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SumTools[Hypergeometric]

  

DefiniteSum

  

compute the definite sum

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

DefiniteSum(T, n, k, l..u)

Parameters

T

-

function of n

n

-

name

k

-

name

l..u

-

range for k

Description

• 

For a specified hypergeometric term T of n and k, the DefiniteSum(T, n, k, l..u) command computes, if it exists, a closed form for the definite sum fn=k=luT.

• 

Let r, s, u, v be integers. The DefiniteSum command computes closed forms for four types of definite sums. They are k=nr+snu+vTn,k, k=nr+sTn,k, k=nu+vTn,k, and k=Tn,k.

• 

A closed form is defined as one that can be represented as a sum of hypergeometric terms or as a d'Alembertian term.

• 

If the input T is a definite sum of a hypergeometric term, and if the environment variable _EnvDoubleSum is set to true, then DefiniteSum tries to find a closed form for the specified definite sum of T. Note that this operation can be very expensive.

  

For more information on the construction of the minimal Z-pair for T, see ExtendedZeilberger.

  

Note: If you set infolevel[DefiniteSum] to 3, Maple prints diagnostics.

Examples

withSumTools[Hypergeometric]:

T1kbinomial2n,kbinomial2nk,n22n+12n+1+k:

k=0nT=DefiniteSumT,n,k,0..n

k=0n1kbinomial2n,kbinomial2nk,n22n+12n+1+k=6427Γ2n+32Γn+3231024n729n2n+12Γn+432Γn+232Γn+1

(1)

T1kk+1binomial2n,k:

Set the infolevel to 3.

infolevelDefiniteSum3:

k=02n1T=DefiniteSumT,n,k,0..2n1

DefiniteSum:   "try algorithms for definite sum"
Definite:   "Construct the Zeilberger's recurrence"
Definite:   "Solve the recurrence equation ..."
Definite:   "Find hypergeometric solutions"
Definite:   "Find a particular d'Alembertian solution"
Definite:   "Solve the homogeneous linear recurrence equation"
Definite:   "Construction of the general solution successful"
Definite:   "Solve the initial-condition problem"

k=02n11kk+1binomial2n,k=182n+12Ψ1,n+126Ψ1,n+1

(2)

infolevelDefiniteSum0:

T1kbinomialn,k1binomialx+k,k:

Tm=0nTn=m|Tn=m

T:=m=0n1kbinomialm,kbinomialx+k,k

(3)

_EnvDoubleSumtrue

_EnvDoubleSum:=true

(4)

k=0nT=DefiniteSumT,n,k,0..n

k=0nm=0n1kbinomialm,kbinomialx+k,k=1+xΨx+n+1xΨx+1

(5)

References

  

Abramov, S.A., and Zima, E.V. "D'Alembertian Solutions of Inhomogeneous Linear Equations (differential, difference, and some other)." Proceedings ISSAC'96, pp. 232-240. 1996.

  

Petkovsek, M. "Hypergeometric Solutions of Linear Recurrences with Polynomial Coefficients." Journal of Symbolic Computing. Vol. 14. (1992): 243-264.

  

van Hoeij, M. "Finite Singularities and Hypergeometric Solutions of Linear Recurrence Equations." Journal of Pure and Applied Algebra. Vol. 139. (1999): 109-131.

  

Zeilberger, D. "The Method of Creative Telescoping." Journal of Symbolic Computing. Vol. 11. (1991): 195-204.

See Also

infolevel

LinearOperators[dAlembertianSolver]

LREtools[hypergeomsols]

sum

SumTools[Hypergeometric]

SumTools[Hypergeometric][ExtendedZeilberger]

SumTools[Hypergeometric][IndefiniteSum]

SumTools[Hypergeometric][Zeilberger]

 


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