return the polynomial solution of linear difference equation depending on a hypergeometric term
PolynomialSolution(eq, var, term)
linear difference equation depending on a hypergeometric term
function variable for which to solve, for example, z(n)
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 y⁡x in Qxt,t−1 , that is, in the form y=yd⁢td+...+yg⁢tg where d≤g and yd,...,yg are in Q⁡x.
The solution is the function, corresponding to var. The solution involves arbitrary constants of the form, for example, _c1 and _c2.
eq ≔ y⁡n+2−t+n⁢y⁡n+1+n⁢t−1⁢y⁡n
eq ≔ y⁡n+2−n!+n⁢y⁡n+1+n⁢n!−1⁢y⁡n
eq ≔ t+n2⁢z⁡n+1−2⁢n⁢t+2⁢t+n2+2⁢n+1⁢z⁡n
eq ≔ 45⁢y⁡x−9⁢y⁡x⁢x−18⁢y⁡x+3+9⁢y⁡x+3⁢x
Bronstein, M. "On solutions of Linear Ordinary Difference Equations in their Coefficients Field." INRIA Research Report. No. 3797. November 1999.
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