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LREtools[HypergeometricTerm]

  

UniversalDenominator

  

return the universal denominator of the rational solutions for a difference equation depending on a hypergeometric term

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

UniversalDenominator(eq, var, term)

Parameters

eq

-

linear difference equation depending on a hypergeometric term

var

-

function variable for which to solve, for example, z(n)

term

-

hypergeometric term

Description

• 

The PolynomialSolution(eq, var, term) command returns the universal denominator of the rational solution of the linear difference equation eq.

• 

The hypergeometric term in the linear difference equation is specified by a name, for example, t. The meaning of the term is defined by the parameter term. It can be specified directly in the form of an equation, for example, t=n!, or specified as a list consisting of the name of the term variable and the consecutive term ratio, for example, t,n+1.

• 

The term "rational solution" means a solution yx in Qxt. (See PolynomialSolution for the meaning of "polynomial solution".) Here we use the term "denominator" which is q in Qxt to mean that qy is in Qxt,t1 .

• 

The search for a rational solution is based on finding a universal denominator which is u in Qxt such that uy is in Qxt,t1 for any rational solution y. By replacing y with Yu in the given equation, we reduce the problem to searching for a polynomial solution.

Examples

withLREtoolsHypergeometricTerm:

eqyn+11+n+1tyn1+t

eq:=yn+11+n+1tyn1+t

(1)

UniversalDenominatoreq,yn,t=n!

1+t

(2)

eqnumeryn+2n+2+n+2n+1t1+n+2n+1tynn+t1+t

eq:=yn+2n2t2ynn3tynn2t2+yn+2n2t+3yn+2nt23ynn2t3ynnt2+4yn+2nt+2yn+2t22ynnt2ynt2+yn+2n+4yn+2tynnynt+2yn+2

(3)

UniversalDenominatoreq,yn,t=n!

n+t

(4)

eqyn+12t+n+1ynt+n

eq:=yn+12t+n+1ynn+t

(5)

UniversalDenominatoreq,yn,t=2n

n+t

(6)

References

  

Abramov, S.A., and Bronstein, M. "Hypergeometric dispersion and the orbit problem." Proc. ISSAC 2000.

  

Bronstein, M. "On solutions of Linear Ordinary Difference Equations in their Coefficients Field." INRIA Research Report. No. 3797. November 1999.

See Also

LREtools[HypergeometricTerm]

LREtools[HypergeometricTerm][HGDispersion]

LREtools[HypergeometricTerm][PolynomialSolution]

LREtools[HypergeometricTerm][RationalSolution]

 


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