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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
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[1] Abramov, S.; Barkatou, M.; Khmelnov, D., On full rank differential systems with power series coefficients, J. Symb. Comput., 68, 120-137, (2015) · Zbl 1308.34019
[2] Abramov, S.; Khmelnov, D., Regular solutions of linear differential systems with power series coefficients, Program. Comput. Softw., 40, 2, 98-106, (2014) · Zbl 1317.34019
[3] Andrews, G.; Askey, R.; Roy, R., Special Functions, Encyclopedia of Mathematics and Its Applications, (1999), Cambridge University Press
[4] Chyzak, F., Gröbner bases, symbolic summation and symbolic integration, (Gröbner Bases and Applications. Gröbner Bases and Applications, Linz, 1998. Gröbner Bases and Applications. Gröbner Bases and Applications, Linz, 1998, London Math. Soc. Lecture Note Ser., vol. 251, (1998), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 32-60 · Zbl 0898.68040
[5] DLMF, (Olver, f. W.J.; Olde Daalhuis, A. B.; Lozier, D. W.; Schneider, B. I.; Boisvert, R. F.; Clark, C. W.; Miller, B. R.; Saunders, B. V., NIST Digital Library of Mathematical Functions, (2017)), Release 1.0.16 of 2017-09-18
[6] Flajolet, P.; Gerhold, S.; Salvy, B., On the non-holonomic character of logarithms, powers, and the nth prime function, Electron. J. Comb., 11, 2, (2005) · Zbl 1076.05004
[7] Horn, R.; Johnson, C., Matrix Analysis, (1985), Cambridge Univ. Press · Zbl 0576.15001
[8] Jiménez-Pastor, A.; Pillwein, V., Algorithmic Arithmetics with DD-Finite Functions, (Schost, Éric, Proceedings of ISSAC 2018, (2018), ACM: ACM New York, NY, USA), 231-237
[9] Kauers, M., Guessing Handbook, (2009), Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, RISC Report Series 09-07
[10] Kauers, M., The Holonomic Toolkit, (Blümlein, J.; Schneider, C., Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, (2013), Springer), 119-144 · Zbl 1308.81101
[11] Kauers, M.; Jaroschek, M.; Johansson, F., Ore Polynomials in Sage, (Gutierrez, J.; Schicho, J.; Weimann, M., Computer Algebra and Polynomials. Computer Algebra and Polynomials, Lecture Notes in Computer Science, (2014)), 105-125 · Zbl 1439.16049
[12] Kauers, M.; Paule, P., The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates, (2011), Springer Publishing Company, Incorporated · Zbl 1225.00001
[13] Koutschan, C., Advanced Applications of the Holonomic Systems Approach, (September 2009), Johannes Kepler University, Ph.D. thesis, RISC-Linz
[14] Mallinger, C., Algorithmic Manipulations and Transformations of Univariate Holonomic Functions and Sequences, (August 1996), RISC, J. Kepler University, Master’s thesis
[15] McLachlan, N. W., Theory and Application of Mathieu Functions, (1964), Dover Publications, Inc.: Dover Publications, Inc. New York · Zbl 0128.29603
[16] Pillwein, V., Computer Algebra Tools for Special Functions in High Order Finite Element Methods, (2008), Johannes Kepler University Linz, Ph.D. thesis
[17] Rainville, E., Special Functions, (1971), Chelsea Publishing Co.: Chelsea Publishing Co. Bronx, NY · Zbl 0231.33001
[18] Rainville, E.; Bedient, P., Elementary Differential Equations, (1969), The Macmillan Company/Collier-Macmillan Limited: The Macmillan Company/Collier-Macmillan Limited New York/London
[19] Salvy, B.; Zimmermann, P., Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Softw., 20, 2, 163-177, (1994) · Zbl 0888.65010
[20] Stanley, R., Differentiably finite power series, Eur. J. Comb., 1, 2, 175-188, (1980) · Zbl 0445.05012
[21] Stanley, R., Enumerative Combinatorics, vol. 2, (1999), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0928.05001
[22] Stein, W., Sage Mathematics Software, (2017), (Version 8.1). The Sage Development Team
[23] van Hoeij, M., Formal solutions and factorization of differential operators with power series coefficients, J. Symb. Comput., 24, 1, 1-30, (1997) · Zbl 0924.12005
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