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 ) = ∑ n = 0 ∞ a n z n {\displaystyle 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 ∈ K [ z , d z ] , L ≠ 0 {\displaystyle L\in K[z,d_{z}],L\neq 0}, such that L . f = 0 {\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 a 0 , … , a n {\displaystyle a_{0},\dots ,a_{n}} (the denominator of an algebraic number x {\displaystyle x} is the smallest positive integer m {\displaystyle m} such m x {\displaystyle mx} is an algebraic integer) are ≤ c n {\displaystyle \leq c^{n}} and the algebraic conjugates of a n {\displaystyle a_{n}} have their absolute value bounded by c n {\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.