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{{for|the generalization of hypergeometric series |MacRobert E function}}
In [[mathematics]], '''E-functions''' are a type of [[power series]] that satisfy particular arithmetic conditions on the coefficients. They are of interest in [[transcendental number theory]], and are closely related to [[G-function (power series)|G-function]]s.

==Definition==

A power series with coefficients in the field of algebraic numbers

:<math>f(x)=\sum_{n=0}^\infty c_n \frac{x^n}{n!} \in \overline{\mathbb{Q}}[\![x]\!]</math>

is called an '''{{math|1=''E''}}-function'''<ref>Carl Ludwig Siegel, ''Transcendental Numbers'', p.33, Princeton University Press, 1949.</ref> if it satisfies the following three conditions:

* It is a solution of a non-zero linear differential equation with polynomial coefficients (this implies that all the coefficients {{math|1=''c<sub>n</sub>''}} belong to the same [[algebraic number field]], {{math|1=''K''}}, which has [[Degree of a field extension|finite degree]] over the rational numbers);
* For all <math> \varepsilon>0</math>,&nbsp;&nbsp;&nbsp;<math>\overline{\left|c_n\right|}=O\left(n^{n\varepsilon}\right),</math>
: where the left hand side represents the maximum of the absolute values of all the [[Conjugate element (field theory)|algebraic conjugates]] of {{math|1=''c<sub>n</sub>''}};
* For all <math> \varepsilon>0</math> there is a sequence of natural numbers {{math|1=''q''<sub>0</sub>, ''q''<sub>1</sub>, ''q''<sub>2</sub>,...}} such that {{math|1=''q<sub>n</sub>c<sub>k</sub>''}} is an [[algebraic integer]] in {{math|1=''K''}} for {{math|1=''k'' = 0, 1, 2,..., ''n''}}, and {{math|1=''n'' = 0, 1, 2,...}} and for which <math>q_n=O\left(n^{n\varepsilon}\right). </math>

The second condition implies that {{math|1=''f''}} is an [[entire function]] of {{math|1=''x''}}.

==Uses==

{{math|1=''E''}}-functions were first studied by [[Carl Ludwig Siegel|Siegel]] in 1929.<ref>C.L. Siegel, ''Über einige Anwendungen diophantischer Approximationen'', Abh. Preuss. Akad. Wiss. '''1''', 1929.</ref>  He found a method to show that the values taken by certain {{math|1=''E''}}-functions were [[algebraically independent]]. This was a result which established the algebraic independence of classes of numbers rather than just linear independence.<ref>Alan Baker, ''Transcendental Number Theory'', pp.109-112, Cambridge University Press, 1975.</ref> Since then these functions have proved somewhat useful in [[number theory]] and in particular they have application in [[Transcendental numbers|transcendence]] proofs and [[differential equations]].<ref>[[Serge Lang]], ''Introduction to Transcendental Numbers'', pp.76-77, Addison-Wesley Publishing Company, 1966.</ref>

==The Siegel–Shidlovsky theorem==

Perhaps the main result connected to {{math|1=''E''}}-functions is the Siegel–Shidlovsky theorem (also known as the Siegel and Shidlovsky theorem), named after [[Carl Ludwig Siegel]] and Andrei Borisovich Shidlovsky.

Suppose that we are given {{math|1=''n''}} {{math|1=''E''}}-functions, {{math|1=''E''<sub>1</sub>(''x''),...,''E''<sub>''n''</sub>(''x'')}}, that satisfy a system of homogeneous linear differential equations
:<math>y^\prime_i=\sum_{j=1}^n f_{ij}(x)y_j\quad(1\leq i\leq n)</math>
where the {{math|1=''f<sub>ij</sub>''}} are rational functions of {{math|1=''x''}}, and the coefficients of each {{math|1=''E''}} and {{math|1=''f''}} are elements of an algebraic number field {{math|1=''K''}}.  Then the theorem states that if {{math|1=''E''<sub>1</sub>(''x''),...,''E''<sub>''n''</sub>(''x'')}} are algebraically independent over {{math|1=''K''(''x'')}}, then for any non-zero algebraic number {{math|1=α}} that is not a pole of any of the {{math|1=''f<sub>ij</sub>''}} the numbers {{math|1=''E''<sub>1</sub>(α),...,''E''<sub>''n''</sub>(α)}} are algebraically independent.

==Examples==

# Any polynomial with algebraic coefficients is a simple example of an {{math|1=''E''}}-function.
# The [[exponential function]] is an {{math|1=''E''}}-function, in its case {{math|1=''c<sub>n</sub>''&nbsp;=&nbsp;1}} for all of the {{math|1=''n''}}.
# If {{math|1=λ}} is an algebraic number then the [[Bessel function]] {{math|1=''J''<sub>λ</sub>}} is an {{math|1=''E''}}-function.
# The sum or product of two {{math|1=''E''}}-functions is an {{math|1=''E''}}-function.  In particular {{math|1=''E''}}-functions form a [[Ring (mathematics)|ring]].
# If {{math|1=''a''}} is an algebraic number and {{math|1=''f''(''x'')}} is an {{math|1=''E''}}-function then {{math|1=''f''(''ax'')}} will be an {{math|1=''E''}}-function.
# If {{math|1=''f''(''x'')}} is an {{math|1=''E''}}-function then the derivative and integral of {{math|1=''f''}} are also {{math|1=''E''}}-functions.

==References==

<references/>
* {{mathworld|title=E-Function|urlname=E-Function}}

[[Category:Transcendental numbers]]
[[Category:Algebraic number theory]]
[[Category:Analytic functions]]
[[Category:Analytic number theory]]