
of equations in Zeilberger’s algorithm.
Default value: true
When warnings is true,
functions in the zeilberger package print
warning messages during execution.
Default value: true
When Gosper_in_Zeilberger is true,
the Zeilberger function calls Gosper before calling parGosper.
Otherwise, Zeilberger goes immediately to parGosper.
Default value: true
When trivial_solutions is true,
Zeilberger returns solutions
which have certificate equal to zero, or all coefficients equal to zero.
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