return the polynomial solution of linear difference equation depending on a hypergeometric term - Maple Help

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LREtools[HypergeometricTerm][PolynomialSolution] - return the polynomial solution of linear difference equation depending on a hypergeometric term

Calling Sequence

PolynomialSolution(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 polynomial solution of the linear difference equation eq. If such a solution does not exist, the function returns NULL.

• 

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 term variable and the consecutive term ratio, for example, t,n+1.

• 

If the third parameter is omitted, then the input equation can contain a hypergeometric term directly (not a name). In this case, the procedure extracts the term from the equation, transforms the equation to the form with a name representing a hypergeometric term, and then solves the transformed equation.

• 

The term "polynomial solution" means a solution yx in Qxt,t1 , that is, in the form y=ydtd+...+ygtg where dg and yd,...,yg are in Qx.

• 

The solution is the function, corresponding to var. The solution involves arbitrary constants of the form, for example, _c1 and _c2.

Examples

withLREtools[HypergeometricTerm]:

eq:=yn+2t+nyn+1+nt1yn

eq:=yn+2t+nyn+1+nt1yn

(1)

PolynomialSolutioneq,yn,t=n!

t_C1n,t,n+1

(2)

eq:=yn+2n!+nyn+1+nn!1yn

eq:=yn+2n!+nyn+1+nn!1yn

(3)

PolynomialSolutioneq,yn

t_C1n,t,n+1

(4)

eq:=t+n2zn+12nt+2t+n2+2n+1zn

eq:=n2+tzn+1n2+2nt+2n+2t+1zn

(5)

PolynomialSolutioneq,zn,t=2nn!

n2_C1+t_C1,t,2n+2

(6)

eq:=45yx9yxx18yx+3+9yx+3x

eq:=45yx9yxx18yx+3+9yx+3x

(7)

PolynomialSolutioneq,yx,t,91107x8x2

_C1x5,t,98x27x+10

(8)

See Also

LREtools[HypergeometricTerm], LREtools[HypergeometricTerm][RationalSolution], LREtools[HypergeometricTerm][SubstituteTerm]

References

  

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


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