Zeros and Asymptotics of Holonomic Sequences

Abstract

In this thesis we study the zeros and asymptotics of sequences that satisfy linear recurrence relations with generally nonconstant coefficients.

By the theorem of Skolem-Mahler-Lech, the set of zero terms of a sequence that satisfies a linear recurrence relation with constant coefficients taken from a field of characteristic zero is comprised of the union of finitely many arithmetic progressions together with a finite exceptional set. Further, in the nondegenerate case, we can eliminate the possibility of arithmetic progressions and conclude that there are only finitely many zero terms. For generally nonconstant coefficients, there are generalizations of this theorem due to Bézivin and to Methfessel that imply, under fairly general conditions, that we obtain a finite union of arithmetic progressions together with an exceptional set of density zero. Further, a condition is given under which one can exclude the possibility of arithmetic progressions and obtain a set of zero terms of density zero. In this thesis, it is shown that this condition reduces to the nondegeneracy condition in the case of constant coefficients. This allows for a consistent definition of nondegeneracy valid for generally nonconstant coefficients and a unified result is obtained.

The asymptotic theory of sequences that satisfy linear recurrence relations with generally nonconstant coefficients begins with the basic theorems of Poincaré and Perron. There are some generalizations of these theorems that hold in greater generality, but if we restrict the coefficient sequences of our linear recurrences to be polynomials in the index, we obtain full asymptotic expansions of a predictable form for the solution sequences. These expansions can be obtained by applying a transfer method of Flajolet and Sedgewick or, in some cases, by applying a bivariate method of Pemantle and Wilson. In this thesis, these methods are applied to a family of binomial sums and full asymptotic expansions are obtained. The leading terms of the expansions are obtained explicitly in all cases, while in some cases a field containing the asymptotic coefficients is obtained and some divisibility properties for the asymptotic coefficients are obtained using a generalization of a method of Stoll and Haible.