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:

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

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

See also [1], [2], [3], [8].

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