Siegel 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
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, such that ;
  2. the projective height of the first n coefficients is O for some fixed constant c > 0. That is, the denominators of are and the algebraic conjugates of have their absolute value bounded by.
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.