|
sumtools[sumrecursion] - Zeilberger's algorithm
|
|
Calling Sequence
|
|
sumrecursion(f, k, s(n))
|
|
|
Parameters
|
|
|
f
|
-
|
expression
|
|
k
|
-
|
name, summation variable
|
|
n
|
-
|
name, recurrence variable
|
|
s
|
-
|
name, recurrence function
|
|
|
|
|
|
Description
|
|
|
•
|
This function is an implementation of Koepf's extension of Zeilberger's algorithm, calculating a (downward) recurrence equation for the sum
|
|
|
the sum to be taken over all integers k, with respect to n if f is an (m,l)-fold hypergeometric term with respect to (n,k) for some m and l. The minimal values for m, and l are determined automatically.
|
|
•
|
The output is a recurrence which equals zero. The recurrence is a function of n the recurrence variable and  .
|
|
•
|
An expression f is called (m,l)-fold hypergeometric term with respect to (n,k) if
|
|
|
are rational with respect to n and k. This is typically the case for ratios of products of rational functions, exponentials, factorials, binomial coefficients, and Pochhammer symbols that are rational-linear in their arguments. The implementation supports this type of input.
|
|
•
|
The command with(sumtools,sumrecursion) allows the use of the abbreviated form of this command.
|
|
|
|
Examples
|
|
|
>
|
|
|
![[Hypersum, Sumtohyper, extended_gosper, gosper, hyperrecursion, hypersum, hyperterm, simpcomb, sumrecursion, sumtohyper]](/support/helpjp/helpview.aspx?si=6530/file04499/math134.png)
| (1) |
|
>
|
|
|
| (2) |
|
>
|
|
|
| (3) |
Dougall's identity
|
>
|
![f := hyperterm([a, 1+(1/2)*a, b, c, d, 1+2*a-b-c-d+n, -n], [(1/2)*a, 1+a-b, 1+a-c, 1+a-d, 1+a-1-2*a+b+c+d-n, 1+a+n], 1, k)/hyperterm([1+a, 1+a-b-c, 1+a-b-d, 1+a-c-d, 1], [1+a-b, 1+a-c, 1+a-d, 1+a-b-c-d], 1, n)](/support/helpjp/helpview.aspx?si=6530/file04499/math154.png)
|
|
| (4) |
|
>
|
|
|
| (5) |
|
|
Download Help Document
Was this information helpful?
