Gosper's algorithm for summation - Maple Help

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sumtools[extended_gosper] - Gosper's algorithm for summation

Calling Sequence

extended_gosper(f, k)

extended_gosper(f, k=m..n)

extended_gosper(f, k, j)

Parameters

f

-

expression

k

-

name, summation variable

m, n

-

expressions, representing upper and lower summation bounds

j

-

integer

Description

• 

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+1gk

• 

An expression g is called j-fold upward antidifference of f if

fk=gk+jgk

• 

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

k=mnfk

  

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.

Examples

withsumtools:

see (SIAM Review, 1994, Problem 94-2)

extended_gosper1k+14k+12k!k!4k2k1k+1!,k

2k+11k+12k!k!4k2k1k+1!

(1)

extended_gosperbinomialn,k2nbinomialn1,k2n1,k

kbinomialn,k2nbinomialn1,k2n12kn

(2)

extended_gosperpochhammerb,k2k2!,k

12kpochhammerb,12kb12k!+12k+1pochhammerb,12k+12b12k+12!

(3)

extended_gosperk2!,k

FAIL

(4)

extended_gosperkk2!,k

212k!+212k+12!

(5)

extended_gosperkk2!,k,2

212k!

(6)

extended_gosperkk2!,k=1..n

212n+12!+212n+1!212!21!

(7)

See Also

sumtools, sumtools[gosper], SumTools[Hypergeometric][ExtendedGosper]


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