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''Kummer function, Pochhammer function''

A solution of the [[Confluent hypergeometric equation|confluent hypergeometric equation]]

$$ \tag{1 }
zw  ^ {\prime\prime} + ( \gamma - z) w  ^  \prime  - \alpha w  =  0.
$$

The function may be defined using the so-called Kummer series

$$ \tag{2 }
\Phi ( \alpha ;  \gamma ;  z)  = \ 
{} _ {1} F _ {1} ( \alpha , \gamma ;  z) =
$$

$$ 
= \ 
1 + {
\frac \alpha  \gamma 
 } {
\frac{z}{1!}
 } + 
\frac{
\alpha ( \alpha + 1) }{\gamma ( \gamma + 1) }
 
\frac{z  ^ {2} }{2! }
 + \dots ,
$$

where  $  \alpha $
and  $  \gamma $
are parameters which assume any real or complex values except for  $  \gamma = 0, - 1, - 2 \dots $
and  $  z $
is a complex variable. The function  $  \Psi ( \alpha ;  \gamma ;  z ) $
is called the confluent hypergeometric function of the first kind. The second linearly independent solution of equation (1),

$$ 
\Psi ( \alpha ;  \gamma ;  z)  = \ 

\frac{\Gamma ( \alpha - \gamma + 1) \Gamma ( \gamma - 1) }{\Gamma ( \alpha ) \Gamma ( 1 - \gamma ) }

z ^ {1 - \gamma } \Phi ( \alpha - \gamma + 1 ;  2 - \gamma ;  z),
$$

$$ 
\gamma  \neq  0 , - 1 , - 2 \dots \  |  \mathop{\rm arg}  z |  <  \pi ,
$$

is called the confluent hypergeometric function of the second kind.

The confluent hypergeometric function  $  \Phi ( \alpha ;  \gamma ;  z ) $
is an entire analytic function in the entire complex  $  z $-
plane; if  $  z $
is fixed, it is an entire function of  $  \alpha $
and a meromorphic function of  $  \gamma $
with simple poles at the points  $  \gamma = 0, - 1 , - 2 ,\dots $. 
The confluent hypergeometric function  $  \Psi ( \alpha ;  \gamma ;  z ) $
is an analytic function in the complex  $  z $-
plane with the slit  $  ( - \infty , 0 ) $
and an entire function of  $  \alpha $
and  $  \gamma $.

The confluent hypergeometric function  $  \Phi ( \alpha ;  \gamma ;  z ) $
is connected with the [[Hypergeometric function|hypergeometric function]]  $  F ( \alpha , \beta , \gamma ;  z ) $
by the relation

$$ 
\Phi ( \alpha ;  \gamma ;  z)  = \ 
\lim\limits _ {\beta \rightarrow \infty }  F
\left (  \alpha , \beta , \gamma ; \ 
{
\frac{z} \beta 
 } \right ) .
$$

Elementary relationships. The four functions  $  \Phi ( \alpha \pm  1 ;  \gamma ;  z ) $, 
$  \Phi ( \alpha ;  \gamma \pm  1 ;  z ) $
are called adjacent (or contiguous) to the function  $  \Phi ( \alpha ;  \gamma ;  z ) $. 
There is a linear relationship between  $  \Phi ( \alpha ;  \gamma ;  z ) $
and any two functions adjacent to it, e.g.

$$ 
\gamma \Phi ( \alpha ;  \gamma ;  z) -
\gamma \Phi ( \alpha - 1 ;  \gamma ;  z) -
z \Phi ( \alpha ;  \gamma + 1 ;  z)  =  0.
$$

Six formulas of this type may be obtained from the relations between adjacent functions for hypergeometric functions. The successive use of these recurrence formulas yields linear relations connecting the function  $  \Phi ( \alpha ;  \gamma ;  z ) $
with the associated functions  $  \Phi ( \alpha + m ;  \gamma + n ;  z) $, 
where  $  m $
and  $  n $
are integers.

Differentiation formulas:

$$ 

\frac{d  ^ {n} }{dz  ^ {n} }

\Phi ( \alpha ;  \gamma ;  z)  = \ 

\frac{\alpha \dots ( \alpha + n - 1 ) }{\gamma \dots ( \gamma + n - 1 ) }

\Phi ( \alpha + n ;  \gamma + n ;  z),
$$

$$ 
n  =  1 , 2 , .  .  . .
$$

Basic integral representations.

$$ 
\Phi ( \alpha ;  \gamma ;  z) =
$$

$$ 
= \ 

\frac{\Gamma ( \gamma ) }{\Gamma ( \alpha ) \Gamma ( \gamma - \alpha
) }
 \int\limits _ { 0 } ^ { 1 }  e ^ {zt } t ^ {\alpha - 1
 } ( 1 - t) ^ {\gamma - \alpha - 1 }  dt ,\   \mathop{\rm Re}  \gamma >  \mathop{\rm Re}  \alpha > 0 ;
$$

$$ 
\Psi ( \alpha ;  \gamma ;  z) =
$$

$$ 
= \ 
{
\frac{1}{\Gamma ( \alpha ) }
 } \int\limits _ { 0 } ^  \infty   e ^ {- zt } t ^ 
{\alpha - 1 } ( 1 + t) ^ {\gamma - \alpha - 1 }  dt ,\   \mathop{\rm Re}  \alpha > 0 ,\   \mathop{\rm Re}  z > 0 .
$$

The asymptotic behaviour of confluent hypergeometric functions as  $  z \rightarrow \infty $
can be studied using the integral representations [[#References|[1]]], [[#References|[2]]], [[#References|[3]]]. If  $  \gamma \rightarrow \infty $, 
while  $  \alpha $
and  $  z $
are bounded, the behaviour of the function  $  \Phi ( \alpha ;  \gamma ;  z) $
is described by formula (2). In particular, for large  $  \gamma $
and bounded  $  \alpha $
and  $  z $:

$$ 
\Phi ( \alpha ;  \gamma ;  z)  = \ 
1 + O ( | \gamma |  ^ {-} 1 ) .
$$

Representations of functions by confluent hypergeometric functions.

Bessel functions:

$$ 
J _  \nu  ( z)  = \ 
{
\frac{1}{\Gamma ( 1 + \nu ) }
 }
\left ( {
\frac{z}{2}
 } \right )  ^  \nu 
e ^ {- iz } \Phi \left (
\nu + {
\frac{1}{2}
 } ;  2 \nu + 1 ;  2iz \right ) ,
$$

$$ 
I _  \nu  ( z)  =  {
\frac{1}{\Gamma ( 1 + \nu ) }
 } \left (
{
\frac{z}{2}
 } \right )  ^  \nu  e ^ {- z } \Phi \left ( \nu +
{
\frac{1}{2}
 } ;  2 \nu + 1 ;  2z \right ) ,
$$

$$ 
K _  \nu  ( z)  =  \sqrt \pi e ^ {- z } ( 2z)  ^  \nu  \Psi \left (
\nu + {
\frac{1}{2}
 } ;  2 \nu + 1 ;  2z \right ) .
$$

Laguerre polynomials:

$$ 
L _ {n} ^ {( \alpha ) } ( z)  = \ 

\frac{( \alpha + 1) _ {n} }{n! }
 \Phi  (- n ;  \alpha + 1 ;  z).
$$

Probability integrals:

$$ 
 \mathop{\rm erf} ( z)  = \ 

\frac{2z }{\sqrt \pi }
 \Phi
\left ( {
\frac{1}{2}
 } ;  {
\frac{3}{2}
 } ;  - z  ^ {2} \right ) ,
$$

$$ 
 \mathop{\rm erf}  c ( z)  =  {
\frac{1}{\sqrt \pi}
 } e ^ {- x  ^ {2} } \Psi
\left ( {
\frac{1}{2}
 } ;  {
\frac{1}{2}
 } ;  z  ^ {2} \right ) .
$$

The exponential integral function:

$$ 
-  \mathop{\rm Ei} (- z)  = \ 
e  ^ {-} z \Psi ( 1 ;  1 ;  z) .
$$

The [[logarithmic integral]] function:

$$ 
 \mathop{\rm li} ( z)  = \ 
z \Psi ( 1 ;  1 ;  -  \mathop{\rm ln}  z) .
$$

Gamma-functions:

$$ 
\Gamma ( \alpha , z)  = \ 
e ^ {- z } \Psi ( 1 - \alpha ;  1 - \alpha ;  z) .
$$

Elementary functions:

$$ 
e  ^ {z}  =  \Phi ( \alpha ;  \alpha ;  z) ,
$$

$$ 
\sin  z  =  e ^ {iz } z \Phi ( 1 ;  2 ;  - 2iz) .
$$

See also [[#References|[1]]], [[#References|[2]]], [[#References|[3]]], [[#References|[8]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.)  et al. (ed.) , ''Higher transcendental functions'' , '''2. Bessel functions, parabolic cylinder functions, orthogonal polynomials''' , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.S. Gradshtein,   I.M. Ryzhik,   "Table of integrals, series and products" , Acad. Press  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Abramowitz,   I.A. Stegun,   "Handbook of mathematical functions" , Dover, reprint  (1964)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.T. Whittaker,   G.N. Watson,   "A course of modern analysis" , Cambridge Univ. Press  (1952)  pp. Chapt. 6</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.L. Lebedev,   R.M. Fedorova,   "Handbook of mathematical tables" , Moscow  (1956)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  N.M. Burunova,   "Handbook of mathematical tables" , Moscow  (1959)  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.A. Fletcher,   J.C.P. Miller,   L. Rosenhead,   L.J. Comrie,   "An index of mathematical tables" , '''1–2''' , Oxford Univ. Press  (1962)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  N.N. Lebedev,   "Special functions and their applications" , Prentice-Hall  (1965)  (Translated from Russian)</TD></TR></table>