# Difference between revisions of "E-function"

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− | + | {{TEX|done}}{{MSC|11J91}} | |

− | + | 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]]. | |

− | + | 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 ordinary differential equation|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 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}\,, | ||

+ | $$ | ||

− | + | 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]]. | |

− | + | See also [[G-function|$G$-function]]. | |

− | |||

− | |||

− | |||

− | |||

− | |||

− | See also [[G-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) | + | <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