# Confluent hypergeometric function

Kummer function, Pochhammer function

A solution of the 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 $F ( \alpha , \beta , \gamma ; z )$ 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 [1], [2], [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:

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

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