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
:$f(z)=\sum_^\infty a_n z^n$
where the coefficients ''a_n'' all belong to the same algebraic number field, ''K'', and with the following two properties.

# ''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 ,d_z L\neq 0, such that $L.f=0$;
# 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)

Analytic number theory
Algebraic number theory
Ordinary differential equations
Transcendental numbers
Analytic functions

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