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

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

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

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)