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.


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
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