Laguerre polynomials explained

In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation:$$xy$$ + (1 - x)y' + ny = 0,\ y = y(x)which is a second-order linear differential equation. This equation has nonsingular solutions only if is a non-negative integer.

Sometimes the name Laguerre polynomials is used for solutions of$$xy$$ + (\alpha + 1 - x)y' + ny = 0~.where is still a non-negative integer.Then they are also named generalized Laguerre polynomials, as will be done here (alternatively associated Laguerre polynomials or, rarely, Sonine polynomials, after their inventor^[1] Nikolay Yakovlevich Sonin).

More generally, a Laguerre function is a solution when is not necessarily a non-negative integer.

The Laguerre polynomials are also used for Gauss–Laguerre quadrature to numerically compute integrals of the form$$\backslash int\_0^\backslash infty\; f(x)\; e^\; \backslash ,\; dx.$$

These polynomials, usually denoted, , ..., are a polynomial sequence which may be defined by the Rodrigues formula,

$$L\_n(x)=\backslash frac\backslash frac\backslash left(e^\; x^n\backslash right)\; =\backslash frac\; \backslash left(\backslash frac\; -1\; \backslash right)^n\; x^n,$$reducing to the closed form of a following section.

They are orthogonal polynomials with respect to an inner product$$\backslash langle\; f,g\; \backslash rangle\; =\; \backslash int\_0^\backslash infty\; f(x)\; g(x)\; e^\backslash ,dx.$$

The rook polynomials in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables. Further see the Tricomi–Carlitz polynomials.

The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of the Schrödinger equation for a one-electron atom. They also describe the static Wigner functions of oscillator systems in quantum mechanics in phase space. They further enter in the quantum mechanics of the Morse potential and of the 3D isotropic harmonic oscillator.

Physicists sometimes use a definition for the Laguerre polynomials that is larger by a factor of n! than the definition used here. (Likewise, some physicists may use somewhat different definitions of the so-called associated Laguerre polynomials.)

The first few polynomials

These are the first few Laguerre polynomials:

n	L _{n(x)}
0	1
1	- x +1
2	\tfrac{ 1 }{ 2 } (x ^{2-4x+2)}
3	\tfrac{ 1 }{ 6 } ( - x ^{3+9x} ^{2-18x+6)}
4	\tfrac{ 1 }{ 24 } (x ^{4-16x} ^{3+72x} ^{2-96x+24)}
5	\tfrac{ 1 }{ 120 } ( - x ^{5+25x} ^{4-200x} ^{3+600x} ^{2-600x+120)}
6	\tfrac{ 1 }{ 720 } (x ^{6-36x} ^{5+450x} ^{4-2400x} ^{3+5400x} ^{2-4320x+720)}
7	\tfrac{ 1 }{ 5040 } ( - x ^{7+49x} ^{6-882x} ^{5+7350x} ^{4-29400x} ^{3+52920x} ^{2-35280x+5040)}
8	\tfrac{ 1 }{ 40320 } (x ^{8-64x} ^{7+1568x} ^{6-18816x} ^{5+117600x} ^{4-376320x} ^{3+564480x} ^{2-322560x+40320)}
9	\tfrac{ 1 }{ 362880 } ( - x ^{9+81x} ^{8-18144x} ^{7+42336x} ^{6-381024x} ^{5+1905120x} ^{4-5080320x} ^{3+6531840x} ^{2-3265920x+362880)}
10	\tfrac{ 1 }{ 3628800 } (x ^{10} -100 x ^{9+4050x} ^{8-86400x} ^{7+1058400x} ^{6-7620480x} ^{5+31752000x} ^{4-72576000x} ^{3+81648000x} ^{2-36288000x+3628800)}
n	\tfrac{ 1 }{n!} (( - x) ^{n} + n ^{2(-x)} ^{n-1} + ... + n({n!})( - x) + n!)

Recursive definition, closed form, and generating function

One can also define the Laguerre polynomials recursively, defining the first two polynomials as$$L\_0(x)\; =\; 1$$$$L\_1(x)\; =\; 1\; -\; x$$and then using the following recurrence relation for any :$$L\_(x)\; =\; \backslash frac.$$Furthermore,$$x\; L\text{'}\_n(x)\; =\; nL\_n\; (x)\; -\; nL\_(x).$$

In solution of some boundary value problems, the characteristic values can be useful:$$L\_(0)\; =\; 1,\; L\_\text{'}(0)\; =\; -k.$$

The closed form is$$L\_n(x)=\backslash sum\_^n\; \backslash binom\backslash frac\; x^k\; .$$

The generating function for them likewise follows, $$\backslash sum\_^\backslash infty\; t^n\; L\_n(x)=\; \backslash frac\; e^.$$The operator form is$$L\_n(x)\; =\; \backslash frace^x\; \backslash frac\; (x^n\; e^)$$

Polynomials of negative index can be expressed using the ones with positive index:$$L\_(x)=e^xL\_(-x).$$

Generalized Laguerre polynomials

For arbitrary real α the polynomial solutions of the differential equation^[2] $$x\backslash ,y$$ + \left(\alpha +1 - x\right) y' + n\,y = 0are called generalized Laguerre polynomials, or associated Laguerre polynomials.

One can also define the generalized Laguerre polynomials recursively, defining the first two polynomials as$$L^\_0(x)\; =\; 1$$$$L^\_1(x)\; =\; 1\; +\; \backslash alpha\; -\; x$$

and then using the following recurrence relation for any :$$L^\_(x)\; =\; \backslash frac.$$

The simple Laguerre polynomials are the special case of the generalized Laguerre polynomials:$$L^\_n(x)\; =\; L\_n(x).$$

The Rodrigues formula for them is$$L\_n^(x)\; =\; \backslash left(e^\; x^\backslash right)=\; \backslash frac\backslash left(\backslash frac-1\backslash right)^nx^.$$

The generating function for them is$$\backslash sum\_^\backslash infty\; t^n\; L^\_n(x)=\; \backslash frac\; e^.$$

Explicit examples and properties of the generalized Laguerre polynomials

• Laguerre functions are defined by confluent hypergeometric functions and Kummer's transformation as^[3] $$L\_n^(x)\; :=\; M(-n,\backslash alpha+1,x).$$ where $$ is a generalized binomial coefficient. When is an integer the function reduces to a polynomial of degree . It has the alternative expression^[4] $$L\_n^(x)=\; \backslash frac\; U(-n,\backslash alpha+1,x)$$ in terms of Kummer's function of the second kind.
• The closed form for these generalized Laguerre polynomials of degree is^[5] $$L\_n^\; (x)\; =\; \backslash sum\_^n\; (-1)^i\; \backslash frac$$ derived by applying 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

D = (d)/(dx) and consider the differential operator M = xD ^{2+(\alpha+1)D} . Then \exp( - tM)x ^{n=(-1)} ^{nt} ^{nn!L} ^{(\alpha)} ^{n\left( (x)/(t)} \right) .

• The first few generalized Laguerre polynomials are:

n	L ^{(\alpha)} _{n} (x)
0	1
1	- x + \alpha +1
2	\tfrac{ 1 }{ 2 } (x ^{2-2\left(} \alpha +2 \right) x + \left( \alpha +1 \right) \left( \alpha +2 \right))
3	\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))
4	\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) … \left( \alpha +4 \right) x + \left( \alpha +1 \right) … \left( \alpha +4 \right))
5	\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) … \left( \alpha +5 \right) x ^{2-5\left(} \alpha +2 \right) … \left( \alpha +5 \right) x + \left( \alpha +1 \right) … \left( \alpha +5 \right))
6	\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) … \left( \alpha +6 \right) x ^{3+15\left(} \alpha +3 \right) … \left( \alpha +6 \right) x ^{2-6\left(} \alpha +2 \right) … \left( \alpha +6 \right) x + \left( \alpha +1 \right) … \left( \alpha +6 \right))
7	\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) … \left( \alpha +7 \right) x ^{4-35\left(} \alpha +4 \right) … \left( \alpha +7 \right) x ^{3+21\left(} \alpha +3 \right) … \left( \alpha +7 \right) x ^{2-7\left(} \alpha +2 \right) … \left( \alpha +7 \right) x + \left( \alpha +1 \right) … \left( \alpha +7 \right))
8	\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) … \left( \alpha +8 \right) x ^{5+70\left(} \alpha +5 \right) … \left( \alpha +8 \right) x ^{4-56\left(} \alpha +4 \right) … \left( \alpha +8 \right) x ^{3+28\left(} \alpha +3 \right) … \left( \alpha +8 \right) x ^{2-8\left(} \alpha +2 \right) … \left( \alpha +8 \right) x + \left( \alpha +1 \right) … \left( \alpha +8 \right))
9	\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) … \left( \alpha +9 \right) x ^{6-126\left(} \alpha +6 \right) … \left( \alpha +9 \right) x ^{5+126\left(} \alpha +5 \right) … \left( \alpha +9 \right) x ^{4-84\left(} \alpha +4 \right) … \left( \alpha +9 \right) x ^{3+36\left(} \alpha +3 \right) … \left( \alpha +9 \right) x ^{2-9\left(} \alpha +2 \right) … \left( \alpha +9 \right) x + \left( \alpha +1 \right) … \left( \alpha +9 \right))
10	\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) … \left( \alpha +10 \right) x ^{7+210\left(} \alpha +7 \right) … \left( \alpha +10 \right) x ^{6-252\left(} \alpha +6 \right) … \left( \alpha +10 \right) x ^{5+210\left(} \alpha +5 \right) … \left( \alpha +10 \right) x ^{4-120\left(} \alpha +4 \right) … \left( \alpha +10 \right) x ^{3+45\left(} \alpha +3 \right) … \left( \alpha +10 \right) x ^{2-10\left(} \alpha +2 \right) … \left( \alpha +10 \right) x + \left( \alpha +1 \right) … \left( \alpha +10 \right))

• The coefficient of the leading term is

Notes and References

1. Recherches sur les fonctions cylindriques et le développement des fonctions continues en séries. N. Sonine. Math. Ann.. 1880. 16. 1. 1–80. 10.1007/BF01459227. 121602983.
2. A&S p. 781
3. A&S p. 509
4. A&S p. 510
5. A&S p. 775