D-finite Numbers

TL;DR

This paper introduces D-finite numbers, a class of limits of P-recursive sequences related to D-finite functions, encompassing many mathematical constants and algebraic numbers, and explores their properties and evaluations.

Contribution

It defines D-finite numbers based on subrings of complex numbers and shows their connection to limits of D-finite functions at specific points, expanding understanding of these constants.

Findings

D-finite numbers include many known constants and algebraic numbers.

They can be characterized as limits of D-finite functions at the point one.

Evaluating D-finite functions at non-singular algebraic points yields D-finite numbers.

Abstract

D-finite functions and P-recursive sequences are defined in terms of linear differential and recurrence equations with polynomial coefficients. In this paper, we introduce a class of numbers closely related to D-finite functions and P-recursive sequences. It consists of the limits of convergent P-recursive sequences. Typically, this class contains many well-known mathematical constants in addition to the algebraic numbers. Our definition of the class of D-finite numbers depends on two subrings of the field of complex numbers. We investigate how different choices of these two subrings affect the class. Moreover, we show that D-finite numbers are essentially limits of D-finite functions at the point one, and evaluating D-finite functions at non-singular algebraic points typically yields D-finite numbers. This result makes it easier to recognize certain numbers to be D-finite.