sumtools
hyperrecursion
Zeilberger-Koepf's hyperrecursion algorithm
Calling Sequence
Parameters
Description
Examples
hyperrecursion(U, L, z, s(n))
U, L
-
lists of the upper and lower parameters
z
evaluation point
n
name, recurrence variable
s
name, recurrence function
This function is an implementation of Koepf's extension of Zeilberger's algorithm, calculating a (downward) recurrence equation for the sum
∑khypertermU,L,k
the sum to be taken over all integers k, with respect to n. Here, U and L denote the lists of upper and lower parameters, and z is the evaluation point. The arguments of U and L are assumed to be rational-linear with respect to n. The resulting expression equals zero.
The output is a recurrence which equals zero. The recurrence is output as a function of n, the recurrence variable, and sn,sn−1,....
The command with(sumtools,hyperrecursion) allows the use of the abbreviated form of this command.
withsumtools:
hyperrecursion−n,a,b,1,fn
−n+a−b+1fn−1+b+n−1fn
Dougall's identity
hyperrecursiona,1+a2,b,c,d,1+2a−b−c−d+n,−n,a2,1+a−b,1+a−c,1+a−d,1+a−1+2a−b−c−d+n,1+a+n,1,sn
−a+na−c−d+na−b−d+na−b−c+nsn−1+a−d+na−c+na−b+na−b−c−d+nsn
hyperrecursiona+12,a,b,1−b,−n,2a+13+n,a2+1,12,2a−b+33,2a+b+23,−3n,2a+1+3n,a2,1,sn
b−2+3nb+1−3n2a−1+3n2a+3nsn−1+3n−13n−23n−1+b+2a3n−b+2asn
See Also
sum
sumtools[gosper]
SumTools[Hypergeometric][Zeilberger]
sumtools[hypersum]
sumtools[hyperterm]
sumtools[sumrecursion]
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