Editing Laguerre polynomials (section)

=== Properties ===
* Laguerre functions are defined by [[confluent hypergeometric function]]s and Kummer's transformation as<ref>A&S p. 509</ref> <math display="block"> L_n^{(\alpha)}(x) := {n+ \alpha \choose n} M(-n,\alpha+1,x).</math> where <math display="inline">{n+ \alpha \choose n}</math> is a generalized [[binomial coefficient]]. When {{mvar|n}} is an integer the function reduces to a polynomial of degree {{mvar|n}}. It has the alternative expression<ref>A&S p. 510</ref> <math display="block">L_n^{(\alpha)}(x)= \frac {(-1)^n}{n!} U(-n,\alpha+1,x)</math> in terms of [[confluent hypergeometric function|Kummer's function of the second kind]].
* The closed form for these generalized Laguerre polynomials of degree {{mvar|n}} is<ref>A&S p. 775</ref> <math display="block"> L_n^{(\alpha)} (x) = \sum_{i=0}^n (-1)^i {n+\alpha \choose n-i} \frac{x^i}{i!} </math> derived by applying [[Leibniz rule (generalized product rule)|Leibniz's theorem for differentiation of a product]] to Rodrigues' formula.
* Laguerre polynomials have a differential operator representation, much like the closely related Hermite polynomials. Namely, let  <math>D = \frac{d}{dx}</math> and consider the differential operator <math>M=xD^2+(\alpha+1)D</math>. Then <math>\exp(-tM)x^n=(-1)^nt^nn!L^{(\alpha)}_n\left(\frac{x}{t}\right)</math>.{{citation needed|date=April 2023}}
* The first few generalized Laguerre polynomials are:

{| class="wikitable" style="margin:0.5em auto"
|-
! width="20%"| ''n''
! <math>L_n^{(\alpha)}(x)\,</math>
|-
| align="center" | 0
| <math>1\,</math>
|-
| align="center" | 1
| <math>-x+\alpha +1\,</math>
|-
| align="center" | 2
| <math> \tfrac{1}{2} (x^2-2\left( \alpha +2 \right) x+\left( \alpha +1 \right) \left( \alpha +2 \right)) \,</math>
|-
| align="center" | 3
| <math>\tfrac{1}{6} (-x^3+3\left( \alpha +3 \right) x^2-3\left( \alpha +2 \right) \left( \alpha +3 \right) x+\left( \alpha +1 \right) \left( \alpha +2 \right) \left( \alpha +3 \right)) \,</math>
|-
| align="center" | 4
| <math>\tfrac{1}{24} (x^4-4\left( \alpha +4 \right) x^3+6\left( \alpha +3 \right) \left( \alpha +4 \right) x^2-4\left( \alpha +2 \right) \cdots \left( \alpha +4 \right) x+\left( \alpha +1 \right) \cdots \left( \alpha +4 \right)) \,</math>
|-
| align="center" | 5
| <math>\tfrac{1}{120} (-x^5+5\left( \alpha +5 \right) x^4-10\left( \alpha +4 \right) \left( \alpha +5 \right) x^3+10\left( \alpha +3 \right) \cdots \left( \alpha +5 \right) x^2-5\left( \alpha +2 \right) \cdots \left( \alpha +5 \right) x+\left( \alpha +1 \right) \cdots \left( \alpha +5 \right)) \,</math>
|-
| align="center" | 6
| <math>\tfrac{1}{720} (x^6-6\left( \alpha +6 \right) x^5+15\left( \alpha +5 \right) \left( \alpha +6 \right) x^4-20\left( \alpha +4 \right) \cdots \left( \alpha +6 \right) x^3+15\left( \alpha +3 \right) \cdots \left( \alpha +6 \right) x^2-6\left( \alpha +2 \right) \cdots \left( \alpha +6 \right) x+\left( \alpha +1 \right) \cdots \left( \alpha +6 \right)) \,</math>
|-
| align="center" | 7
| <math>\tfrac{1}{5040} (-x^7+7\left( \alpha +7 \right) x^6-21\left( \alpha +6 \right) \left( \alpha +7 \right) x^5+35\left( \alpha +5 \right) \cdots \left( \alpha +7 \right) x^4-35\left( \alpha +4 \right) \cdots \left( \alpha +7 \right) x^3+21\left( \alpha +3 \right) \cdots \left( \alpha +7 \right) x^2-7\left( \alpha +2 \right) \cdots \left( \alpha +7 \right) x+\left( \alpha +1 \right) \cdots \left( \alpha +7 \right)) \,</math>
|-
| align="center" | 8
| <math>\tfrac{1}{40320} (x^8-8\left( \alpha +8 \right) x^7+28\left( \alpha +7 \right) \left( \alpha +8 \right) x^6-56\left( \alpha +6 \right) \cdots \left( \alpha +8 \right) x^5+70\left( \alpha +5 \right) \cdots \left( \alpha +8 \right) x^4-56\left( \alpha +4 \right) \cdots \left( \alpha +8 \right) x^3+28\left( \alpha +3 \right) \cdots \left( \alpha +8 \right) x^2-8\left( \alpha +2 \right) \cdots \left( \alpha +8 \right) x+\left( \alpha +1 \right) \cdots \left( \alpha +8 \right)) \,</math>
|-
| align="center" | 9
| <math>\tfrac{1}{362880} (-x^9+9\left( \alpha +9 \right) x^8-36\left( \alpha +8 \right) \left( \alpha +9 \right) x^7+84\left( \alpha +7 \right) \cdots \left( \alpha +9 \right) x^6-126\left( \alpha +6 \right) \cdots \left( \alpha +9 \right) x^5+126\left( \alpha +5 \right) \cdots \left( \alpha +9 \right) x^4-84\left( \alpha +4 \right) \cdots \left( \alpha +9 \right) x^3+36\left( \alpha +3 \right) \cdots \left( \alpha +9 \right) x^2-9\left( \alpha +2 \right) \cdots \left( \alpha +9 \right) x+\left( \alpha +1 \right) \cdots \left( \alpha +9 \right)) \,</math>
|-
| align="center" | 10
| <math>\tfrac{1}{3628800} (x^{10}-10\left( \alpha +10 \right) x^9+45\left( \alpha +9 \right) \left( \alpha +10 \right) x^8-120\left( \alpha +8 \right) \cdots \left( \alpha +10 \right) x^7+210\left( \alpha +7 \right) \cdots \left( \alpha +10 \right) x^6-252\left( \alpha +6 \right) \cdots \left( \alpha +10 \right) x^5+210\left( \alpha +5 \right) \cdots \left( \alpha +10 \right) x^4-120\left( \alpha +4 \right) \cdots \left( \alpha +10 \right) x^3+45\left( \alpha +3 \right) \cdots \left( \alpha +10 \right) x^2-10\left( \alpha +2 \right) \cdots \left( \alpha +10 \right) x+\left( \alpha +1 \right) \cdots \left( \alpha +10 \right)) \,</math>
|}

* The [[coefficient]] of the leading term is {{math|(−1)<sup>''n''</sup>/''n''<nowiki>!</nowiki>}};
* The [[constant term]], which is the value at&nbsp;0, is <math display="block">L_n^{(\alpha)}(0) = {n+\alpha\choose n} = \frac{\Gamma(n + \alpha + 1)}{n!\, \Gamma(\alpha + 1)};</math>
<!-- \frac{n^\alpha}{\Gamma(\alpha+1)} + O\left(n^{\alpha-1}\right);</math> -->
* If {{math|''α''}} is non-negative, then ''L''<sub>''n''</sub><sup>(''α'')</sup> has ''n'' [[real number|real]], strictly positive [[Root of a function|roots]] (notice that <math>\left((-1)^{n-i} L_{n-i}^{(\alpha)}\right)_{i=0}^n</math> is a [[Sturm chain]]), which are all in the [[Interval (mathematics)|interval]] <math>\left( 0, n+\alpha+ (n-1) \sqrt{n+\alpha} \, \right].</math>{{citation needed|date=September 2011}}
* The polynomials' asymptotic behaviour for large {{mvar|n}}, but fixed {{mvar|α}} and {{math|''x'' > 0}}, is given by<ref>Szegő, p. 198.</ref><ref>D. Borwein, J. M. Borwein, R. E. Crandall, "Effective Laguerre asymptotics", ''SIAM J. Numer. Anal.'', vol. 46 (2008), no. 6, pp. 3285–3312 {{doi|10.1137/07068031X}}</ref> <math display="block">
\begin{align}
& L_n^{(\alpha)}(x) = \frac{n^{\frac{\alpha}{2}-\frac{1}{4}}}{\sqrt{\pi}} \frac{e^{\frac{x}{2}}}{x^{\frac{\alpha}{2}+\frac{1}{4}}} \sin\left(2 \sqrt{nx}- \frac{\pi}{2}\left(\alpha-\frac{1}{2} \right) \right)+O\left(n^{\frac{\alpha}{2}-\frac{3}{4}}\right), \\[6pt]
& L_n^{(\alpha)}(-x) = \frac{(n+1)^{\frac{\alpha}{2}-\frac{1}{4}}}{2\sqrt{\pi}} \frac{e^{-x/2}}{x^{\frac{\alpha}{2}+\frac{1}{4}}} e^{2 \sqrt{x(n+1)}} \cdot\left(1+O\left(\frac{1}{\sqrt{n+1}}\right)\right),
\end{align}
</math> and summarizing by <math display="block">\frac{L_n^{(\alpha)}\left(\frac x n\right)}{n^\alpha}\approx e^{x/ 2n} \cdot \frac{J_\alpha\left(2\sqrt x\right)}{\sqrt x^\alpha},</math> where <math>J_\alpha</math> is the [[Bessel function#Asymptotic forms|Bessel function]].