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SumTools[Hypergeometric]

  

ConjugateRTerm

  

construct r-terms conjugate to a bivariate hypergeometric term

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

ConjugateRTerm[1](T, n, k, 'listform')

ConjugateRTerm[2](T, n, k, 'listform')

Parameters

T

-

hypergeometric term of n and k

n

-

name

k

-

name

'listform'

-

(optional) specify output as a list

Description

• 

For a specified bivariate hypergeometric term Tn,k in n and k, the ConjugateRTerm[1](T, n, k) and ConjugateRTerm[2](T, n, k) commands construct two r-terms conjugate to Tn,k.

• 

The output is a bivariate hypergeometric term, called an r-term, conjugate to Tn,k, that is, it can be written as Rn,kTpn,k where Rn,k is a rational function of n and k, and Tpn,k=unvki=1skbi+nai+gi!i=s+1tai+bi+gi!, a_i, b_i are integers, gcdai,bi=1, 0ai, s, t are non-negative integers, and g_i, u, v are complex numbers. Tpn,k is called a factorial term.

• 

A polynomial pn,k is integer-linear if it has the form an+bk+c where a, b are integers, and c is a complex number.

  

For the first constructed r-term, all the integer-linear polynomials in the numerator and the denominator of the rational function Rn,k are moved into the factorial term Tpn,k.

  

For the second r-term, the integer-linear polynomials are moved from the factorial term Tpn,k to the rational function Rn,k, that is, for ij such that ai=aj, bi=bj, then gigj is not an integer; and in the case that gigj=0, either i,js or i,s+1j.

• 

If the optional argument 'listform' is specified, the output is a list Rn,k,Tpn,k.

• 

A sequence Tn,k is a bivariate hypergeometric term of n and k if there are nonzero polynomials f0, f_1, g_0, g_1 of n and k such that

f[1]n,kTn+1,k=f[0]n,kTn,k,g[1]n,kTn,k+1=g[0]n,kTn,k

  

for all non-negative integers n, k. Two hypergeometric terms T_1, T_2 are conjugate if they satisfy the above two relations with the same f_0, f_1, g_0, g_1.

• 

Note: The ConjugateRTerm command replaces the CanonicalRepresentation command.

Examples

withSumTools[Hypergeometric]:

T2kbinomial2n+k,n1+94n+83kn+k288n53

T:=2kbinomial2n+k,n1+94n+83kn+k288n53

(1)

ConjugateRTerm[1]T,n,k,'listform'

537304,1+94n+83k!n+k3!12kn14188!2n+k!2+94n+83k!n+k2!n!n5388!n+k!

(2)

ConjugateRTerm[2]T,n,k,'listform'

280973041+94n+83k88n53n+k2,12k2n+k!n!n+k!

(3)

References

  

Abramov, S.A., and Petkovsek, M. "Canonical Representations of Hypergeometric Terms." Proceedings FPSAC'2001. pp. 1-10. 2001.

  

Abramov, S.A., and Petkovsek, M. "Proof of a Conjecture of Wilf and Zeilberger." University of Ljubljana, Preprint series. Vol. 39. (2001): 748.

See Also

SumTools[Hypergeometric]

SumTools[Hypergeometric][IsHolonomic]

SumTools[Hypergeometric][IsProperHypergeometricTerm]

SumTools[Hypergeometric][RationalCanonicalForm]

 


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