compute closed forms of indefinite sums of hypergeometric terms

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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 with respect to .

•	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 , where is the closed form of the indefinite sum of w.r.t. , as above, and are lists of points where does not exist or the computed sum is undefined or improper, respectively (see SumTools[IndefiniteSum][Indefinite] for more detailed help).

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

Examples

Gosper's algorithm:

f := (4*n-1)*binomial(2*n, n)^2/((2*n-1)^2*4^(2*n))

(1)

-4*n^2*binomial(2*n, n)^2/((2*n-1)^2*4^(2*n))

(2)

The points where the telescoping equation fails:

(3)

(4)

Error, numeric exception: division by zero

Koepf's extension to Gosper's algorithm:

(5)

((1/3)*n-j-k)*binomial(m, j)^2*binomial(m, k)^2*binomial(2*m+(1/3)*n-j-k, 2*m)/(2*m+1)+((1/3)*n+1/3-j-k)*binomial(m, j)^2*binomial(m, k)^2*binomial(2*m+(1/3)*n+1/3-j-k, 2*m)/(2*m+1)+((1/3)*n+2/3-j-k)*binomial(m, j)^2*binomial(m, k)^2*binomial(2*m+(1/3)*n+2/3-j-k, 2*m)/(2*m+1)

(6)

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

f := (n^2-2*n-1)*2^n/((n+1)*n^2*factorial(n+3))

(7)

(1/12)*(n+3)*(Product(2/(_i+4), _i = 1 .. n-1))/n+sum((1/12)*(n^2+2*n-1)*(Product(2/(_i+4), _i = 1 .. n-1))/n^2, n)

(8)

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

•	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.