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SumTools[Hypergeometric]

  

ExtendedZeilberger

  

construct a minimal Z-pair

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

ExtendedZeilberger(V, n, k, En)

Parameters

V

-

definite sum of hypergeometric term

n

-

name

k

-

name

En

-

name denoting the shift operator with respect to n

Description

• 

Let Tm,k be a hypergeometric term of m and k. Let Vn,k=m=n+bn+dTm,k where b and d are integers. The ExtendedZeilberger(V, n, k, En) command, an extension to Zeilberger's algorithm, constructs the minimal Z-pair for Vn,k provided that it exists.

• 

It can be shown that a Z-pair for Vn,k exists if and only if a Z-pair for the hypergeometric term Tn,k exists.

Examples

withSumTools[Hypergeometric]:

T1kbinomialm,kbinomial2k,k122k

T:=1kbinomialm,kbinomial2k,k22k

(1)

Vm=0nT

V:=m=0n1kbinomialm,kbinomial2k,k22k

(2)

ExtendedZeilbergerV,n,k,En

2n+4En2+4n7En+2n+3,2k21kbinomialn+1,kbinomial2k,kn2+k22k

(3)

_EnvDoubleSumtrue

_EnvDoubleSum:=true

(4)

k=0nV=DefiniteSumV,n,k,0..n

k=0nm=0n1kbinomialm,kbinomial2k,k22k=2Γn+32πΓn+1

(5)

Try the Maple command sum:

k=0nm=0nT

124n6+21n+1Γn+32n+1n+2πΓn124n6+21n+1Γn+32πΓ3+n1+1nΓn+32n+2πΓn+1+k=0nk1k{1k=0sinπkπkotherwisebinomial2k,kk+122k

(6)

References

  

Le, H.Q. "Computing the Minimal Telescoper for Sums of Hypergeometric Terms." SIGSAM Bulletin: Communications on Computer Algebra. Vol. 35 No. 3. (2001): 2-10.

See Also

sum

SumTools[Hypergeometric][DefiniteSum]

SumTools[Hypergeometric][IsZApplicable]

SumTools[Hypergeometric][Zeilberger]

SumTools[Hypergeometric][ZeilbergerRecurrence]

 


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