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| 80.1 Introduction to zeilberger | ||
| 80.2 Functions and Variables for zeilberger |
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zeilberger is a implementation of Zeilberger's algorithm
for definite hypergeometric summation, and also
Gosper's algorithm for indefinite hypergeometric
summation.
zeilberger makes use of the "filtering" optimization method developed by Axel Riese.
zeilberger was developed by Fabrizio Caruso.
load (zeilberger) loads this package.
Categories: Sums and products Share packages Package zeilberger
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zeilberger implements Gosper's algorithm for indefinite hypergeometric summation.
Given a hypergeometric term F_k in k we want to find its hypergeometric
anti-difference, that is, a hypergeometric term f_k such that
F_k = f_(k+1) - f_k.
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zeilberger implements Zeilberger's algorithm for definite hypergeometric summation.
Given a proper hypergeometric term (in n and k)
F_(n,k)
and a positive integer d we want to find a d-th order linear
recurrence with polynomial coefficients (in n) for
F_(n,k)
and a rational function R in n and k such that
a_0 F_(n,k) + ... + a_d F_(n+d),k = Delta_k(R(n,k) F_(n,k)),
where Delta_k is the k-forward difference operator, i.e., Delta_k(t_k) := t_(k+1) - t_k.
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There are also verbose versions of the commands which are called by adding one of the following prefixes:
SummaryJust a summary at the end is shown
VerboseSome information in the intermidiate steps
VeryVerboseMore information
ExtraEven more information including information on the linear system in Zeilberger's algorithm
For example:
GosperVerbose, parGosperVeryVerbose,
ZeilbergerExtra, AntiDifferenceSummary.
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Returns the hypergeometric anti-difference of F_k, if it exists.
Otherwise AntiDifference returns no_hyp_antidifference.
Categories: Package zeilberger
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.
Categories: Package zeilberger
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
Categories: Package zeilberger
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.
Categories: Package zeilberger
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.
Categories: Package zeilberger
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Default value: 5
MAX_ORD is the maximum recurrence order attempted by Zeilberger.
Categories: Package zeilberger
Default value: false
When simplified_output is true,
functions in the zeilberger package attempt
further simplification of the solution.
Categories: Package zeilberger
Default value: linsolve
linear_solver names the solver which is used to solve the system
of equations in Zeilberger's algorithm.
Categories: Package zeilberger
Default value: true
When warnings is true,
functions in the zeilberger package print
warning messages during execution.
Categories: Package zeilberger
Default value: true
When Gosper_in_Zeilberger is true,
the Zeilberger function calls Gosper before calling parGosper.
Otherwise, Zeilberger goes immediately to parGosper.
Categories: Package zeilberger
Default value: true
When trivial_solutions is true,
Zeilberger returns solutions
which have certificate equal to zero, or all coefficients equal to zero.
Categories: Package zeilberger
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Default value: false
When mod_test is true,
parGosper executes a
modular test for discarding systems with no solutions.
Categories: Package zeilberger
Default value: linsolve
modular_linear_solver names the linear solver used by the modular test in parGosper.
Categories: Package zeilberger
Default value: big_primes[10]
ev_point is the value at which the variable n is evaluated
when executing the modular test in parGosper.
Categories: Package zeilberger
Default value: big_primes[1]
mod_big_prime is the modulus used by the modular test in parGosper.
Categories: Package zeilberger
Default value: 4
mod_threshold is the
greatest order for which the modular test in parGosper is attempted.
Categories: Package zeilberger
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