Sequence of differential equation solutions
Complex color plot of the Laguerre polynomial L n(x) with n as -1 divided by 9 and x as z to the power of 4 from -2-2i to 2+2i In mathematics , the Laguerre polynomials , named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation:
x y ″ + ( 1 − x ) y ′ + n y = 0 , y = y ( x ) {\displaystyle xy''+(1-x)y'+ny=0,\ y=y(x)} which is a second-order
linear differential equation . This equation has
nonsingular solutions only if
n is a non-negative integer.
Sometimes the name Laguerre polynomials is used for solutions of
x y ″ + ( α + 1 − x ) y ′ + n y = 0 . {\displaystyle xy''+(\alpha +1-x)y'+ny=0~.} where
n 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 n is not necessarily a non-negative integer.
The Laguerre polynomials are also used for Gauss–Laguerre quadrature to numerically compute integrals of the form
∫ 0 ∞ f ( x ) e − x d x . {\displaystyle \int _{0}^{\infty }f(x)e^{-x}\,dx.} These polynomials, usually denoted L 0 , L 1 , ..., are a polynomial sequence which may be defined by the Rodrigues formula ,
L n ( x ) = e x n ! d n d x n ( e − x x n ) = 1 n ! ( d d x − 1 ) n x n , {\displaystyle L_{n}(x)={\frac {e^{x}}{n!}}{\frac {d^{n}}{dx^{n}}}\left(e^{-x}x^{n}\right)={\frac {1}{n!}}\left({\frac {d}{dx}}-1\right)^{n}x^{n},} reducing to the closed form of a following section.
They are orthogonal polynomials with respect to an inner product
⟨ f , g ⟩ = ∫ 0 ∞ f ( x ) g ( x ) e − x d x . {\displaystyle \langle f,g\rangle =\int _{0}^{\infty }f(x)g(x)e^{-x}\,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 [ edit ] These are the first few Laguerre polynomials:
n L n ( x ) {\displaystyle L_{n}(x)\,} 0 1 {\displaystyle 1\,} 1 − x + 1 {\displaystyle -x+1\,} 2 1 2 ( x 2 − 4 x + 2 ) {\displaystyle {\tfrac {1}{2}}(x^{2}-4x+2)\,} 3 1 6 ( − x 3 + 9 x 2 − 18 x + 6 ) {\displaystyle {\tfrac {1}{6}}(-x^{3}+9x^{2}-18x+6)\,} 4 1 24 ( x 4 − 16 x 3 + 72 x 2 − 96 x + 24 ) {\displaystyle {\tfrac {1}{24}}(x^{4}-16x^{3}+72x^{2}-96x+24)\,} 5 1 120 ( − x 5 + 25 x 4 − 200 x 3 + 600 x 2 − 600 x + 120 ) {\displaystyle {\tfrac {1}{120}}(-x^{5}+25x^{4}-200x^{3}+600x^{2}-600x+120)\,} 6 1 720 ( x 6 − 36 x 5 + 450 x 4 − 2400 x 3 + 5400 x 2 − 4320 x + 720 ) {\displaystyle {\tfrac {1}{720}}(x^{6}-36x^{5}+450x^{4}-2400x^{3}+5400x^{2}-4320x+720)\,} n 1 n ! ( ( − x ) n + n 2 ( − x ) n − 1 + ⋯ + n ( n ! ) ( − x ) + n ! ) {\displaystyle {\tfrac {1}{n!}}((-x)^{n}+n^{2}(-x)^{n-1}+\dots +n({n!})(-x)+n!)\,}
The first six Laguerre polynomials. Recursive definition, closed form, and generating function [ edit ] One can also define the Laguerre polynomials recursively, defining the first two polynomials as
L 0 ( x ) = 1 {\displaystyle L_{0}(x)=1} L 1 ( x ) = 1 − x {\displaystyle L_{1}(x)=1-x} and then using the following
recurrence relation for any
k ≥ 1:
L k + 1 ( x ) = ( 2 k + 1 − x ) L k ( x ) − k L k − 1 ( x ) k + 1 . {\displaystyle L_{k+1}(x)={\frac {(2k+1-x)L_{k}(x)-kL_{k-1}(x)}{k+1}}.} Furthermore,
x L n ′ ( x ) = n L n ( x ) − n L n − 1 ( x ) . {\displaystyle xL'_{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 . {\displaystyle L_{k}(0)=1,L_{k}'(0)=-k.} The closed form is
L n ( x ) = ∑ k = 0 n ( n k ) ( − 1 ) k k ! x k . {\displaystyle L_{n}(x)=\sum _{k=0}^{n}{\binom {n}{k}}{\frac {(-1)^{k}}{k!}}x^{k}.} The generating function for them likewise follows,
∑ n = 0 ∞ t n L n ( x ) = 1 1 − t e − t x / ( 1 − t ) . {\displaystyle \sum _{n=0}^{\infty }t^{n}L_{n}(x)={\frac {1}{1-t}}e^{-tx/(1-t)}.} The operator form is
L n ( x ) = 1 n ! e x d n d x n ( x n e − x ) {\displaystyle 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 x L n − 1 ( − x ) . {\displaystyle L_{-n}(x)=e^{x}L_{n-1}(-x).} Generalized Laguerre polynomials [ edit ] For arbitrary real α the polynomial solutions of the differential equation[2]
x y ″ + ( α + 1 − x ) y ′ + n y = 0 {\displaystyle 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 0 ( α ) ( x ) = 1 {\displaystyle L_{0}^{(\alpha )}(x)=1} L 1 ( α ) ( x ) = 1 + α − x {\displaystyle L_{1}^{(\alpha )}(x)=1+\alpha -x} and then using the following recurrence relation for any k ≥ 1 :
L k + 1 ( α ) ( x ) = ( 2 k + 1 + α − x ) L k ( α ) ( x ) − ( k + α ) L k − 1 ( α ) ( x ) k + 1 . {\displaystyle L_{k+1}^{(\alpha )}(x)={\frac {(2k+1+\alpha -x)L_{k}^{(\alpha )}(x)-(k+\alpha )L_{k-1}^{(\alpha )}(x)}{k+1}}.} The simple Laguerre polynomials are the special case α = 0 of the generalized Laguerre polynomials:
L n ( 0 ) ( x ) = L n ( x ) . {\displaystyle L_{n}^{(0)}(x)=L_{n}(x).} The Rodrigues formula for them is
L n ( α ) ( x ) = x − α e x n ! d n d x n ( e − x x n + α ) = x − α n ! ( d d x − 1 ) n x n + α . {\displaystyle 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)^{n}x^{n+\alpha }.} The generating function for them is
∑ n = 0 ∞ t n L n ( α ) ( x ) = 1 ( 1 − t ) α + 1 e − t x / ( 1 − t ) . {\displaystyle \sum _{n=0}^{\infty }t^{n}L_{n}^{(\alpha )}(x)={\frac {1}{(1-t)^{\alpha +1}}}e^{-tx/(1-t)}.} The first few generalized Laguerre polynomials, Ln (k ) (x ) Explicit examples and properties of the generalized Laguerre polynomials [ edit ] Laguerre functions are defined by confluent hypergeometric functions and Kummer's transformation as[3] L n ( α ) ( x ) := ( n + α n ) M ( − n , α + 1 , x ) . {\displaystyle L_{n}^{(\alpha )}(x):={n+\alpha \choose n}M(-n,\alpha +1,x).} where ( n + α n ) {\textstyle {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[4] L n ( α ) ( x ) = ( − 1 ) n n ! U ( − n , α + 1 , x ) {\displaystyle 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[5] L n ( α ) ( x ) = ∑ i = 0 n ( − 1 ) i ( n + α n − i ) x i i ! {\displaystyle 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 = d d x {\displaystyle D={\frac {d}{dx}}} and consider the differential operator M = x D 2 + ( α + 1 ) D {\displaystyle M=xD^{2}+(\alpha +1)D} . Then exp ( − t M ) x n = ( − 1 ) n t n n ! L n ( α ) ( x t ) {\displaystyle \exp(-tM)x^{n}=(-1)^{n}t^{n}n!L_{n}^{(\alpha )}\left({\frac {x}{t}}\right)} .[citation needed ] The first few generalized Laguerre polynomials are: L 0 ( α ) ( x ) = 1 L 1 ( α ) ( x ) = − x + ( α + 1 ) L 2 ( α ) ( x ) = x 2 2 − ( α + 2 ) x + ( α + 1 ) ( α + 2 ) 2 L 3 ( α ) ( x ) = − x 3 6 + ( α + 3 ) x 2 2 − ( α + 2 ) ( α + 3 ) x 2 + ( α + 1 ) ( α + 2 ) ( α + 3 ) 6 {\displaystyle {\begin{aligned}L_{0}^{(\alpha )}(x)&=1\\L_{1}^{(\alpha )}(x)&=-x+(\alpha +1)\\L_{2}^{(\alpha )}(x)&={\frac {x^{2}}{2}}-(\alpha +2)x+{\frac {(\alpha +1)(\alpha +2)}{2}}\\L_{3}^{(\alpha )}(x)&={\frac {-x^{3}}{6}}+{\frac {(\alpha +3)x^{2}}{2}}-{\frac {(\alpha +2)(\alpha +3)x}{2}}+{\frac {(\alpha +1)(\alpha +2)(\alpha +3)}{6}}\end{aligned}}} The coefficient of the leading term is (−1)n /n ! ; The constant term , which is the value at 0, is L n ( α ) ( 0 ) = ( n + α n ) = Γ ( n + α + 1 ) n ! Γ ( α + 1 ) ; {\displaystyle L_{n}^{(\alpha )}(0)={n+\alpha \choose n}={\frac {\Gamma (n+\alpha +1)}{n!\,\Gamma (\alpha +1)}};} If α is non-negative, then L n (α ) has n real , strictly positive roots (notice that ( ( − 1 ) n − i L n − i ( α ) ) i = 0 n {\displaystyle \left((-1)^{n-i}L_{n-i}^{(\alpha )}\right)_{i=0}^{n}} is a Sturm chain ), which are all in the interval ( 0 , n + α + ( n − 1 ) n + α ] . {\displaystyle \left(0,n+\alpha +(n-1){\sqrt {n+\alpha }}\,\right].} [citation needed ] The polynomials' asymptotic behaviour for large n , but fixed α and x > 0 , is given by[6] [7] L n ( α ) ( x ) = n α 2 − 1 4 π e x 2 x α 2 + 1 4 sin ( 2 n x − π 2 ( α − 1 2 ) ) + O ( n α 2 − 3 4 ) , L n ( α ) ( − x ) = ( n + 1 ) α 2 − 1 4 2 π e − x / 2 x α 2 + 1 4 e 2 x ( n + 1 ) ⋅ ( 1 + O ( 1 n + 1 ) ) , {\displaystyle {\begin{aligned}&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{aligned}}} and summarizing by L n ( α ) ( x n ) n α ≈ e x / 2 n ⋅ J α ( 2 x ) x α , {\displaystyle {\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 }}},} where J α {\displaystyle J_{\alpha }} is the Bessel function . As a contour integral [ edit ] Given the generating function specified above, the polynomials may be expressed in terms of a contour integral
L n ( α ) ( x ) = 1 2 π i ∮ C e − x t / ( 1 − t ) ( 1 − t ) α + 1 t n + 1 d t , {\displaystyle 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 [ edit ] The addition formula for Laguerre polynomials:[8]
L n ( α + β + 1 ) ( x + y ) = ∑ i = 0 n L i ( α ) ( x ) L n − i ( β ) ( y ) . {\displaystyle L_{n}^{(\alpha +\beta +1)}(x+y)=\sum _{i=0}^{n}L_{i}^{(\alpha )}(x)L_{n-i}^{(\beta )}(y).} Laguerre's polynomials satisfy the recurrence relations
L n ( α ) ( x ) = ∑ i = 0 n L n − i ( α + i ) ( y ) ( y − x ) i i ! , {\displaystyle L_{n}^{(\alpha )}(x)=\sum _{i=0}^{n}L_{n-i}^{(\alpha +i)}(y){\frac {(y-x)^{i}}{i!}},} in particular
L n ( α + 1 ) ( x ) = ∑ i = 0 n L i ( α ) ( x ) {\displaystyle L_{n}^{(\alpha +1)}(x)=\sum _{i=0}^{n}L_{i}^{(\alpha )}(x)} and
L n ( α ) ( x ) = ∑ i = 0 n ( α − β + n − i − 1 n − i ) L i ( β ) ( x ) , {\displaystyle L_{n}^{(\alpha )}(x)=\sum _{i=0}^{n}{\alpha -\beta +n-i-1 \choose n-i}L_{i}^{(\beta )}(x),} or
L n ( α ) ( x ) = ∑ i = 0 n ( α − β + n n − i ) L i ( β − i ) ( x ) ; {\displaystyle L_{n}^{(\alpha )}(x)=\sum _{i=0}^{n}{\alpha -\beta +n \choose n-i}L_{i}^{(\beta -i)}(x);} moreover
L n ( α ) ( x ) − ∑ j = 0 Δ − 1 ( n + α n − j ) ( − 1 ) j x j j ! = ( − 1 ) Δ x Δ ( Δ − 1 ) ! ∑ i = 0 n − Δ ( n + α n − Δ − i ) ( n − i ) ( n i ) L i ( α + Δ ) ( x ) = ( − 1 ) Δ x Δ ( Δ − 1 ) ! ∑ i = 0 n − Δ ( n + α − i − 1 n − Δ − i ) ( n − i ) ( n i ) L i ( n + α + Δ − i ) ( x ) {\displaystyle {\begin{aligned}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+\alpha \choose n-\Delta -i}{(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+\alpha -i-1 \choose n-\Delta -i}{(n-i){n \choose i}}}L_{i}^{(n+\alpha +\Delta -i)}(x)\end{aligned}}} They can be used to derive the four 3-point-rules
L n ( α ) ( x ) = L n ( α + 1 ) ( x ) − L n − 1 ( α + 1 ) ( x ) = ∑ j = 0 k ( k j ) L n − j ( α + k ) ( x ) , n L n ( α ) ( x ) = ( n + α ) L n − 1 ( α ) ( x ) − x L n − 1 ( α + 1 ) ( x ) , or x k k ! L n ( α ) ( x ) = ∑ i = 0 k ( − 1 ) i ( n + i i ) ( n + α k − i ) L n + i ( α − k ) ( x ) , n L n ( α + 1 ) ( x ) = ( n − x ) L n − 1 ( α + 1 ) ( x ) + ( n + α ) L n − 1 ( α ) ( x ) x L n ( α + 1 ) ( x ) = ( n + α ) L n − 1 ( α ) ( x ) − ( n − x ) L n ( α ) ( x ) ; {\displaystyle {\begin{aligned}L_{n}^{(\alpha )}(x)&=L_{n}^{(\alpha +1)}(x)-L_{n-1}^{(\alpha +1)}(x)=\sum _{j=0}^{k}{k \choose j}L_{n-j}^{(\alpha +k)}(x),\\[10pt]nL_{n}^{(\alpha )}(x)&=(n+\alpha )L_{n-1}^{(\alpha )}(x)-xL_{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]nL_{n}^{(\alpha +1)}(x)&=(n-x)L_{n-1}^{(\alpha +1)}(x)+(n+\alpha )L_{n-1}^{(\alpha )}(x)\\[10pt]xL_{n}^{(\alpha +1)}(x)&=(n+\alpha )L_{n-1}^{(\alpha )}(x)-(n-x)L_{n}^{(\alpha )}(x);\end{aligned}}} combined they give this additional, useful recurrence relations
L n ( α ) ( x ) = ( 2 + α − 1 − x n ) L n − 1 ( α ) ( x ) − ( 1 + α − 1 n ) L n − 2 ( α ) ( x ) = α + 1 − x n L n − 1 ( α + 1 ) ( x ) − x n L n − 2 ( α + 2 ) ( x ) {\displaystyle {\begin{aligned}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{aligned}}} Since L n ( α ) ( x ) {\displaystyle L_{n}^{(\alpha )}(x)} is a monic polynomial of degree n {\displaystyle n} in α {\displaystyle \alpha } , there is the partial fraction decomposition
n ! L n ( α ) ( x ) ( α + 1 ) n = 1 − ∑ j = 1 n ( − 1 ) j j α + j ( n j ) L n ( − j ) ( x ) = 1 − ∑ j = 1 n x j α + j L n − j ( j ) ( x ) ( j − 1 ) ! = 1 − x ∑ i = 1 n L n − i ( − α ) ( x ) L i − 1 ( α + 1 ) ( − x ) α + i . {\displaystyle {\begin{aligned}{\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{aligned}}} The second equality follows by the following identity, valid for integer
i and
n and immediate from the expression of
L n ( α ) ( x ) {\displaystyle L_{n}^{(\alpha )}(x)} in terms of
Charlier polynomials :
( − x ) i i ! L n ( i − n ) ( x ) = ( − x ) n n ! L i ( n − i ) ( x ) . {\displaystyle {\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 of generalized Laguerre polynomials [ edit ] Differentiating the power series representation of a generalized Laguerre polynomial k times leads to
d k d x k L n ( α ) ( x ) = { ( − 1 ) k L n − k ( α + k ) ( x ) if k ≤ n , 0 otherwise. {\displaystyle {\frac {d^{k}}{dx^{k}}}L_{n}^{(\alpha )}(x)={\begin{cases}(-1)^{k}L_{n-k}^{(\alpha +k)}(x)&{\text{if }}k\leq n,\\0&{\text{otherwise.}}\end{cases}}} This points to a special case (α = 0 ) of the formula above: for integer α = k the generalized polynomial may be written
L n ( k ) ( x ) = ( − 1 ) k d k L n + k ( x ) d x k , {\displaystyle L_{n}^{(k)}(x)=(-1)^{k}{\frac {d^{k}L_{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:
1 k ! d k d x k x α L n ( α ) ( x ) = ( n + α k ) x α − k L n ( α − k ) ( x ) , {\displaystyle {\frac {1}{k!}}{\frac {d^{k}}{dx^{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 ( α ′ ) ( x ) = ( α ′ − α ) ( α ′ + n α ′ − α ) ∫ 0 x t α ( x − t ) α ′ − α − 1 x α ′ L n ( α ) ( t ) d t . {\displaystyle 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,[9]
d d α L n ( α ) ( x ) = ∑ i = 0 n − 1 L i ( α ) ( x ) n − i . {\displaystyle {\frac {d}{d\alpha }}L_{n}^{(\alpha )}(x)=\sum _{i=0}^{n-1}{\frac {L_{i}^{(\alpha )}(x)}{n-i}}.} This is evident from the contour integral representation below.
The generalized Laguerre polynomials obey the differential equation
x L n ( α ) ′ ′ ( x ) + ( α + 1 − x ) L n ( α ) ′ ( x ) + n L n ( α ) ( x ) = 0 , {\displaystyle xL_{n}^{(\alpha )\prime \prime }(x)+(\alpha +1-x)L_{n}^{(\alpha )\prime }(x)+nL_{n}^{(\alpha )}(x)=0,} which may be compared with the equation obeyed by the
k th derivative of the ordinary Laguerre polynomial,
x L n [ k ] ′ ′ ( x ) + ( k + 1 − x ) L n [ k ] ′ ( x ) + ( n − k ) L n [ k ] ( x ) = 0 , {\displaystyle xL_{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 ) ≡ d k L n ( x ) d x k {\displaystyle L_{n}^{[k]}(x)\equiv {\frac {d^{k}L_{n}(x)}{dx^{k}}}} for this equation only.
In Sturm–Liouville form the differential equation is
− ( x α + 1 e − x ⋅ L n ( α ) ( x ) ′ ) ′ = n ⋅ x α e − x ⋅ L n ( α ) ( x ) , {\displaystyle -\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 L (α) n is an eigenvector for the eigenvalue n .
Orthogonality [ edit ] The generalized Laguerre polynomials are orthogonal over [0, ∞) with respect to the measure with weighting function xα e −x :[10]
∫ 0 ∞ x α e − x L n ( α ) ( x ) L m ( α ) ( x ) d x = Γ ( n + α + 1 ) n ! δ n , m , {\displaystyle \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
∫ 0 ∞ x α ′ − 1 e − x L n ( α ) ( x ) d x = ( α − α ′ + n n ) Γ ( α ′ ) . {\displaystyle \int _{0}^{\infty }x^{\alpha '-1}e^{-x}L_{n}^{(\alpha )}(x)dx={\alpha -\alpha '+n \choose n}\Gamma (\alpha ').} If Γ ( x , α + 1 , 1 ) {\displaystyle \Gamma (x,\alpha +1,1)} denotes the gamma distribution then the orthogonality relation can be written as
∫ 0 ∞ L n ( α ) ( x ) L m ( α ) ( x ) Γ ( x , α + 1 , 1 ) d x = ( n + α n ) δ n , m , {\displaystyle \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 )[citation needed ]
K n ( α ) ( x , y ) := 1 Γ ( α + 1 ) ∑ i = 0 n L i ( α ) ( x ) L i ( α ) ( y ) ( α + i i ) = 1 Γ ( α + 1 ) L n ( α ) ( x ) L n + 1 ( α ) ( y ) − L n + 1 ( α ) ( x ) L n ( α ) ( y ) x − y n + 1 ( n + α n ) = 1 Γ ( α + 1 ) ∑ i = 0 n x i i ! L n − i ( α + i ) ( x ) L n − i ( α + i + 1 ) ( y ) ( α + n n ) ( n i ) ; {\displaystyle {\begin{aligned}K_{n}^{(\alpha )}(x,y)&:={\frac {1}{\Gamma (\alpha +1)}}\sum _{i=0}^{n}{\frac {L_{i}^{(\alpha )}(x)L_{i}^{(\alpha )}(y)}{\alpha +i \choose i}}\\[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{aligned}}} recursively
K n ( α ) ( x , y ) = y α + 1 K n − 1 ( α + 1 ) ( x , y ) + 1 Γ ( α + 1 ) L n ( α + 1 ) ( x ) L n ( α ) ( y ) ( α + n n ) . {\displaystyle 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)}{\alpha +n \choose n}}.} Moreover,[clarification needed Limit as n goes to infinity? ]
y α e − y K n ( α ) ( ⋅ , y ) → δ ( y − ⋅ ) . {\displaystyle y^{\alpha }e^{-y}K_{n}^{(\alpha )}(\cdot ,y)\to \delta (y-\cdot ).} Turán's inequalities can be derived here, which is
L n ( α ) ( x ) 2 − L n − 1 ( α ) ( x ) L n + 1 ( α ) ( x ) = ∑ k = 0 n − 1 ( α + n − 1 n − k ) n ( n k ) L k ( α − 1 ) ( x ) 2 > 0. {\displaystyle L_{n}^{(\alpha )}(x)^{2}-L_{n-1}^{(\alpha )}(x)L_{n+1}^{(\alpha )}(x)=\sum _{k=0}^{n-1}{\frac {\alpha +n-1 \choose n-k}{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 ,
∫ 0 ∞ x α + 1 e − x [ L n ( α ) ( x ) ] 2 d x = ( n + α ) ! n ! ( 2 n + α + 1 ) . {\displaystyle \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 [ edit ] Let a function have the (formal) series expansion
f ( x ) = ∑ i = 0 ∞ f i ( α ) L i ( α ) ( x ) . {\displaystyle f(x)=\sum _{i=0}^{\infty }f_{i}^{(\alpha )}L_{i}^{(\alpha )}(x).} Then
f i ( α ) = ∫ 0 ∞ L i ( α ) ( x ) ( i + α i ) ⋅ x α e − x Γ ( α + 1 ) ⋅ f ( x ) d x . {\displaystyle f_{i}^{(\alpha )}=\int _{0}^{\infty }{\frac {L_{i}^{(\alpha )}(x)}{i+\alpha \choose i}}\cdot {\frac {x^{\alpha }e^{-x}}{\Gamma (\alpha +1)}}\cdot f(x)\,dx.} The series converges in the associated Hilbert space L 2 [0, ∞) if and only if
‖ f ‖ L 2 2 := ∫ 0 ∞ x α e − x Γ ( α + 1 ) | f ( x ) | 2 d x = ∑ i = 0 ∞ ( i + α i ) | f i ( α ) | 2 < ∞ . {\displaystyle \|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}<\infty .} Further examples of expansions [ edit ] Monomials are represented as
x n n ! = ∑ i = 0 n ( − 1 ) i ( n + α n − i ) L i ( α ) ( x ) , {\displaystyle {\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 n ) = ∑ i = 0 n α i i ! L n − i ( x + i ) ( α ) . {\displaystyle {n+x \choose n}=\sum _{i=0}^{n}{\frac {\alpha ^{i}}{i!}}L_{n-i}^{(x+i)}(\alpha ).} This leads directly to
e − γ x = ∑ i = 0 ∞ γ i ( 1 + γ ) i + α + 1 L i ( α ) ( x ) convergent iff ℜ ( γ ) > − 1 2 {\displaystyle 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
Γ ( α , x ) = x α e − x ∑ i = 0 ∞ L i ( α ) ( x ) 1 + i ( ℜ ( α ) > − 1 , x > 0 ) . {\displaystyle \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).} In quantum mechanics [ edit ] 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.[11]
Vibronic transitions in the Franck-Condon approximation can also be described using Laguerre polynomials.[12]
Multiplication theorems [ edit ] Erdélyi gives the following two multiplication theorems [13]
t n + 1 + α e ( 1 − t ) z L n ( α ) ( z t ) = ∑ k = n ∞ ( k n ) ( 1 − 1 t ) k − n L k ( α ) ( z ) , e ( 1 − t ) z L n ( α ) ( z t ) = ∑ k = 0 ∞ ( 1 − t ) k z k k ! L n ( α + k ) ( z ) . {\displaystyle {\begin{aligned}&t^{n+1+\alpha }e^{(1-t)z}L_{n}^{(\alpha )}(zt)=\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 )}(zt)=\sum _{k=0}^{\infty }{\frac {(1-t)^{k}z^{k}}{k!}}L_{n}^{(\alpha +k)}(z).\end{aligned}}} Relation to Hermite polynomials [ edit ] The generalized Laguerre polynomials are related to the Hermite polynomials :
H 2 n ( x ) = ( − 1 ) n 2 2 n n ! L n ( − 1 / 2 ) ( x 2 ) H 2 n + 1 ( x ) = ( − 1 ) n 2 2 n + 1 n ! x L n ( 1 / 2 ) ( x 2 ) {\displaystyle {\begin{aligned}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!xL_{n}^{(1/2)}(x^{2})\end{aligned}}} where the
H n (x ) are the
Hermite polynomials based on the weighting function
exp(−x 2 ) , the so-called "physicist's version."
Because of this, the generalized Laguerre polynomials arise in the treatment of the quantum harmonic oscillator .
Relation to hypergeometric functions [ edit ] The Laguerre polynomials may be defined in terms of hypergeometric functions , specifically the confluent hypergeometric functions , as
L n ( α ) ( x ) = ( n + α n ) M ( − n , α + 1 , x ) = ( α + 1 ) n n ! 1 F 1 ( − n , α + 1 , x ) {\displaystyle L_{n}^{(\alpha )}(x)={n+\alpha \choose n}M(-n,\alpha +1,x)={\frac {(\alpha +1)_{n}}{n!}}\,_{1}F_{1}(-n,\alpha +1,x)} where
( a ) n {\displaystyle (a)_{n}} is the
Pochhammer symbol (which in this case represents the rising factorial).
Hardy–Hille formula [ edit ] The generalized Laguerre polynomials satisfy the Hardy–Hille formula[14] [15]
∑ n = 0 ∞ n ! Γ ( α + 1 ) Γ ( n + α + 1 ) L n ( α ) ( x ) L n ( α ) ( y ) t n = 1 ( 1 − t ) α + 1 e − ( x + y ) t / ( 1 − t ) 0 F 1 ( ; α + 1 ; x y t ( 1 − t ) 2 ) , {\displaystyle \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)}\,_{0}F_{1}\left(;\alpha +1;{\frac {xyt}{(1-t)^{2}}}\right),} where the series on the left converges for
α > − 1 {\displaystyle \alpha >-1} and
| t | < 1 {\displaystyle |t|<1} . Using the identity
0 F 1 ( ; α + 1 ; z ) = Γ ( α + 1 ) z − α / 2 I α ( 2 z ) , {\displaystyle \,_{0}F_{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
∑ n = 0 ∞ n ! Γ ( 1 + α + n ) L n ( α ) ( x ) L n ( α ) ( y ) t n = 1 ( x y t ) α / 2 ( 1 − t ) e − ( x + y ) t / ( 1 − t ) I α ( 2 x y t 1 − t ) . {\displaystyle \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).} This formula is a generalization of the
Mehler kernel for
Hermite polynomials , which can be recovered from it by using the relations between Laguerre and Hermite polynomials given above.
Physics Convention [ edit ] The generalized Laguerre polynomials are used to describe the quantum wavefunction for hydrogen atom orbitals.[16] [17] [18] The convention used throughout this article expresses the generalized Laguerre polynomials as [19]
L n ( α ) ( x ) = Γ ( α + n + 1 ) Γ ( α + 1 ) n ! 1 F 1 ( − n ; α + 1 ; x ) , {\displaystyle L_{n}^{(\alpha )}(x)={\frac {\Gamma (\alpha +n+1)}{\Gamma (\alpha +1)n!}}\,_{1}F_{1}(-n;\alpha +1;x),} where 1 F 1 ( a ; b ; x ) {\displaystyle \,_{1}F_{1}(a;b;x)} is the confluent hypergeometric function . In the physics literature,[18] the generalized Laguerre polynomials are instead defined as
L ¯ n ( α ) ( x ) = [ Γ ( α + n + 1 ) ] 2 Γ ( α + 1 ) n ! 1 F 1 ( − n ; α + 1 ; x ) . {\displaystyle {\bar {L}}_{n}^{(\alpha )}(x)={\frac {\left[\Gamma (\alpha +n+1)\right]^{2}}{\Gamma (\alpha +1)n!}}\,_{1}F_{1}(-n;\alpha +1;x).} The physics version is related to the standard version by
L ¯ n ( α ) ( x ) = ( n + α ) ! L n ( α ) ( x ) . {\displaystyle {\bar {L}}_{n}^{(\alpha )}(x)=(n+\alpha )!L_{n}^{(\alpha )}(x).} There is yet another, albeit less frequently used, convention in the physics literature [20] [21] [22]
L ~ n ( α ) ( x ) = ( − 1 ) α L ¯ n − α ( α ) . {\displaystyle {\tilde {L}}_{n}^{(\alpha )}(x)=(-1)^{\alpha }{\bar {L}}_{n-\alpha }^{(\alpha )}.} Umbral Calculus Convention [ edit ] Generalized Laguerre polynomials are linked to Umbral calculus by being Sheffer sequences for D / ( D − I ) {\displaystyle D/(D-I)} when multiplied by n ! {\displaystyle n!} . In Umbral Calculus convention,[23] the default Laguerre polynomials are defined to be
L n ( x ) = n ! L n ( − 1 ) ( x ) = ∑ k = 0 n L ( n , k ) ( − x ) k {\displaystyle {\mathcal {L}}_{n}(x)=n!L_{n}^{(-1)}(x)=\sum _{k=0}^{n}L(n,k)(-x)^{k}} where
L ( n , k ) = ( n − 1 k − 1 ) n ! k ! {\textstyle L(n,k)={\binom {n-1}{k-1}}{\frac {n!}{k!}}} are the signless
Lah numbers .
( L n ( x ) ) n ∈ N {\textstyle ({\mathcal {L}}_{n}(x))_{n\in \mathbb {N} }} is a sequence of polynomials of
binomial type ,
ie they satisfy
L n ( x + y ) = ∑ k = 0 n ( n k ) L k ( x ) L n − k ( y ) {\displaystyle {\mathcal {L}}_{n}(x+y)=\sum _{k=0}^{n}{\binom {n}{k}}{\mathcal {L}}_{k}(x){\mathcal {L}}_{n-k}(y)} See also [ edit ] ^ N. Sonine (1880). "Recherches sur les fonctions cylindriques et le développement des fonctions continues en séries" . Math. Ann. 16 (1): 1–80. doi :10.1007/BF01459227 . S2CID 121602983 . ^ A&S p. 781 ^ A&S p. 509 ^ A&S p. 510 ^ A&S p. 775 ^ Szegő, p. 198. ^ 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 ^ A&S equation (22.12.6), p. 785 ^ Koepf, Wolfram (1997). "Identities for families of orthogonal polynomials and special functions". Integral Transforms and Special Functions . 5 (1–2): 69–102. CiteSeerX 10.1.1.298.7657 . doi :10.1080/10652469708819127 . ^ "Associated Laguerre Polynomial" . ^ Ratner, Schatz, Mark A., George C. (2001). Quantum Mechanics in Chemistry . 0-13-895491-7: Prentice Hall. pp. 90–91. {{cite book }}
: CS1 maint: location (link ) CS1 maint: multiple names: authors list (link ) ^ Jong, Mathijs de; Seijo, Luis; Meijerink, Andries; Rabouw, Freddy T. (2015-06-24). "Resolving the ambiguity in the relation between Stokes shift and Huang–Rhys parameter" . Physical Chemistry Chemical Physics . 17 (26): 16959–16969. Bibcode :2015PCCP...1716959D . doi :10.1039/C5CP02093J . hdl :1874/321453 . ISSN 1463-9084 . PMID 26062123 . S2CID 34490576 . ^ C. Truesdell, "On the Addition and Multiplication Theorems for the Special Functions ", Proceedings of the National Academy of Sciences, Mathematics , (1950) pp. 752–757. ^ Szegő, p. 102. ^ W. A. Al-Salam (1964), "Operational representations for Laguerre and other polynomials" , Duke Math J. 31 (1): 127–142. ^ Griffiths, David J. (2005). Introduction to quantum mechanics (2nd ed.). Upper Saddle River, NJ: Pearson Prentice Hall. ISBN 0131118927 . ^ Sakurai, J. J. (2011). Modern quantum mechanics (2nd ed.). Boston: Addison-Wesley. ISBN 978-0805382914 . ^ a b Merzbacher, Eugen (1998). Quantum mechanics (3rd ed.). New York: Wiley. ISBN 0471887021 . ^ Abramowitz, Milton (1965). Handbook of mathematical functions, with formulas, graphs, and mathematical tables . New York: Dover Publications. ISBN 978-0-486-61272-0 . ^ Schiff, Leonard I. (1968). Quantum mechanics (3d ed.). New York: McGraw-Hill. ISBN 0070856435 . ^ Messiah, Albert (2014). Quantum Mechanics . Dover Publications. ISBN 9780486784557 . ^ Boas, Mary L. (2006). Mathematical methods in the physical sciences (3rd ed.). Hoboken, NJ: Wiley. ISBN 9780471198260 . ^ Rota, Gian-Carlo; Kahaner, D; Odlyzko, A (1973-06-01). "On the foundations of combinatorial theory. VIII. Finite operator calculus" . Journal of Mathematical Analysis and Applications . 42 (3): 684–760. doi :10.1016/0022-247X(73)90172-8 . ISSN 0022-247X . References [ edit ] Abramowitz, Milton ; Stegun, Irene Ann , eds. (1983) [June 1964]. "Chapter 22" . Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables . Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 773. ISBN 978-0-486-61272-0 . LCCN 64-60036 . MR 0167642 . LCCN 65-12253 . G. Szegő, Orthogonal polynomials , 4th edition, Amer. Math. Soc. Colloq. Publ. , vol. 23, Amer. Math. Soc., Providence, RI, 1975. Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Orthogonal Polynomials" , in Olver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions , Cambridge University Press, ISBN 978-0-521-19225-5 , MR 2723248 . B. Spain, M.G. Smith, Functions of mathematical physics , Van Nostrand Reinhold Company, London, 1970. Chapter 10 deals with Laguerre polynomials. "Laguerre polynomials" , Encyclopedia of Mathematics , EMS Press , 2001 [1994] Eric W. Weisstein , "Laguerre Polynomial ", From MathWorld—A Wolfram Web Resource. George Arfken and Hans Weber (2000). Mathematical Methods for Physicists . Academic Press. ISBN 978-0-12-059825-0 . External links [ edit ]
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