gfun[`rec+rec`] - termwise sum of two holonomic recurrences

gfun[`rec*rec`] - termwise product of two holonomic recurrences

gfun[cauchyproduct] - Cauchy product of two holonomic recurrences

Calling Sequence

`rec+rec`(rec1, rec2, u(n), <homogeneous=bool>, <operator=name>)

`rec*rec`(rec1, rec2, u(n), <homogeneous=bool>, <operator=name>)

cauchyproduct(rec1, rec2, u(n), <homogeneous=bool>, <operator=name>)

Parameters

rec1, rec2 - two linear recurrences with polynomial coefficients

u, n - variable and index of the recurrence

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

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

Description

If a(n) and b(n) are the sequences defined respectively by rec1 and rec2, gfun[`rec+rec`] outputs a recurrence for a(n)+b(n), gfun[`rec*rec`] outputs a recurrence for a(n)*b(n), and gfun[cauchyproduct] outputs a recurrence for their Cauchy product or convolution c(n) = sum(a(i)*b(n-i),i=0..n).
With operator=name, initial conditions are not computed and the result is a polynomial in that name, or L(name)=q for an inhomogeneous recurrence.

Examples

> with(gfun):
 rec1:=u(n+1)=(n+1)*u(n):
 rec2 := u(n+1)=2*u(n):
 `rec+rec`(rec1,rec2,u(n));

\[\{\left(2 n^{2}+2 n \right) u \! \left(n \right)+\left(-n^{2}-3 n +2\right) u \! \left(n +1\right)+\left(n -1\right) u \! \left(n +2\right), u \! \left(0\right) = \_C_{0}, u \! \left(1\right) = \_C_{1}, u \! \left(3\right) = 4 \_C_{0}+2 \_C_{1}\}\]

> `rec*rec`(rec1,rec2,u(n));

\[\left(-2 n -2\right) u \! \left(n \right)+u \! \left(n +1\right)\]

> cauchyproduct(rec1,rec2,u(n));

\[\{\left(2 n +4\right) u \! \left(n \right)+\left(-4-n \right) u \! \left(n +1\right)+u \! \left(n +2\right), u \! \left(0\right) = \_C_{0}, u \! \left(1\right) = 3 \_C_{0}\}\]