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- W2182087884 abstract "Holonomic functions (respectively sequences) satisfy linear ordinary differential equations (respectively recurrences) with polynomial coefficients. This class can be generalized to functions of several continuous or discrete variables, thus encompassing most special functions that occur in applications, for instance in mathematical physics. In particular, all hypergeometric functions are holonomic. This work makes several contributions to the theory of holonomic functions and sequences. In the first part, new methods are introduced to show that a given function or sequence is not holonomic. First, number-theoretic methods are applied, and connections to the theory of transcendental numbers are pointed out. A new application of the saddle point method from asymptotic analysis to a concrete function is given, which proves its non-holonomicity. The second part addresses questions of positivity of holonomic (and more general) sequences. First, two new methods for proving positivity of sequences algorithmically are presented. The first one is limited to holonomic sequences and is based on the signs of the recurrence coefficients. The second method is applicable to a class much larger than the holonomic sequences. Its main idea is the construction of an inductive proof. To perform the induction step, the involved sequences and their shifts are replaced by real variables. The induction step is thus reduced to a (sufficient) system of polynomial equations and inequalities over the reals. Its satisfiability is known to be decidable by Cylindrical Algebraic Decomposition. Our procedure does not terminate in general, but succeeds in automatically proving numerous non-trivial examples from standard textbooks on inequalities. Finally, solutions of linear recurrences with constant coefficients are considered from the viewpoint of positivity. We show that such sequences, called C-finite, oscillate in certain non-trivial cases, i.e., are neither eventually positive nor eventually negative. To this end, a result from Diophantine geometry, viz. about lattice points in certain regions of the plane, is provided. Furthermore, we investigate the asymptotic density of the positivity set of an arbitrary C-finite sequence. Its existence is established, and its possible values are determined. The methods we use for this belong to the theory of equidistributed sequences." @default.
- W2182087884 created "2016-06-24" @default.
- W2182087884 creator A5032284258 @default.
- W2182087884 date "2008-04-03" @default.
- W2182087884 modified "2023-09-27" @default.
- W2182087884 title "Combinatorial Sequences: Non-Holonomicity and Inequalities" @default.
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