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 k ≥ 1: L_{k + 1}(x) = \frac{(2k + 1 - x)L_k(x) - k L_{k - 1}(x)}{k + 1}. Furthermore, x L'n(x) = nL_n (x) - nL{n-1}(x).

In solution of some boundary value problems, the characteristic values can be useful: L_{k}(0) = 1, L_{k}'(0) = -k.

The closed form is L_n(x)=\sum_{k=0}^n \binom{n}{k}\frac{(-1)^k}{k!} x^k .

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

Polynomials of negative index can be expressed using the ones with positive index: L_{-n}(x)=e^xL_{n-1}(-x).

Generalized Laguerre polynomials

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

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

and then using the following recurrence relation for any k ≥ 1: L^{(\alpha)}_{k + 1}(x) = \frac{(2k + 1 + \alpha - x)L^{(\alpha)}k(x) - (k + \alpha) L^{(\alpha)}{k - 1}(x)}{k + 1}.

The simple Laguerre polynomials are the special case of the generalized Laguerre polynomials: L^{(0)}_n(x) = L_n(x).

The Rodrigues formula for them is L_n^{(\alpha)}(x) = {x^{-\alpha} e^x \over n!}{d^n \over dx^n} \left(e^{-x} x^{n+\alpha}\right) = \frac{x^{-\alpha}}{n!}\left( \frac{d}{dx}-1\right)^nx^{n+\alpha}.

The generating function for them is \sum_{n=0}^\infty t^n L^{(\alpha)}_n(x)= \frac{1}{(1-t)^{\alpha+1}} e^{-tx/(1-t)}.

Properties

• Laguerre functions are defined by confluent hypergeometric functions and Kummer's transformation as L_n^{(\alpha)}(x) := {n+ \alpha \choose n} M(-n,\alpha+1,x). where {n+ \alpha \choose n} is a generalized binomial coefficient. When n is an integer the function reduces to a polynomial of degree n. It has the alternative expression L_n^{(\alpha)}(x)= \frac {(-1)^n}{n!} U(-n,\alpha+1,x) in terms of Kummer's function of the second kind.
• The closed form for these generalized Laguerre polynomials of degree n is L_n^{(\alpha)} (x) = \sum_{i=0}^n (-1)^i {n+\alpha \choose n-i} \frac{x^i}{i!} 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 = \frac{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(\frac{x}{t}\right).
• The first few generalized Laguerre polynomials are:

• The coefficient of the leading term is (−1)n/n!;
• --The constant term, which is the value at 0, is L_n^{(\alpha)}(0) = {n+\alpha\choose n} = \frac{\Gamma(n + \alpha + 1)}{n!, \Gamma(\alpha + 1)};
• The discriminant is\operatorname{Disc}\left(L_n^{(\alpha)}\right)=\prod_{j=1}^n j^{j-2 n+2}(j+\alpha)^{j-1}

As a contour integral

Given the generating function specified above, the polynomials may be expressed in terms of a contour integral L_n^{(\alpha)}(x)=\frac{1}{2\pi i}\oint_C\frac{e^{-xt/(1-t)}}{(1-t)^{\alpha+1},t^{n+1}} ; dt, where the contour circles the origin once in a counterclockwise direction without enclosing the essential singularity at 1

Recurrence relations

The addition formula for Laguerre polynomials: L^{(\alpha_{1}+\dots+\alpha_{r}+r-1)}{n}\left(x{1}+\dots+x_{r}\right)=\sum_{m_{1}+\dots+m_{r}=n}L^{(\alpha_{1})}{m{1}}\left(x_{1}\right)\cdots L^{(\alpha_{r})}{m{r}}\left(x_{r}\right).Laguerre's polynomials satisfy the recurrence relations L_n^{(\alpha)}(x)= \sum_{i=0}^n L_{n-i}^{(\alpha+i)}(y)\frac{(y-x)^i}{i!}, in particular L_n^{(\alpha+1)}(x)= \sum_{i=0}^n L_i^{(\alpha)}(x) and L_n^{(\alpha)}(x)= \sum_{i=0}^n {\alpha-\beta+n-i-1 \choose n-i} L_i^{(\beta)}(x), or L_n^{(\alpha)}(x)=\sum_{i=0}^n {\alpha-\beta+n \choose n-i} L_i^{(\beta- i)}(x); moreover \begin{align} L_n^{(\alpha)}(x)- \sum_{j=0}^{\Delta-1} {n+\alpha \choose n-j} (-1)^j \frac{x^j}{j!}&= (-1)^\Delta\frac{x^\Delta}{(\Delta-1)!} \sum_{i=0}^{n-\Delta} \frac{(n-i){n \choose i}}L_i^{(\alpha+\Delta)}(x)\[6pt] &=(-1)^\Delta\frac{x^\Delta}{(\Delta-1)!} \sum_{i=0}^{n-\Delta} \frac{(n-i){n \choose i}}L_i^{(n+\alpha+\Delta-i)}(x) \end{align}

They can be used to derive the four 3-point-rules \begin{align} L_n^{(\alpha)}(x) &= L_n^{(\alpha+1)}(x) - L_{n-1}^{(\alpha+1)}(x) = \sum_{j=0}^k {k \choose j}(-1)^j L_{n-j}^{(\alpha+k)}(x), \[10pt] n L_n^{(\alpha)}(x) &= (n + \alpha )L_{n-1}^{(\alpha)}(x) - x L_{n-1}^{(\alpha+1)}(x), \[10pt] & \text{or } \ \frac{x^k}{k!}L_n^{(\alpha)}(x) &= \sum_{i=0}^k (-1)^i {n+i \choose i} {n+\alpha \choose k-i} L_{n+i}^{(\alpha-k)}(x), \[10pt] n L_n^{(\alpha+1)}(x) &= (n-x) L_{n-1}^{(\alpha+1)}(x) + (n+\alpha)L_{n-1}^{(\alpha)}(x) \[10pt] x L_n^{(\alpha+1)}(x) &= (n+\alpha)L_{n-1}^{(\alpha)}(x)-(n-x)L_n^{(\alpha)}(x); \end{align}

combined they give this additional, useful recurrence relations\begin{align} L_n^{(\alpha)}(x)&= \left(2+\frac{\alpha-1-x}n \right)L_{n-1}^{(\alpha)}(x)- \left(1+\frac{\alpha-1}n \right)L_{n-2}^{(\alpha)}(x)\[10pt] &= \frac{\alpha+1-x}n L_{n-1}^{(\alpha+1)}(x)- \frac x n L_{n-2}^{(\alpha+2)}(x) \end{align}

Since L_n^{(\alpha)}(x) is a monic polynomial of degree n in \alpha, there is the partial fraction decomposition \begin{align} \frac{n!,L_n^{(\alpha)}(x)}{(\alpha+1)n} &= 1- \sum{j=1}^n (-1)^j \frac{j}{\alpha + j} {n \choose j}L_n^{(-j)}(x) \ &= 1- \sum_{j=1}^n \frac{x^j}{\alpha + j},,\frac{L_{n-j}^{(j)}(x)}{(j-1)!} \ &= 1-x \sum_{i=1}^n \frac{L_{n-i}^{(-\alpha)}(x) L_{i-1}^{(\alpha+1)}(-x)}{\alpha +i}. \end{align} The second equality follows by the following identity, valid for integer i and n and immediate from the expression of L_n^{(\alpha)}(x) in terms of Charlier polynomials: \frac{(-x)^i}{i!} L_n^{(i-n)}(x) = \frac{(-x)^n}{n!} L_i^{(n-i)}(x). For the third equality apply the fourth and fifth identities of this section.

Derivatives

Differentiating the power series representation of a generalized Laguerre polynomial k times leads to \frac{d^k}{d x^k} L_n^{(\alpha)} (x) = \begin{cases} (-1)^k L_{n-k}^{(\alpha+k)}(x) & \text{if } k\le n, \ 0 & \text{otherwise.} \end{cases}

This points to a special case () of the formula above: for integer the generalized polynomial may be written L_n^{(k)}(x)=(-1)^k\frac{d^kL_{n+k}(x)}{dx^k}, the shift by k sometimes causing confusion with the usual parenthesis notation for a derivative.

Moreover, the following equation holds: \frac{1}{k!} \frac{d^k}{d x^k} x^\alpha L_n^{(\alpha)} (x) = {n+\alpha \choose k} x^{\alpha-k} L_n^{(\alpha-k)}(x), which generalizes with Cauchy's formula to L_n^{(\alpha')}(x) = (\alpha'-\alpha) {\alpha'+ n \choose \alpha'-\alpha} \int_0^x \frac{t^\alpha (x-t)^{\alpha'-\alpha-1}}{x^{\alpha'}} L_n^{(\alpha)}(t),dt.

The derivative with respect to the second variable α has the form, \frac{d}{d \alpha}L_n^{(\alpha)}(x)= \sum_{i=0}^{n-1} \frac{L_i^{(\alpha)}(x)}{n-i}. The generalized Laguerre polynomials obey the differential equation x L_n^{(\alpha) \prime\prime}(x) + (\alpha+1-x)L_n^{(\alpha)\prime}(x) + n L_n^{(\alpha)}(x)=0, which may be compared with the equation obeyed by the kth derivative of the ordinary Laguerre polynomial,

x L_n^{[k] \prime\prime}(x) + (k+1-x)L_n^{[k]\prime}(x) + (n-k) L_n^{[k]}(x)=0, where L_n^{[k]}(x)\equiv\frac{d^kL_n(x)}{dx^k} for this equation only.

In Sturm–Liouville form the differential equation is

-\left(x^{\alpha+1} e^{-x}\cdot L_n^{(\alpha)}(x)^\prime\right)' = n\cdot x^\alpha e^{-x}\cdot L_n^{(\alpha)}(x),

which shows that is an eigenvector for the eigenvalue n.

Orthogonality

The generalized Laguerre polynomials are orthogonal over with respect to the measure with weighting function xα e−x:

\int_0^\infty x^\alpha e^{-x} L_n^{(\alpha)}(x)L_m^{(\alpha)}(x)dx=\frac{\Gamma(n+\alpha+1)}{n!} \delta_{n,m},

which follows from

\int_0^\infty x^{\alpha'-1} e^{-x} L_n^{(\alpha)}(x)dx= {\alpha-\alpha'+n \choose n} \Gamma(\alpha').

If \Gamma(x,\alpha+1,1) denotes the gamma distribution then the orthogonality relation can be written as

\int_0^{\infty} L_n^{(\alpha)}(x)L_m^{(\alpha)}(x)\Gamma(x,\alpha+1,1) dx={n+ \alpha \choose n}\delta_{n,m}.

The associated, symmetric kernel polynomial has the representations (Christoffel–Darboux formula)

\begin{align} K_n^{(\alpha)}(x,y) &:= \frac{1}{\Gamma(\alpha+1)} \sum_{i=0}^n \frac{L_i^{(\alpha)}(x) L_i^{(\alpha)}(y)}\[4pt] & =\frac{1}{\Gamma(\alpha+1)} \frac{L_n^{(\alpha)}(x) L_{n+1}^{(\alpha)}(y) - L_{n+1}^{(\alpha)}(x) L_n^{(\alpha)}(y)}{\frac{x-y}{n+1} {n+\alpha \choose n}} \[4pt] &= \frac{1}{\Gamma(\alpha+1)}\sum_{i=0}^n \frac{x^i}{i!} \frac{L_{n-i}^{(\alpha+i)}(x) L_{n-i}^{(\alpha+i+1)}(y)}; \end{align}

recursively

K_n^{(\alpha)}(x,y)=\frac{y}{\alpha+1} K_{n-1}^{(\alpha+1)}(x,y)+ \frac{1}{\Gamma(\alpha+1)} \frac{L_n^{(\alpha+1)}(x) L_n^{(\alpha)}(y)}.

Moreover,

y^\alpha e^{-y} K_n^{(\alpha)}(\cdot, y) \to \delta(y- \cdot).

Turán's inequalities can be derived here, which is L_n^{(\alpha)}(x)^2- L_{n-1}^{(\alpha)}(x) L_{n+1}^{(\alpha)}(x)= \sum_{k=0}^{n-1} \frac{n{n\choose k}} L_k^{(\alpha-1)}(x)^20.

The following integral is needed in the quantum mechanical treatment of the hydrogen atom,

\int_0^{\infty}x^{\alpha+1} e^{-x} \left[L_n^{(\alpha)} (x)\right]^2 dx= \frac{(n+\alpha)!}{n!}(2n+\alpha+1).

Series expansions

Let a function have the (formal) series expansion f(x)= \sum_{i=0}^\infty f_i^{(\alpha)} L_i^{(\alpha)}(x).

Then f_i^{(\alpha)}=\int_0^\infty \frac{L_i^{(\alpha)}(x)} \cdot \frac{x^\alpha e^{-x}}{\Gamma(\alpha+1)} \cdot f(x) ,dx .

The series converges in the associated Hilbert space L2[0, ∞) if and only if

| f |{L^2}^2 := \int_0^\infty \frac{x^\alpha e^{-x}}{\Gamma(\alpha+1)} | f(x)|^2 , dx = \sum{i=0}^\infty {i+\alpha \choose i} |f_i^{(\alpha)}|^2

Further examples of expansions

Monomials are represented as \frac{x^n}{n!}= \sum_{i=0}^n (-1)^i {n+ \alpha \choose n-i} L_i^{(\alpha)}(x), while binomials have the parametrization {n+x \choose n}= \sum_{i=0}^n \frac{\alpha^i}{i!} L_{n-i}^{(x+i)}(\alpha).

This leads directly to e^{-\gamma x}= \sum_{i=0}^\infty \frac{\gamma^i}{(1+\gamma)^{i+\alpha+1}} L_i^{(\alpha)}(x) \qquad \text{convergent iff } \Re(\gamma) -\tfrac{1}{2} for the exponential function. The incomplete gamma function has the representation \Gamma(\alpha,x)=x^\alpha e^{-x} \sum_{i=0}^\infty \frac{L_i^{(\alpha)}(x)}{1+i} \qquad \left(\Re(\alpha)-1 , x 0\right).

Asymptotics

In terms of elementary functions

Perron's formulaFor any fixed positive integer M, fixed real number \alpha, fixed and bounded interval [c, d] \subset (0, + \infty) , uniformly for x \in [c, d] , at n \to \infty :L^{(\alpha)}{n}\left(x\right)=\frac{n^{\frac{1}{2}\alpha-\frac{1}{4}}{\mathrm{e}}^{\frac{1}{2}x}}x^{\frac{1}{2}\alpha+\frac{1}{4}}}\left(\cos\theta{n}^{(\alpha)}(x)\left(\sum_{m=0}^{M-1}\frac{a_{m}(x)}{n^{\frac{1}{2}m}}+O\left(\frac{1}{n^{\frac{1}{2}M}}\right)\right)+\sin\theta_{n}^{(\alpha)}(x)\left(\sum_{m=1}^{M-1}\frac{b_{m}(x)}{n^{\frac{1}{2}m}}+O\left(\frac{1}{n^{\frac{1}{2}M}}\right)\right)\right) where \theta_{n}^{(\alpha)}(x) :=2(nx)^{\frac{1}{2}}-\left(\tfrac{1}{2}\alpha+\tfrac{1}{4}\right)\pi. and a_0, b_1, a_1, b_2, \dots are functions depending on \alpha, x but not n , and regular for x 0 . The first few ones are:\begin{aligned} & a_0(x)=1 \ & a_1(x)=0 \ & b_1(x)=\frac{1}{48 x^{\frac{1}{2}}}\left(4 x^2-24 (\alpha + 1) x+3-12 \alpha^2\right) \end{aligned} This is Perron's formula. There is also a generalization for x \in \mathbb C \setminus 0, \infty) . [Fejér's formula is a special case of Perron's formula with M = 1.

In terms of Bessel functions

The Mehler–Heine formula states:

: \lim_{n \to \infty} n^{-\alpha}L_n^{(\alpha)}\left(\frac{z^2}{4n}\right) = \left(\frac{z}{2}\right)^{-\alpha} J_\alpha(z),

where J_\alpha is a Bessel function of the first kind.

See also:.

In terms of Airy functions

Let \nu =4 n+2 \alpha+2. Let \operatorname{Ai} be the Airy function. Let \alpha be arbitrary and real, \epsilon and \omega be positive and fixed.

The Plancherel–Rotach asymptotics formulas:

• for x =\nu\cos^2\varphi and \epsilon\leq \varphi \leq \tfrac{\pi}{2} -\epsilon n^{-1/2}, uniformly at n \to \infty :

:: e^{-x/2}L^{(\alpha )}_n(x) =(-1)^{n}(\pi \sin \varphi)^{-1/2}x^{-\alpha/2-1/4}n^{\alpha/2-1/4} \big{\sin\left[\left(n+\tfrac{\alpha+1}{2}\right)(\sin 2\varphi-2\varphi)+3\pi/4\right] +(nx)^{-1/2}\mathcal{O}(1)\big}

• for x = \nu\cosh^2\varphi and \epsilon\leq \varphi \leq \omega, uniformly at n \to \infty :

:: e^{-x/2}L^{(\alpha )}_n(x) =\tfrac{1}{2}(-1)^{n}(\pi \sinh \varphi )^{-1/2}x^{-\alpha/2-1/4}n^{\alpha /2-1/4} \exp\left[\left(n+\tfrac{\alpha+1}{2}\right)(2\varphi-\sinh 2\varphi)\right] {1+\mathcal{O}\left(n^{-1}\right)}

• for x =\nu -2(2n/3)^{1/3}t and t complex and bounded, uniformly at n \to \infty :

:: e^{-x/2}L^{(\alpha)}_n(x) =(-1)^n\pi^{-1}2^{-\alpha-1/3}3^{1/3}n^{-1/3} \bigg{\pi \operatorname{Ai}(-3^{-1/3}t)+\mathcal{O}\left(n^{-2/3}\right)\bigg}

See DLMF for higher-order terms.

Zeroes

Notation

j_{\alpha, m} is the m-th positive zero of the Bessel function J_\alpha(x).

a_m is the m-th zero of the Airy function \operatorname{Ai}(x), in descending order: 0 a_1 a_2 \cdots.

\nu =4 n+2 \alpha+2.

If \alpha -1, then L_n^{(\alpha)} has n real roots. Thus in this section we assume \alpha -1 by default.

x_1 are the real roots of L_n^{(\alpha)}.

Note that \left((-1)^{n-i} L_{n-i}^{(\alpha)}\right)_{i=0}^n is a Sturm chain.

Inequalities

For \alpha -1, we have these bounds:

• x_1
• x_1
• x_1
• x_n \leq 2 n+\alpha-1+2 \sqrt{(n-2)(n+\alpha-1)} when n \geq 2
• x_n 4n + \alpha - 16 \sqrt{2n}
• x_n 3n-4
• x_n 2n + \alpha - 1
• x_n 2 n+\alpha-2+\sqrt{n^2-2 n+\alpha n+2}
• \begin{aligned} & (n+2) x_1 &\geq\left(n-1-\sqrt{n^2+(n+2)(\alpha+1)}\right)^2-1 \ & (n+2) x_n &\leq\left(n-1+\sqrt{n^2+(n+2)(\alpha+1)}\right)^2-1 \end{aligned}
• \begin{aligned} x_1 & \frac 12 \nu - 3 -\sqrt{1+4(n-1)(n+\alpha-1)} \ x_n & \end{aligned}

For fixed k = 1, \dots, n,\begin{aligned} \nu x_k&j_{\alpha, k}^2 \ x_k& j_{\alpha, k}\ x_k & x_k & \end{aligned}For fixed k, we have \lim_{n \to \infty} \nu x_k = j_{\alpha, k}^2, so the first inequality is sharp.

See also.

Electrostatics

The zeroes satisfy the Stieltjes relations:\begin{aligned} \sum_{1 \leq j \leq n, i \neq j} \frac{1}{x_{i} - x_{j}} &= \frac 12 \left(1 - \frac{\alpha + 1}{x_i}\right)\ \sum_{1 \leq j \leq n} \frac{1}{x_{j}} &= \frac{n}{\alpha + 1}\ \sum_{1 \leq j \leq n, i \neq j} \frac{1}{(x_{i} - x_{j})^2} &= -\frac{(\alpha + 1)(\alpha + 5)}{12 x_i^2} + \frac{2n + \alpha + 1}{6x_i}- \frac{1}{12}\ \sum_{1 \leq j \leq n, i \neq j} \frac{1}{(x_{i} - x_{j})^3} &= -\frac{(\alpha + 1)(\alpha + 3)}{8 x_i^3} + \frac{2n + \alpha + 1}{8x_i^2}\ \end{aligned}The first relation can be interpreted physically. Fix an electric particle at origin with charge +\frac{\alpha + 1}{2}, and produce a constant electric field of strength -\frac 12 . Then, place n electric particles with charge +1 . The first relation states that the zeroes of L_n^{(\alpha)} are the equilibrium positions of the particles.

As the zeroes specify the polynomial up to scaling, this provides an alternative way to uniquely characterize the Laguerre polynomials.

The zeroes also satisfy\sum_{i=1}^n \frac{1}{x-x_i }=-\sum_{k=0}^{\infty} S_{k+1} x^k, \quad S_k := \sum_{i = 1}^n x_i^{-k}which allows the following boundS_m^{-1 / m}

Limit distribution

Let F_n(t) := \frac 1n #{i : x_{i} \leq t} be the cumulative distribution function for the roots, then we have the limit law\lim_{n \to \infty} F_n(4n t) = \frac 2\pi \int_{0}^t \sqrt{\frac{1-s}{s}} ds \quad \forall t \in (0, 1] which can be interpreted as the limit distribution of the Wishart ensemble spectrum.

For fixed \alpha -1 and fixed k, as n \to \infty,\begin{aligned} x_{n+1-k}= & \nu+2^{2 / 3} a_k \nu^{1 / 3}+\frac{1}{5} 2^{4 / 3} a_k^2 \nu^{-1 / 3}+\left(\frac{11}{35}-\alpha^2-\frac{12}{175} a_k^3\right) \nu^{-1} \ & +\left(\frac{16}{1575} a_k+\frac{92}{7875} a_k^4\right) 2^{2 / 3} \nu^{-5 / 3}-\left(\frac{15152}{3031875} a_k^5+\frac{1088}{121275} a_k^2\right) 2^{1 / 3} \nu^{-7 / 3}+\mathcal{O}\left(\nu^{-3}\right), \end{aligned}

For \alpha \in (-1, 0),\begin{aligned} x_1=\frac{\alpha+1}{n} & +\frac{n-1}{2}\left(\frac{\alpha+1}{n}\right)^2-\frac{n^2+3 n-4}{12}\left(\frac{\alpha+1}{n}\right)^3 \ & +\frac{7 n^3+6 n^2+23 n-36}{144}\left(\frac{\alpha+1}{n}\right)^4 \ & -\frac{293 n^4+210 n^3+235 n^2+990 n-1728}{8640}\left(\frac{\alpha+1}{n}\right)^5+\cdots \end{aligned}

In quantum mechanics

In quantum mechanics the Schrödinger equation for the hydrogen-like atom is exactly solvable by separation of variables in spherical coordinates. The radial part of the wave function is a (generalized) Laguerre polynomial.

Vibronic transitions in the Franck-Condon approximation can also be described using Laguerre polynomials.

Multiplication theorems

Erdélyi gives the following two multiplication theorems

\begin{align} & t^{n+1+\alpha} e^{(1-t) z} L_n^{(\alpha)}(z t)=\sum_{k=n}^\infty {k \choose n}\left(1-\frac 1 t\right)^{k-n} L_k^{(\alpha)}(z), \[6pt] & e^{(1-t)z} L_n^{(\alpha)}(z t)=\sum_{k=0}^\infty \frac{(1-t)^k z^k}{k!}L_n^{(\alpha+k)}(z). \end{align}

Relation to Hermite polynomials

The generalized Laguerre polynomials are related to the Hermite polynomials: \begin{align} H_{2n}(x) &= (-1)^n 2^{2n} n! L_n^{(-1/2)} (x^2) \[4pt] H_{2n+1}(x) &= (-1)^n 2^{2n+1} n! x L_n^{(1/2)} (x^2) \end{align} where the H**n(x) are the Hermite polynomials based on the weighting function exp(−x2), the so-called "physicist's version."

Because of this, the generalized Laguerre polynomials arise in the treatment of the quantum harmonic oscillator.

Applying the addition formula,(-1)^n 2^{2n} n! , L^{\left(\frac{r}{2}-1\right)}{n}\Bigl(z_1^2+\cdots+z_r^2\Bigr) =\sum{m_1+\cdots+m_r=n} \prod_{i=1}^r H_{2m_i}(z_i).

Relation to hypergeometric functions

The Laguerre polynomials may be defined in terms of hypergeometric functions, specifically the confluent hypergeometric functions, as L^{(\alpha)}_n(x) = {n+\alpha \choose n} M(-n,\alpha+1,x) =\frac{(\alpha+1)_n} {n!} ,_1F_1(-n,\alpha+1,x) where (a)_n is the Pochhammer symbol (which in this case represents the rising factorial).

Hardy–Hille formula

The generalized Laguerre polynomials satisfy the Hardy–Hille formula \sum_{n=0}^\infty \frac{n!,\Gamma\left(\alpha + 1\right)}{\Gamma\left(n+\alpha+1\right)}L_n^{(\alpha)}(x)L_n^{(\alpha)}(y)t^n=\frac{1}{(1-t)^{\alpha + 1}}e^{-(x+y)t/(1-t)},0F_1\left(;\alpha + 1;\frac{xyt}{(1-t)^2}\right), where the series on the left converges for \alpha-1 and |t|. Using the identity ,0F_1(;\alpha + 1;z)=,\Gamma(\alpha + 1) z^{-\alpha/2} I\alpha\left(2\sqrt{z}\right), (see generalized hypergeometric function), this can also be written as \sum{n=0}^\infty \frac{n!}{\Gamma(1+\alpha+n)}L_n^{(\alpha)}(x)L_n^{(\alpha)}(y) t^n = \frac{1}{(xyt)^{\alpha/2}(1-t)}e^{-(x+y)t/(1-t)} I_\alpha \left(\frac{2\sqrt{xyt}}{1-t}\right).where I_\alpha denotes the modified Bessel function of the first kind, defined as I_\alpha(z) = \sum_{k=0}^\infty \frac{1}{k!, \Gamma(k+\alpha+1)} \left(\frac{z}{2}\right)^{2k+\alpha} This formula is a generalization of the Mehler kernel for Hermite polynomials, which can be recovered from it by setting the Hermite polynomials as a special case of the associated Laguerre polynomials.

Substitute t \mapsto -t/y and take the y \to \infty limit, we obtain \sum_{n=0}^\infty \frac{t^n}{\Gamma(n+1+\alpha)} L_n^{(\alpha)}(x)

\frac{e^t}{(-xt)^{\alpha/2}}I_{\alpha}(2\sqrt{-xt}).The formula is named after G. H. Hardy and Einar Hille.

Physics convention

The generalized Laguerre polynomials are used to describe the quantum wavefunction for hydrogen atom orbitals. The convention used throughout this article expresses the generalized Laguerre polynomials as

L_n^{(\alpha)}(x) = \frac{\Gamma(\alpha + n + 1)}{\Gamma(\alpha + 1) n!} ,_1F_1(-n; \alpha + 1; x),

where ,_1F_1(a;b;x) is the confluent hypergeometric function. In the physics literature, the generalized Laguerre polynomials are instead defined as

\bar{L}_n^{(\alpha)}(x) = \frac{\left[\Gamma(\alpha + n + 1)\right]^2}{\Gamma(\alpha + 1)n!} ,_1F_1(-n; \alpha + 1; x).

The physics version is related to the standard version by

\bar{L}_n^{(\alpha)}(x) = (n+\alpha)! L_n^{(\alpha)}(x).

There is yet another, albeit less frequently used, convention in the physics literature

\tilde{L}n^{(\alpha)}(x) = (-1)^{\alpha}\bar{L}{n-\alpha}^{(\alpha)}.

Umbral calculus convention

Generalized Laguerre polynomials are linked to Umbral calculus by being Sheffer sequences for D/(D-I) when multiplied by n!. In Umbral Calculus convention, the default Laguerre polynomials are defined to be\mathcal L_n(x) = n!L_n^{(-1)}(x) = \sum_{k=0}^n L(n,k) (-x)^kwhere L(n,k) = \binom{n-1}{k-1} \frac{n!}{k!} are the signless Lah numbers. (\mathcal L_n(x)){n\in\N} is a sequence of polynomials of binomial type, ie they satisfy\mathcal L_n(x+y) = \sum{k=0}^n \binom{n}{k} \mathcal L_k(x) \mathcal L_{n-k}(y)

Notes

References

References

1. N. Sonine. (1880). "Recherches sur les fonctions cylindriques et le développement des fonctions continues en séries". [[Math. Ann.]].
2. A&S p. 781
3. A&S p. 509
4. A&S p. 510
5. A&S p. 775
6. "DLMF: §18.16 Zeros ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials".
7. "DLMF: §18.18 Sums ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials".
8. (1997). "Identities for families of orthogonal polynomials and special functions". Integral Transforms and Special Functions.
9. "Associated Laguerre Polynomial".
10. "DLMF: §18.15 Asymptotic Approximations ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials".
11. Perron, Oskar. (1921-01-01). "Über das Verhalten einer ausgearteten hypergeometrischen Reihe bei unbegrenztem Wachstum eines Parameters.".
12. Szegő, p. 198.
13. Turán, Pál. (1970). "Asymptotikus Értékek Meghatározásáról". Birkhäuser Basel.
14. 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
15. Szegő, pp. 200–201
16. Driver, K.. (January 2013). "Inequalities for Extreme Zeros of Some Classical Orthogonal andq-orthogonal Polynomials". Mathematical Modelling of Natural Phenomena.
17. Gatteschi, Luigi. (2002-07-01). "Asymptotics and bounds for the zeros of Laguerre polynomials: a survey". Journal of Computational and Applied Mathematics.
18. Dimitrov, Dimitar K.. (2009-12-01). "Monotonicity of zeros of Laguerre polynomials". Journal of Computational and Applied Mathematics.
19. {{Harvard citation. Szegő. 1975
20. Marcellán, F.. (2007-10-15). "Electrostatic models for zeros of polynomials: Old, new, and some open problems". Journal of Computational and Applied Mathematics.
21. {{Harvard citation. Szegő. 1975
22. Gupta, Dharma P.. (2007). "Inequalities for the smallest zeros of Laguerre polynomials and their -analogues.". JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only].
23. Gawronski, Wolfgang. (1987-07-01). "On the asymptotic distribution of the zeros of Hermite, Laguerre, and Jonquière polynomials". Journal of Approximation Theory.
24. Ratner, Schatz, Mark A., George C.. (2001). "Quantum Mechanics in Chemistry". Prentice Hall.
25. (2015-06-24). "Resolving the ambiguity in the relation between Stokes shift and Huang–Rhys parameter". Physical Chemistry Chemical Physics.
26. C. Truesdell, "[http://www.pnas.org/cgi/reprint/36/12/752.pdf On the Addition and Multiplication Theorems for the Special Functions]", ''Proceedings of the National Academy of Sciences, Mathematics'', (1950) pp. 752–757.
27. Szegő, p. 102.
28. Al-Salam, W. A.. (1964-03-01). "Operational representations for the Laguerre and other polynomials". Duke Mathematical Journal.
29. Szegő, page 102, Equation (5.1.16)
30. G. H. Hardy, “Summation of a series of polynomials of Laguerre,” J. London Math. Soc., v. 7, 1932, pp. 138–139; addendum, 192.
31. E. Hille, “On Laguerre’s series. I, II, III,” Proc. Nat. Acad. Sci. U.S.A., v. 12, 1926, pp. 261–269; 348–352.
32. (2005). "Introduction to quantum mechanics". Pearson Prentice Hall.
33. (2011). "Modern quantum mechanics". Addison-Wesley.
34. (1998). "Quantum mechanics". Wiley.
35. (1965). "Handbook of mathematical functions, with formulas, graphs, and mathematical tables". Dover Publications.
36. (1968). "Quantum mechanics". McGraw-Hill.
37. (2014). "Quantum Mechanics.". Dover Publications.
38. (2006). "Mathematical methods in the physical sciences". Wiley.
39. (1973-06-01). "On the foundations of combinatorial theory. VIII. Finite operator calculus". Journal of Mathematical Analysis and Applications.