Talk at the Maple conference 2022: Symbolic powers of functions defined by second-order linear ODEs.
What is new in FPS (Maple version): AddHolonomicDE, MulHolonomicDE, SelfOpHolonomicDE, and the option partialwrt in HolonomicDE.
Assume you have n (valued positive integer) holonomic DEs DE1, ..., DEn, in the dependent variables y1(t), ...,yn(t), respectively. Then you can compute a differential equation for the sum z(t)=y1(t)+y2(t)+...+yn(t) as follows:
Similarly for the product z(t)=y1(t)*y2(t)*...*yn(t), one uses
Now, if you have a differential equation DE in the dependent variable y(t), and you want to compute a differential equation for a polynomial expression z=p(y(t)), you can proceed as follows:
A start for the multivariate case: for computing partial holonomic differential equations with respect to a variable xk for a given multivariate D-finite (holonomic in a sense) function f in x1, ..., xk, ..., xn, one proceeds as follows
FPS:-HolonomicDE(f, F(x1,...xn), partialwrt=xk);
When the argument partialwrt is not specified, the output is a list of n partial differential equations that can be used to write down a canonical representation of the input expression as a holonomic function.
More to come in FPS very soon...