gfun[`diffeq+diffeq`] - determine the differential equation satisfied by the sum of two holonomic functions

gfun[`diffeq*diffeq`] - determine the differential equation satisfied by the Cauchy product of two holonomic functions

gfun[hadamardproduct] - determine the differential equation satisfied by the Hadamard product of two holonomic functions

Calling Sequence

`diffeq+diffeq`(eq1, eq2, y(z), <homogeneous=bool>, <operator=name>)

`diffeq*diffeq`(eq1, eq2, y(z), <homogeneous=bool>, <operator=name>)

hadamardproduct(eq1, eq2, y(z), <homogeneous=bool>, <operator=name>)

Parameters

eq1, eq2 - two linear differential equations with polynomial coefficients

y,z - name of the holonomic function and itsXS generic variable

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

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

Description

If f (resp. g) is a holonomic function solution of eq1 (resp. eq2), gfun[`diffeq+diffeq`] outputs a linear differential equation verified by f+g, gfun[`diffeq*diffeq`] outputs a linear differential equation verified by f*g, and gfun[hadamardproduct] outputs a linear differential equation verified by the Hadamard product of f and g (the function whose coefficient of z^n in the Taylor expansion around 0 is the product of the corresponding coefficients of f and g).
The differential order of the output equation is at most the sum of the input equations differential orders for gfun[`diffeq+diffeq`], and their product for gfun[`diffeq*diffeq`].
With operator=name, initial conditions are not computed and the result is an operator polynomial, or L=q for an inhomogeneous equation.

Examples

> with(gfun):
eq1 := D(y)(x)-y(x):
eq2 := (1+x)*(D@@2)(y)(x)+D(y)(x):
`diffeq+diffeq`(eq1,eq2,y(x));

\[\left(x^{2}+3 x +2\right) \left(\frac{d^{3}}{d x^{3}}y \! \left(x \right)\right)+\left(-x^{2}-2 x +1\right) \left(\frac{d^{2}}{d x^{2}}y \! \left(x \right)\right)+\left(-x -3\right) \left(\frac{d}{d x}y \! \left(x \right)\right)\]

> `diffeq*diffeq`(eq1,eq2,y(x));

\[\left(1+x \right) \left(\frac{d^{2}}{d x^{2}}y \! \left(x \right)\right)+\left(-2 x -1\right) \left(\frac{d}{d x}y \! \left(x \right)\right)+y \! \left(x \right) x\]

> hadamardproduct(eq1,eq2,y(x));

\[\{\left(1+x \right) \left(\frac{d}{d x}y \! \left(x \right)\right)+\left(\frac{d^{2}}{d x^{2}}y \! \left(x \right)\right) x -\_C_{1}, y \! \left(0\right) = \_C_{0}\}\]