sumtools
extended_gosper
Gosper's algorithm for summation
Calling Sequence
Parameters
Description
Examples
extended_gosper(f, k)
extended_gosper(f, k=m..n)
extended_gosper(f, k, j)
f
-
expression
k
name, summation variable
m, n
expressions, representing upper and lower summation bounds
j
integer
This function is an implementation of an extension of Gosper's algorithm, and calculates a closed-form (upward) antidifference of a j-fold hypergeometric expression f whenever such an antidifference exists. In this case, the procedure can be used to calculate definite sums
∑k=mnfk
whenever f does not depend on variables occurring in m and n.
An expression f is called a j-fold hypergeometric expression with respect to k if
fk+jfk
is rational with respect to 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.
An expression g is called an upward antidifference of f if
fk=gk+1−gk
An expression g is called j-fold upward antidifference of f if
fk=gk+j−gk
If the second argument k is a name, and extended_gosper is invoked with two arguments, then extended_gosper returns the closed form (upward) antidifference of f with respect to k, if applicable.
If the second argument has the form k=m..n then the definite sum
is determined if Gosper's algorithm applies.
If extended_gosper is invoked with three arguments then the third argument is taken as the integer j, and a j-fold upward antidifference of f is returned whenever it is a j-fold hypergeometric term.
If the result FAIL occurs, then the implementation has proved either that the input function f is no j-fold hypergeometric term, or that no j-fold hypergeometric antidifference exists.
The command with(sumtools,extended_gosper) allows the use of the abbreviated form of this command.
withsumtools:
see (SIAM Review, 1994, Problem 94-2)
extended_gosper−1k+14k+12k!k!4k2k−1k+1!,k
−2k+1−1k+12k!k!4k2k−1k+1!
extended_gosperbinomialn,k2n−binomialn−1,k2n−1,k
−knk2n−n−1k2n−12k−n
extended_gosperpochhammerb,k2k2!,k
kpochhammerb,k22bk2!+k+1pochhammerb,k2+122bk2+12!
extended_gosperk2!,k
FAIL
extended_gosperkk2!,k
2k2!+2k2+12!
extended_gosperkk2!,k,2
2k2!
extended_gosperkk2!,k=1..n
2n2+12!+2n2+1!−212!−21!
See Also
sumtools[gosper]
SumTools[Hypergeometric][ExtendedGosper]
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