Laguerre functions

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 or in terms of Whittaker functions. For $ n = 0 , 1 \dots $ the solutions of equation (*) are called 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

[1]	E. Jahnke, F. Emde, "Tables of functions with formulae and curves" , Dover, reprint (1945) (Translated from German)