View source for Laguerre polynomials

''Chebyshev–Laguerre polynomials''

Polynomials that are orthogonal on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057310/l0573101.png" /> with weight function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057310/l0573102.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057310/l0573103.png" />. The standardized Laguerre polynomials are defined by the formula

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Their representation by means of the [[Gamma-function|gamma-function]] is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057310/l0573105.png" /></td> </tr></table>

In applications the most important formulas are:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057310/l0573106.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057310/l0573107.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057310/l0573108.png" /></td> </tr></table>

The polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057310/l0573109.png" /> satisfies the differential equation (Laguerre equation)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057310/l05731010.png" /></td> </tr></table>

The generating function of the Laguerre polynomials has the form

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The orthonormal Laguerre polynomials can be expressed in terms of the standardized polynomials as follows:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057310/l05731012.png" /></td> </tr></table>

The set of all Laguerre polynomials is dense in the space of functions whose square is integrable with weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057310/l05731013.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057310/l05731014.png" />.

Laguerre polynomials are most frequently used under the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057310/l05731015.png" />; these were investigated by E. Laguerre [[#References|[1]]], and are denoted in this case by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057310/l05731016.png" /> (in contrast to them, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057310/l05731017.png" /> are sometimes known as generalized Laguerre polynomials). The first few Laguerre polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057310/l05731018.png" /> have the form

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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057310/l05731020.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057310/l05731021.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057310/l05731022.png" /></td> </tr></table>

The Laguerre polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057310/l05731023.png" /> is sometimes denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057310/l05731024.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Laguerre,   "Sur le transformations des fonctions elliptiques"  ''Bull. Soc. Math. France'' , '''6'''  (1878)  pp. 72–78</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.A. Steklov,   ''Izv. Imp. Akad. Nauk.'' , '''10'''  (1916)  pp. 633–642</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Szegö,   "Orthogonal polynomials" , Amer. Math. Soc.  (1975)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.K. Suetin,   "Classical orthogonal polynomials" , Moscow  (1979)  (In Russian)</TD></TR></table>



====Comments====
Laguerre polynomials can be written as confluent hypergeometric functions (cf. [[Confluent hypergeometric function|Confluent hypergeometric function]]) and belong to the [[Classical orthogonal polynomials|classical orthogonal polynomials]]. They have a close connection with the [[Heisenberg representation|Heisenberg representation]]: as matrix elements of irreducible representations and as spherical functions on certain Gel'fand pairs (cf. [[Gel'fand representation|Gel'fand representation]]) associated with the Heisenberg group. See the references given in [[#References|[a1]]], Chapt. 1, §9.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.B. Folland,   "Harmonic analysis in phase space" , Princeton Univ. Press  (1989)</TD></TR></table>