Difference between revisions of "Laguerre functions"
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Functions that are solutions of the equation | Functions that are solutions of the equation | ||
| − | + | $$ \tag{* } | |
| + | x y ^ {\prime\prime} + ( \alpha - x + 1 ) y ^ \prime + n y = 0 , | ||
| + | $$ | ||
| − | where | + | 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 | + | where $ L _ {n} ^ \alpha ( x) $ |
| + | is a Laguerre polynomial, is sometimes also called a Laguerre function. | ||
====References==== | ====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> | <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> | ||
Latest revision as of 22:15, 5 June 2020
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) |
How to Cite This Entry:
Laguerre functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Laguerre_functions&oldid=47565
Laguerre functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Laguerre_functions&oldid=47565
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article