## A computable extension for D-finite functions: DD-finite functions.(English)Zbl 1427.13034

Let $$K$$ be a field of characteristic zero, $$K[[x]]$$ the ring of formal power series over $$K$$ and $$\partial$$ the standard derivation in $$K[[x]]$$. In addition, let $$R$$ be a non-trivial differential subring of $$K[[x]]$$ and $$R[\partial]$$ the ring of linear differential operators over $$R$$. A power series $$f \in K[[x]]$$ is called differentially definable over $$R$$ if there is a non-zero operator $$A \in R[\partial]$$ such that $$A \cdot f = 0$$. Furthermore, if $$R$$ is the polynomial ring $$K[x]$$, then $$f$$ is called D-finite. Finally, if $$R$$ is the set of D-finite functions, then $$f$$ is called DD-finite.
D-finite functions satisfy several closure properties. In this paper, the authors derive the analogous closure properties for DD-finite functions. In addition, it is proved that the function $$\tan(x)$$ is DD-finite and illustrated the execution of closure properties for this function. At the end, they address the issue of initial values $$( f (0), f^{\prime} (0), f ^{\prime \prime} (0), \ldots )$$ to define the solution within $$K[[x]]$$ of the given linear differential equation uniquely.

### MSC:

 13N15 Derivations and commutative rings 68W30 Symbolic computation and algebraic computation 34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. 13F25 Formal power series rings

### Software:

gfun; ore_algebra; GeneratingFunctions; SageMath; DLMF
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### References:

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