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
- [math]\displaystyle{ f(z)=\sum_{n=0}^\infty a_n z^n }[/math]
where the coefficients an 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;
- the projective height of the first n coefficients is O(cn) for some fixed constant c > 0.
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
- Hazewinkel, Michiel, ed. (2001), "G-function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=g/g110010
- C. L. Siegel, "Über einige Anwendungen diophantischer Approximationen", Ges. Abhandlungen, I, Springer (1966)
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