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SumTools[Hypergeometric]

  

WZMethod

  

perform Wilf-Zeilberger's algorithm

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

WZMethod(f,r,n,k,cert)

Parameters

f

-

function of n and k

r

-

function of n

n

-

variable

k

-

variable

cert

-

(optional) name; assigned the computed WZ certificate

Description

• 

The WZMethod(f,r,n,k,cert) command certifies identities of the form kfn,k=rn.

• 

Let Fn,k=fn,krn if rn0 and Fn,k=fn,k, otherwise. If the method succeeds in certifying the given identity, the output is a list of two elements F,G representing the WZ-pair F,G such that Fn+1,kFn,k=Gn,k+1Gn,k. Otherwise, it returns the error message "WZ method fails".

• 

If the method is successful and if the fifth optional argument cert is given, cert is assigned the WZ certificate Rn,k=Gn,kFn,k.

• 

It is assumed that for each integer 0n, limkGn,k=0 and limkGn,k=0.

Examples

withSumToolsHypergeometric:

Proof of Gauss's 2F1 identity:

fn+k!b+k!cn1!cb1!c+k!n1!cnb1!k+1!b1!

fn+k!b+k!cn1!cb1!c+k!n1!cnb1!k+1!b1!

(1)

r1

r1

(2)

WZpairWZMethodf,r,n,k,cert:

FWZpair1

Fn+k!b+k!cn1!cb1!c+k!n1!cnb1!k+1!b1!

(3)

GWZpair2

Gcn2b!n!n+k!cn1!cn2!n+1+k!n1!cnb1!cb1!b+k!c+kk+1cnb1!n1!b1!k+1!cn2b!n!c+k!kb+nbkc+kn+bc+k+n+1

(4)

cert

k+1c+kcn1n

(5)

Proof of Dixon's identity:

F1kbinomialn+b,n+kbinomialn+c,c+kbinomialb+c,b+k

F−1kn+bn+kn+cc+kb+cb+k

(6)

rn+b+c!n!b!c!

rn+b+c!n!b!c!

(7)

WZpairWZMethodF,r,n,k,certificate:

FWZpair1

F−1kn+bn+kn+cc+kb+cb+kn!b!c!n+b+c!

(8)

GWZpair2

Gn+1+bn+1+kn+1+cc+kn+1!n+b+c!n+1+b+c!n+bn+kn+cc+kn!c!b!b+cb+k−1kc+kn+1+kb+k2n+b+c!n+1+b+c!bcn+bk2+ck2+k2n+bc+k2

(9)

certificate

c+kb+k2n+1+b+cn+k1

(10)

References

  

Wilf, H., and Zeilberger, D. "Rational function certify combinatorial identities." J. Amer. Math. Soc. Vol. 3. (1990): 147-158.

See Also

limit

Sum

SumTools[Hypergeometric]

SumTools[Hypergeometric][Gosper]