compute closed forms of indefinite sums of hypergeometric terms - Maple Help

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SumTools[IndefiniteSum][Hypergeometric] - compute closed forms of indefinite sums of hypergeometric terms

Calling Sequence

Hypergeometric(f, k, opt)

Parameters

f

-

hypergeometric term in k

k

-

name

opt

-

(optional) equation of the form failpoints=true or failpoints=false

Description

• 

The Hypergeometric(f, k) command computes a closed form of the indefinite sum of f with respect to k.

• 

The following algorithms are used to handle indefinite sums of hypergeometric terms (see the References section):

– 

Gosper's algorithm,

– 

Koepf's extension to Gosper's algorithm, and

– 

the algorithm to compute additive decompositions of hypergeometric terms developed by Abramov and Petkovsek.

• 

If the option failpoints=true (or just failpoints for short) is specified, then the command returns a pair s,p,q, where s is the closed form of the indefinite sum of f w.r.t. k, as above, and p,q are lists of points where f does not exist or the computed sum s is undefined or improper, respectively (see SumTools[IndefiniteSum][Indefinite] for more detailed help).

• 

The command returns FAIL if it is not able to compute a closed form.

Examples

withSumTools[IndefiniteSum]:

Gosper's algorithm:

f:=4n1binomial2n,n22n1242n

f:=4n1binomial2n,n22n1242n

(1)

Hypergeometricf,n

4n2binomial2n,n22n1242n

(2)

The points where the telescoping equation fails:

f:=binomial2n3,n4n

f:=binomial2n3,n4n

(3)

s,fp:=Hypergeometricf,n,'failpoints'

s,fp:=2nn+1binomial2n3,nn24n,,2

(4)

sn=2|sn=2

Error, numeric exception: division by zero

Koepf's extension to Gosper's algorithm:

f:=binomialm,j2binomialm,k2binomial2m+n3jk,2m

f:=binomialm,j2binomialm,k2binomial2m+13njk,2m

(5)

Hypergeometricf,n

13njkbinomialm,j2binomialm,k2binomial2m+13njk,2m2m+1+13n+13jkbinomialm,j2binomialm,k2binomial2m+13n+13jk,2m2m+1+13n+23jkbinomialm,j2binomialm,k2binomial2m+13n+23jk,2m2m+1

(6)

Abramov and Petkovsek's algorithm (note that the specified summand is not hypergeometrically summable):

f:=n22n12nn+1n2n+3!

f:=n22n12nn+1n2n+3!

(7)

Hypergeometricf,n

112n+3_i=1n12_i+4n+n112n2+2n1_i=1n12_i+4n2

(8)

SumTools[Hypergeometric]Gosperf,n

Error, (in SumTools:-Hypergeometric:-Gosper) no solution found

See Also

SumTools[IndefiniteSum], SumTools[IndefiniteSum][Indefinite]

References

• 

Abramov, S.A., and Petkovsek, M. "Rational Normal Forms and Minimal Decompositions of Hypergeometric Terms." Journal of Symbolic Computing. Vol. 33. (2002): 521-543.

• 

Gosper, R.W., Jr. "Decision Procedure for Indefinite Hypergeometric Summation." Proceedings of the National Academy of Sciences USA. Vol. 75. (1978): 40-42.

• 

Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998.

• 

Abramov, S.A. and Petkovsek, M. "Gosper's Algorithm, Accurate Summation, and the discrete Newton-Leibniz formula." Proceedings ISSAC'05. (2005): 5-12.


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