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 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 , such that ;
2. the projective height of the first n coefficients is O(cⁿ) for some fixed constant c > 0. That is, the denominators of (the denominator of an algebraic number is the smallest positive integer such is an algebraic integer) 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.

References

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