DD-Finite Functions Implemented in Sage

Differentially definable functions: a survey

“…However, in Example 2.3 we did not obtain the corresponding differential equation. Using the implementation of the method for simple D-finite functions in dd_functions we obtain instead the following linear differential equation for h(x): h(x) ←→    q 5 (x)h (5) (x) + q 4 (x)h (4) + q 3 (x)h ′′′ (x) + q 2 (x)h ′′ (x) + q 1 (x)h ′ (x) = 0 h(0) = 1, h ′ (0) = 1, h ′′ (0) = −1, h ′′′ (0) = 3, h (4) (0) = −6 , where the polynomials q 1 (x), q 2 (x), q 3 (x), q 4 (x) have degree at most 12 and q 5 (x) = 5022(x + 1) 2 . Although the new equation for h(x) has bigger coefficients, the only singularity of this differential equation is the expected singularity at x = −1.…”

“…The package dd_functions [5] provides a full implementation of differentially definable functions in SageMath [13]. It allows the user to define the rings D(R) for a given differential ring R, to build functions using the data structure (2), and it provides a user friendly interface for computing closure properties.…”

“…Another example are the solutions to the Mathiéu differential equation [4, Chapter 28]. All computational results that we obtained are implemented in the package dd_functions [5] in the open source computer algebra system SageMath [13].…”

Simple Differentially Definable Functions

“…The algorithms described in this paper are implemented in the open source computer algebra system SageMath [21] and are included within the package dd_functions [9]. This package is a tool for computing with D-finite, DD-finite and more general classes of differentially definable functions.…”