Representation of holonomic functions in SymPy

Class DifferentialOperator is used to represent the annihilator but we create differential operators easily using the function DifferentialOperators(). Class HolonomicFunction represents a holonomic function.

Let’s explain this with an example:

Take \sin(x) for instance, the differential equation satisfied by it is y^{(2)}(x) + y(x) = 0. By definition we conclude it is a holonomic function. The general solution of this ODE is C_{1} \cdot \sin(x) + C_{2} \cdot \cos(x) but to get \sin(x) we need to provide initial conditions i.e. y(0) = 0, y^{(1)}(0) = 1.

To represent the same in this module one needs to provide the differential equation in the form of annihilator. Basically a differential operator is an operator on functions that differentiates them. So D^{n} \cdot y(x) = y^{(n)}(x) where y^{(n)}(x) denotes n times differentiation of y(x) with respect to x.

So the differential equation can also be written as D^{2} \cdot y(x) + y(x) = 0 or (D^{2} + 1) \cdot y(x) = 0. The part left of y(x) is the annihilator i.e. D^{2}+1.

So this is how one will represent \sin(x) as a Holonomic Function:

>>> from sympy.holonomic import DifferentialOperators, HolonomicFunction
>>> from sympy.abc import x
>>> from sympy import ZZ
>>> R, D = DifferentialOperators(ZZ.old_poly_ring(x), 'D')
>>> HolonomicFunction(D**2 + 1, x, 0, [0, 1])
HolonomicFunction((1) + (1)*D**2, x, 0, [0, 1])

The polynomial coefficients will be members of the ring ZZ[x] in the example. The D operator returned by the function DifferentialOperators() can be used to create annihilators just like SymPy expressions. We currently use the older implementations of rings in SymPy for priority mechanism.