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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-functions.

Definition

A power series with coefficients in the field of algebraic numbers

f(x)=\sum _{n=0}^{\infty }c_{n}{\frac {x^{n}}{n!}}\in {\overline {\mathbb {Q} }}[\![x]\!]

is called an $E$ -function^[1] 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 $c n$ belong to the same algebraic number field, $K$ , which has finite degree over the rational numbers);
For all $\varepsilon >0$ , ${\overline {\left|c_{n}\right|}}=O\left(n^{n\varepsilon }\right),$

where the left hand side represents the maximum of the absolute values of all the algebraic conjugates of

c n

;

For all $\varepsilon >0$ there is a sequence of natural numbers $q 0, q 1, q 2,...$ such that $q n c k$ is an algebraic integer in $K$ for $k = 0, 1, 2,..., n$ , and $n = 0, 1, 2,...$ and for which $q_{n}=O\left(n^{n\varepsilon }\right).$

The second condition implies that $f$ is an entire function of $x$ .

Uses

$E$ -functions were first studied by Siegel in 1929.^[2] He found a method to show that the values taken by certain $E$ -functions were algebraically independent. This was a result which established the algebraic independence of classes of numbers rather than just linear independence.^[3] Since then these functions have proved somewhat useful in number theory and in particular they have application in transcendence proofs and differential equations.^[4]

The Siegel–Shidlovsky theorem

Perhaps the main result connected to $E$ -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 $n$ $E$ -functions, $E 1 (x),..., E n (x)$ , that satisfy a system of homogeneous linear differential equations

y_{i}^{\prime }=\sum _{j=1}^{n}f_{ij}(x)y_{j}\quad (1\leq i\leq n)

where the $f ij$ are rational functions of $x$ , and the coefficients of each $E$ and $f$ are elements of an algebraic number field $K$ . Then the theorem states that if $E 1 (x),..., E n (x)$ are algebraically independent over $K (x)$ , then for any non-zero algebraic number $α$ that is not a pole of any of the $f ij$ the numbers $E 1 (α),..., E n (α)$ are algebraically independent.

Examples

Any polynomial with algebraic coefficients is a simple example of an $E$ -function.
The exponential function is an $E$ -function, in its case $c n = 1$ for all of the $n$ .
If $λ$ is an algebraic number then the Bessel function $J λ$ is an $E$ -function.
The sum or product of two $E$ -functions is an $E$ -function. In particular $E$ -functions form a ring.
If $a$ is an algebraic number and $f (x)$ is an $E$ -function then $f (ax)$ will be an $E$ -function.
If $f (x)$ is an $E$ -function then the derivative and integral of $f$ are also $E$ -functions.

References

^ Carl Ludwig Siegel, Transcendental Numbers, p.33, Princeton University Press, 1949.
^ C.L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. 1, 1929.
^ Alan Baker, Transcendental Number Theory, pp.109-112, Cambridge University Press, 1975.
^ Serge Lang, Introduction to Transcendental Numbers, pp.76-77, Addison-Wesley Publishing Company, 1966.

Weisstein, Eric W. "E-Function". MathWorld.

[1] Carl Ludwig Siegel, Transcendental Numbers, p.33, Princeton University Press, 1949.

[2] C.L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. 1, 1929.

[3] Alan Baker, Transcendental Number Theory, pp.109-112, Cambridge University Press, 1975.

[4] Serge Lang, Introduction to Transcendental Numbers, pp.76-77, Addison-Wesley Publishing Company, 1966.

[1]

[2]

[3]

[4]

@@ Line 1: / Line 1: @@
 {{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 more special than [[G-function (power series)|G-function]]s.
+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
-A function ''f''(''x'') is called of '''type ''E''''', or an '''''E''-function''',<ref>Carl Ludwig Siegel, ''Transcendental Numbers'', p.33, Princeton University Press, 1949.</ref> if the power series
-:<math>f(x)=\sum_{n=0}^{\infty} c_{n}\frac{x^{n}}{n!}</math>
+:<math>f(x)=\sum_{n=0}^\infty c_n \frac{x^n}{n!} \in \overline{\mathbb{Q}}[\![x]\!]</math>
-satisfies the following three conditions:
+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:
-* All the coefficients ''c<sub>n</sub>'' belong to the same [[algebraic number field]], ''K'', which has [[Degree of a field extension|finite degree]] over the rational numbers;
+* 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>
-* For all ε&nbsp;&gt;&nbsp;0,
+: 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>''}};
-:<math>\overline{\left|c_{n}\right|}=O\left(n^{n\varepsilon}\right)</math>,
+* 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>
-where the left hand side represents the maximum of the absolute values of all the [[Conjugate element (field theory)|algebraic conjugates]] of ''c<sub>n</sub>'';
-* For all ε&nbsp;&gt;&nbsp;0 there is a sequence of natural numbers ''q''<sub>0</sub>, ''q''<sub>1</sub>, ''q''<sub>2</sub>,... such that ''q<sub>n</sub>c<sub>k</sub>'' is an [[algebraic integer]] in ''K'' for ''k''=0, 1, 2,..., ''n'', and ''n'' = 0, 1, 2,... and for which
-:<math>q_{n}=O\left(n^{n\varepsilon}\right)</math>.
-The second condition implies that ''f'' is an [[entire function]] of ''x''.
+The second condition implies that {{math|1=''f''}} is an [[entire function]] of {{math|1=''x''}}.
 ==Uses==
-''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 ''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>
+{{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 ''E''-functions is the Siegel–Shidlovsky theorem (also known as the Shidlovsky and Shidlovskii theorem), named after [[Carl Ludwig Siegel]] and Andrei Borisovich Shidlovskii.
+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 ''n'' ''E''-functions, ''E''<sub>1</sub>(''x''),...,''E''<sub>''n''</sub>(''x''), that satisfy a system of homogeneous linear differential equations
+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 ''f<sub>ij</sub>'' are rational functions of ''x'', and the coefficients of each ''E'' and ''f'' are elements of an algebraic number field ''K''.  Then the theorem states that if ''E''<sub>1</sub>(''x''),...,''E''<sub>''n''</sub>(''x'') are algebraically independent over ''K''(''x''), then for any non-zero algebraic number α that is not a pole of any of the ''f<sub>ij</sub>'' the numbers ''E''<sub>1</sub>(α),...,''E''<sub>''n''</sub>(α) are algebraically independent.
+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 ''E''-function.
+# Any polynomial with algebraic coefficients is a simple example of an {{math|1=''E''}}-function.
-# The [[exponential function]] is an ''E''-function, in its case ''c<sub>n</sub>''=1 for all of the ''n''.
+# 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 λ is an algebraic number then the [[Bessel function]] ''J''<sub>λ</sub> is an ''E''-function.
+# 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 ''E''-functions is an ''E''-function.  In particular ''E''-functions form a [[Ring (mathematics)|ring]].
+# 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 ''a'' is an algebraic number and ''f''(''x'') is an ''E''-function then ''f''(''ax'') will be an ''E''-function.
+# 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 ''f''(''x'') is an ''E''-function then the derivative and integral of ''f'' are also ''E''-functions.
+# 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==
@@ Line 45: / Line 43: @@
 * {{mathworld|title=E-Function|urlname=E-Function}}
-[[Category:Number theory]]
+[[Category:Transcendental numbers]]
+[[Category:Algebraic number theory]]
+[[Category:Analytic functions]]
+[[Category:Analytic number theory]]