View source for Laguerre functions

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Functions that are solutions of the equation

$$ \tag{* }
x y  ^ {\prime\prime} + ( \alpha - x + 1 ) y  ^  \prime  + n y  =  0 ,
$$

where  $  \alpha $
and  $  n $
are arbitrary parameters. Laguerre functions can be expressed in terms of the [[Degenerate hypergeometric function|degenerate hypergeometric function]] or in terms of [[Whittaker functions|Whittaker functions]]. For  $  n = 0 , 1 \dots $
the solutions of equation (*) are called [[Laguerre polynomials|Laguerre polynomials]]. The function

$$ 
e _ {n} ^ {( \alpha ) } ( x)  =  x ^ {\alpha / 2 }
e ^ {- x / 2 } L _ {n}  ^  \alpha  ( x) ,
$$

where  $  L _ {n}  ^  \alpha  ( x) $
is a Laguerre polynomial, is sometimes also called a Laguerre function.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Jahnke,   F. Emde,   "Tables of functions with formulae and curves" , Dover, reprint  (1945)  (Translated from German)</TD></TR></table>