On the set of zero coefficients of a function satisfying a linear differential equation

Details On the set of zero coefficients of a function satisfying a linear differential equation On the set of zero coefficients of a function satisfying a linear differential equation

2011-05-30

arxiv.org/abs/1105.6078v1

Mathematics

Number Theory

11 pages

Scientific paper

Let $K$ be a field of characteristic zero and suppose that $f:\mathbb{N}\to K$ satisfies a recurrence of the form $$f(n)\ =\ \sum_{i=1}^d P_i(n) f(n-i),$$ for $n$ sufficiently large, where $P_1(z),...,P_d(z)$ are polynomials in $K[z]$. Given that $P_d(z)$ is a nonzero constant polynomial, we show that the set of $n\in \mathbb{N}$ for which $f(n)=0$ is a union of finitely many arithmetic progressions and a finite set. This generalizes the Skolem-Mahler-Lech theorem, which assumes that $f(n)$ satisfies a linear recurrence. We discuss examples and connections to the set of zero coefficients of a power series satisfying a homogeneous linear differential equation with rational function coefficients.

Affiliated with

Also associated with

No associations

Rating

On the set of zero coefficients of a function satisfying a linear differential equation does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the set of zero coefficients of a function satisfying a linear differential equation, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the set of zero coefficients of a function satisfying a linear differential equation will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-679039