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The concept of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e1100102.png" />-functions was introduced by C.L. Siegel in [[#References|[a1]]], p. 223, in his work on generalizations of the Lindemann–Weierstrass theorem.
+
{{TEX|done}}{{MSC|11J91}}
  
Consider a [[Taylor series|Taylor series]] of the form
+
The concept of $E$-functions was introduced by C.L. Siegel in [[#References|[a1]]], p. 223, in his work on generalisations of the [[Lindemann–Weierstrass theorem]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e1100103.png" /></td> </tr></table>
+
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:
  
where the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e1100104.png" /> belong to a fixed algebraic number field (cf. also [[Algebraic number|Algebraic number]]; [[Field|Field]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e1100105.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e1100106.png" />). Suppose it satisfies the following conditions:
+
i) $f$ satisfies a [[Linear ordinary differential equation|linear differential equation]] with polynomial coefficients;
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e1100107.png" /> satisfies a linear differential equation with polynomial coefficients;
+
ii) for any $\epsilon > 0$ one has $H(a_0,\ldots,a_n) = O\left({ n^{\epsilon n} }\right)$.  
  
ii) for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e1100108.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e1100109.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001010.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001011.png" />-function. Here, the notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001012.png" /> stands for the so-called projective height, given by
+
Then $f$ is called an $E$-function. Here, the notation $H(x_0,\ldots,x_n)$ stands for the so-called projective height (cf [[Height, in Diophantine geometry]]), 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 function]]s of the form
 +
$$
 +
\sum_{k=0}\infty \frac{ (\lambda_1)_k\cdots(\lambda_p)_k }{ (\mu_1)_k\cdots(\mu_q)_k } \left({\frac{z}{q-p}}\right)^{(q-p)k}\,,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001013.png" /></td> </tr></table>
+
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 [[#References|[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 [[#References|[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 [[#References|[a4]]]. In recent years, F. Beukers, W.D. Brownawell and G. Heckman studied these problems with the powerful techniques from differential Galois theory, see [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], and also [[Galois differential group]].
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001014.png" />-tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001015.png" />. The product is taken over all valuations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001017.png" /> (cf. also [[Norm on a field|Norm on a field]]). When the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001018.png" /> are rational numbers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001019.png" /> is simply the maximum of the absolute values of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001020.png" /> times their common denominator. As suggested by their name, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001021.png" />-functions are a variation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001022.png" />. A large class of examples is given by the hypergeometric functions (cf. [[Hypergeometric function|Hypergeometric function]]) of the form
+
See also [[G-function|$G$-function]].
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001023.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001025.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001027.png" /> is the Pochhammer symbol, given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001028.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001029.png" />-functions at algebraic points. These results were published in 1929 and later, in 1949, a more systematic account appeared in [[#References|[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 [[#References|[a3]]]. Roughly speaking, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001030.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001031.png" />-functions that are algebraically independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001032.png" /> (cf. [[Algebraic independence|Algebraic independence]]), then the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001033.png" /> are algebraically independent over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001034.png" /> for all algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001035.png" /> excepting a known finite set. Thus, proving the algebraic independence of values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001036.png" />-functions at algebraic points has been reduced to the problem of showing algebraic independence over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001037.png" /> 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 [[#References|[a4]]]. In recent years, F. Beukers, W.D. Brownawell and G. Heckman studied these problems with the powerful techniques from differential Galois theory, see [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], and also [[Galois differential group|Galois differential group]].
 
 
 
See also [[G-function(2)|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e110/e110010/e11001038.png" />-function]].
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.L. Siegel,  "Über einige Anwendungen diophantischer Approximationen" , ''Ges. Abhandlungen'' , '''I''' , Springer  (1966)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.L. Siegel,  "Transcendental numbers" , ''Ann. Math. Studies'' , '''16''' , Princeton Univ. Press  (1949)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.B. Shidlovskii,  "Transcendental numbers" , De Gruyter  (1989) (In Russian)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  F. Beukers,  W.D. Brownawell,  G. Heckman,  "Siegel normality"  ''Ann. of Math.'' , '''127'''  (1988)  pp. 279–308</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  N.M. Katz,  "Differential Galois theory and exponential sums" , ''Ann. Math. Studies'' , Princeton Univ. Press  (1990)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  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</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  C.L. Siegel,  "Über einige Anwendungen diophantischer Approximationen" , ''Ges. Abhandlungen'' , '''I''' , Springer  (1966)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  C.L. Siegel,  "Transcendental numbers" , ''Ann. Math. Studies'' , '''16''' , Princeton Univ. Press  (1949)</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  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</TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top">  A.B. Shidlovskii,  "Transcendental numbers" (In Russian), Nauka (1987)  {{ZBL|0629.10026}} (English translation by N. Koblitz) De Gruyter  (1989) {{ZBL|0689.10043}}</TD></TR>
 +
<TR><TD valign="top">[a5]</TD> <TD valign="top">  F. Beukers,  W.D. Brownawell,  G. Heckman,  "Siegel normality"  ''Ann. of Math.'' , '''127'''  (1988)  pp. 279–308</TD></TR>
 +
<TR><TD valign="top">[a6]</TD> <TD valign="top">  N.M. Katz,  "Differential Galois theory and exponential sums" , ''Ann. Math. Studies'' , Princeton Univ. Press  (1990)</TD></TR>
 +
<TR><TD valign="top">[a7]</TD> <TD valign="top">  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</TD></TR>
 +
</table>

Latest revision as of 18:03, 26 March 2021

2010 Mathematics Subject Classification: Primary: 11J91 [MSN][ZBL]

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 $H(a_0,\ldots,a_n) = O\left({ n^{\epsilon n} }\right)$.

Then $f$ is called an $E$-function. Here, the notation $H(x_0,\ldots,x_n)$ stands for the so-called projective height (cf Height, in Diophantine geometry), 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 $$ \sum_{k=0}\infty \frac{ (\lambda_1)_k\cdots(\lambda_p)_k }{ (\mu_1)_k\cdots(\mu_q)_k } \left({\frac{z}{q-p}}\right)^{(q-p)k}\,, $$

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" (In Russian), Nauka (1987) Zbl 0629.10026 (English translation by N. Koblitz) De Gruyter (1989) Zbl 0689.10043
[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
How to Cite This Entry:
E-function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=E-function&oldid=14699
This article was adapted from an original article by F. Beukers (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article