E-function

The concept of $E$-functions was introduced by C.L. Siegel in [a1], p. 223, in his work on generalisations of the Lindemann–Weierstrass theorem.

Consider a Taylor series of the form $$f(z) = \sum_{n=0}^\infty \frac{a_n}{n!} z^n$$ where the numbers $a_n$ belong to a fixed algebraic number field (cf. also Algebraic number; Field) $K$ ($[K:\mathbb{Q}] < \infty$). Suppose it satisfies the following conditions:

i) $f$ satisfies a linear differential equation with polynomial coefficients;

ii) for any $\epsilon > 0$ one has $H(a_0,\ldots,a_n) < n^{\epsilon n}$.

Then $f$ is called an $E$-function. Here, the notation $H(x_0,\ldots,x_n)$ stands for the so-called projective height, given by $$\prod_\nu \max(|x_0|_\nu,\ldots,|x_n|_\nu)$$ for any $(n+1)$-tuple $(x_0,\ldots,x_n) \in K^{n+1}$. The product is taken over all valuations $\nu$ of $K$ (cf. also Norm on a field). When the $x_i$ are rational numbers, $H(x_0,\ldots,x_n)$ is simply the maximum of the absolute values of the $x_i$ multiplied by their common denominator. As suggested by their name,$E$-functions are a variation on the exponential function $e^z$. A large class of examples is given by the hypergeometric functions of the form $$\sum_{k=0}\infty \frac{ (\lambda_1)_k\cdots(\lambda_p)_k }{ (\mu_1)_k\cdots(\mu_q)_k } \left({\frac{}{}}\right)^{(q-p)k}\,,$$

where $q > p$, $\lambda_i, \mu_j \in \mathbb{Q}$ for all $i,j$ and (x)_k is the Pochhammer symbol, given by $(x)_k = x(x+1)\cdots(x+k-1)$. Motivated by the success of the Lindemann–Weierstrass theorem and techniques of A. Thue and W. Maier, Siegel was the first to define and study them. He found a number of transcendence results on values of $E$-functions at algebraic points. These results were published in 1929 and later, in 1949, a more systematic account appeared in [a2]. Unfortunately, Siegel's main result contains a normality condition on the differential equations which, in practice, seemed very hard to verify. This condition was removed by A.B. Shidlovskii, around 1955 [a3]. Roughly speaking, if $f_1(z),\ldots,f_n(z)$ are $E$-functions that are algebraically independent over $\mathbb{C}(z)$ (cf. Algebraic independence), then the values $f_1(\xi),\ldots,f_n(\xi)$ are algebraically independent over $\mathbb{Q}$ for all algebraic $\xi$ excepting a known finite set. Thus, proving the algebraic independence of values of $E$-functions at algebraic points has been reduced to the problem of showing algebraic independence over $\mathbb{C}(z)$ of functions satisfying linear differential equations. During the last thirty years the latter problem has been the object of study of a school of Russian mathematicians and a few non-Russian mathematicians as well. Many of these results are contained in [a4]. In recent years, F. Beukers, W.D. Brownawell and G. Heckman studied these problems with the powerful techniques from differential Galois theory, see [a5], [a6], [a7], and also Galois differential group.

See also $G$-function.

References

[a1]	C.L. Siegel, "Über einige Anwendungen diophantischer Approximationen" , Ges. Abhandlungen , I , Springer (1966)
[a2]	C.L. Siegel, "Transcendental numbers" , Ann. Math. Studies , 16 , Princeton Univ. Press (1949)
[a3]	A.B. Shidlovskii, "A criterion for algebraic independence of the values of a class of entire functions" Amer. Math. Soc. Transl. Ser. 2 , 22 (1962) pp. 339–370 Izv. Akad. SSSR Ser. Math. , 23 (1959) pp. 35–66
[a4]	A.B. Shidlovskii, "Transcendental numbers" , De Gruyter (1989) (In Russian)
[a5]	F. Beukers, W.D. Brownawell, G. Heckman, "Siegel normality" Ann. of Math. , 127 (1988) pp. 279–308
[a6]	N.M. Katz, "Differential Galois theory and exponential sums" , Ann. Math. Studies , Princeton Univ. Press (1990)
[a7]	F. Beukers, "Differential Galois theory" M. Waldschmidt (ed.) P. Moussa (ed.) J.M. Luck (ed.) C. Itzykson (ed.) , From Number Theory to Physics , Springer (1995) pp. Chapt. 8