Combinatorial Sequences: Non-Holonomicity and Inequalities

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 algorithmicall