Editing Laguerre polynomials (section)

== Recursive definition, closed form, and generating function ==
One can also define the Laguerre polynomials recursively, defining the first two polynomials as
<math display="block">L_0(x) = 1</math>
<math display="block">L_1(x) = 1 - x</math>
and then using the following [[Orthogonal polynomials#Recurrence relations|recurrence relation]] for any {{math|''k'' ≥ 1}}:
<math display="block">L_{k + 1}(x) = \frac{(2k + 1 - x)L_k(x) - k L_{k - 1}(x)}{k + 1}. </math>
Furthermore,
<math display="block">  x L'_n(x) = nL_n (x) - nL_{n-1}(x).</math>

In solution of some boundary value problems, the characteristic values can be useful:
<math display="block">L_{k}(0) = 1, L_{k}'(0) = -k.  </math>

The '''closed form''' is
<math display="block">L_n(x)=\sum_{k=0}^n \binom{n}{k}\frac{(-1)^k}{k!} x^k .</math>

The [[generating function]] for them likewise follows, 
<math display="block">\sum_{n=0}^\infty t^n L_n(x)=  \frac{1}{1-t} e^{-tx/(1-t)}.</math>The operator form is
<math display="block">L_n(x) = \frac{1}{n!}e^x \frac{d^n}{dx^n} (x^n e^{-x}) </math>

Polynomials of negative index can be expressed using the ones with positive index:
<math display="block">L_{-n}(x)=e^xL_{n-1}(-x).</math>

{| class="wikitable" style="margin:0.5em auto"
|+A table of the Laguerre polynomials
|-
! width="20%" | ''n''
! <math>L_n(x)\,</math>
|-
| align="center" | 0 || <math>1\,</math>
|-
| align="center" | 1 || <math>-x+1\,</math>
|-
| align="center" | 2
| <math> \tfrac{1}{2} (x^2-4x+2) \,</math>
|-
| align="center" | 3
| <math>\tfrac{1}{6} (-x^3+9x^2-18x+6) \,</math>
|-
| align="center" | 4
| <math>\tfrac{1}{24} (x^4-16x^3+72x^2-96x+24) \,</math>
|-
| align="center" | 5
| <math>\tfrac{1}{120} (-x^5+25x^4-200x^3+600x^2-600x+120) \,</math>
|-
| align="center" | 6
| <math>\tfrac{1}{720} (x^6-36x^5+450x^4-2400x^3+5400x^2-4320x+720) \,</math>
|-
| align="center" | 7
| <math>\tfrac{1}{5040} (-x^7+49x^6-882x^5+7350x^4-29400x^3+52920x^2-35280x+5040) \,</math>
|-
| align="center" | 8
| <math>\tfrac{1}{40320} (x^8-64x^7+1568x^6-18816x^5+117600x^4-376320x^3+564480x^2-322560x+40320) \,</math>
|-
| align="center" | 9
| <math>\tfrac{1}{362880} (-x^9+81x^8-2592x^7+42336x^6-381024x^5+1905120x^4-5080320x^3+6531840x^2-3265920x+362880) \,</math>
|-
| align="center" | 10
| <math>\tfrac{1}{3628800} (x^{10}-100x^9+4050x^8-86400x^7+1058400x^6-7620480x^5+31752000x^4-72576000x^3+81648000x^2-36288000x+3628800) \,</math>
|-
| align="center" | ''n''
| <math>\tfrac{1}{n!} ((-x)^n + n^2(-x)^{n-1} + \dots + n({n!})(-x) + n!) \,</math>
|}

[[Image:Laguerre poly.svg|thumb|center|600px|The first six Laguerre polynomials.]]