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 an 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

  • C. L. Siegel, "Über einige Anwendungen diophantischer Approximationen", Ges. Abhandlungen, I, Springer (1966)