gfun[poltorec] - determine recurrence satisfied by a polynomial in holonomic sequences

Calling Sequence

poltorec(P, listrec, list_unknowns, u(n), <computeini=bool>, <homogeneous=bool>, <operator=name>)

Parameters

P - polynomial in z and the (possibly shifted) variables in list_unknowns

listrec - list containing, for each of the variables in list_unknowns, either a linear recurrence equation or a set containing the equation together with initial conditions

list_unknowns - list of the unknowns [u1(n), u2(n),...], in the same order as in listrec

u,n - the name of the holonomic sequence and the generic variable

homogeneous - (optional) boolean requesting a homogeneous output recurrence; the default is false

operator - (optional) name used verbatim as the shift operator

Description

If u1(n), u2(n),... are sequences solutions of listrec[1], listrec[2],..., poltorec outputs a linear recurrence equation verified by P(n,u1(n),...).
The option homogeneous=true requests a homogeneous recurrence (the default is false). Affine constants are represented by an invariant auxiliary state during the direct dependence computation. The existing option computeini continues to control initial conditions.
With operator=name, output initial conditions are not constructed. The supplied name is used verbatim; homogeneous output is an operator polynomial and inhomogeneous output has the form L(name)=-q. Input data is still used when available to restore singular-index factors lost during elimination before the operator is returned.

Examples

> with(gfun):
rec1:={u1(n+1)=(n+1)*u1(n),u1(0)=1}:
rec2:={u2(n+2)=2*u2(n+1)-3*n*u2(n),u2(1)=1,u2(0)=1}:
poltorec(u1(n)^2+2*u1(n)*u2(n),[rec1,rec2],[u1(n),u2(n)],u(n));

\[\{\left(-3 n^{7}-39 n^{6}-192 n^{5}-462 n^{4}-579 n^{3}-363 n^{2}-90 n \right) u \! \left(n \right)+\left(5 n^{5}+54 n^{4}+209 n^{3}+354 n^{2}+254 n +60\right) u \! \left(n +1\right)+\left(-n^{4}-12 n^{3}-46 n^{2}-62 n -15\right) u \! \left(n +2\right)+\left(n^{2}+4 n \right) u \! \left(n +3\right), u \! \left(0\right) = 3, u \! \left(1\right) = 3, u \! \left(2\right) = 12, u \! \left(3\right) = 48\}\]

Cassini's identity:

> fib:={F(n+2)=F(n+1)+F(n),F(0)=1,F(1)=1}:
poltorec(F(n+2)*F(n)-F(n+1)^2,[fib],[F(n)],f(n));

\[\{f \! \left(n +1\right)+f \! \left(n \right), f \! \left(0\right) = 1\}\]

gfun[poltorec] - determine recurrence satisfied by a polynomial in holonomic sequences

Calling Sequence

Parameters

Description

Examples

See Also