Siegel G-function

This article is about Siegel G-functions. For the general functions introduced by Cornelius Meijer, see Meijer G-function.

In mathematics, the Siegel G-functions are a class of functions in transcendental number theory introduced by C. L. Siegel. They satisfy a linear differential equation with polynomial coefficients, and the coefficients of their power series expansion lie in a fixed algebraic number field and have heights of at most exponential growth.

Definition

A Siegel G-function is a function given by an infinite power series

${\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}}$ 

where the coefficients a_n all belong to the same algebraic number field, K, and with the following two properties.

1. f is the solution to a linear differential equation with coefficients that are polynomials in z. More precisely, there is a differential operator ${\displaystyle L\in K[z,d_{z}],L\neq 0}$ , such that ${\displaystyle L.f=0}$ ;
2. the projective height of the first n coefficients is O(cⁿ) for some fixed constant c > 0. That is, the denominators of ${\displaystyle a_{0},\dots ,a_{n}}$  (the denominator of an algebraic number ${\displaystyle x}$  is the smallest positive integer ${\displaystyle m}$  such ${\displaystyle mx}$  is an algebraic integer) are ${\displaystyle \leq c^{n}}$  and the algebraic conjugates of ${\displaystyle a_{n}}$  have their absolute value bounded by ${\displaystyle c^{n}}$ .

The second condition means the coefficients of f grow no faster than a geometric series. Indeed, the functions can be considered as generalisations of geometric series, whence the name G-function, just as E-functions are generalisations of the exponential function.

References

• Beukers, F. (2001) [1994], "G-function", Encyclopedia of Mathematics, EMS Press
• C. L. Siegel, "Über einige Anwendungen diophantischer Approximationen", Ges. Abhandlungen, I, Springer (1966)