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 : 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 is an integer the function reduces to a polynomial of degree . 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 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 ; • -->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 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 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 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 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 .
Orthogonality The generalized Laguerre polynomials are
orthogonal over with respect to the measure with
weighting function : \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)}{{\alpha+n \choose n}{n \choose i}}; \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)^2>0. 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 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 For 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}}{{\pi}^{\frac{1}{2}}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:. • 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 ==