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 |
E-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=E-function&oldid=35744