D-finite numbers

作者：Huang Hui^*; Kauers Manuel

来源：International Journal of Number Theory, 2018, 14(7): 1827-1848.

DOI：10.1142/S1793042118501099

• Complex numbers
• D-finite functions
• P-recursive sequences
• algebraic numbers
• evaluation of special functions

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 wellknown 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.

• 出版日期2018-8