Sequence of differential equation solutions
Complex color plot ofL −1/9 (z 4 ) from −2−2i to 2+2i Inmathematics , theLaguerre polynomials , named afterEdmond Laguerre (1834–1886), are nontrivial solutions ofLaguerre's differential equation: x y ″ + ( 1 − x ) y ′ + n y = 0 , y = y ( x ) {\displaystyle xy''+(1-x)y'+ny=0,\ y=y(x)} linear differential equation . This equation hasnonsingular solutions only ifn is a non-negative integer.
Sometimes the nameLaguerre polynomials is used for solutions ofx y ″ + ( α + 1 − x ) y ′ + n y = 0 . {\displaystyle xy''+(\alpha +1-x)y'+ny=0~.} n is still a non-negative integer.Then they are also namedgeneralized Laguerre polynomials , as will be done here (alternativelyassociated Laguerre polynomials or, rarely,Sonine polynomials , after their inventor[ 1] Nikolay Yakovlevich Sonin ).
More generally, aLaguerre function is a solution whenn is not necessarily a non-negative integer.
The Laguerre polynomials are also used forGauss–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 denotedL 0 L 1 polynomial sequence which may be defined by theRodrigues 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},}
They areorthogonal polynomials with respect to aninner 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.}
Therook polynomials in combinatorics are more or less the same as Laguerre polynomials, up to elementary changes of variables. Further see theTricomi–Carlitz polynomials .
The Laguerre polynomials arise in quantum mechanics, in the radial part of the solution of theSchrödinger equation for a one-electron atom. They also describe the static Wigner functions of oscillator systems inquantum mechanics in phase space . They further enter in the quantum mechanics of theMorse potential and of the3D isotropic harmonic oscillator .
Physicists sometimes use a definition for the Laguerre polynomials that is larger by a factor ofn ! than the definition used here. (Likewise, some physicists may use somewhat different definitions of the so-called associated Laguerre polynomials.)
[ edit ] One can also define the Laguerre polynomials recursively, defining the first two polynomials asL 0 ( x ) = 1 {\displaystyle L_{0}(x)=1} L 1 ( x ) = 1 − x {\displaystyle L_{1}(x)=1-x} recurrence relation for anyk ≥ 1L 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}}.} 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.}
Theclosed form isL 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}.}
Thegenerating 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)}.} 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).}
A table of the 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)\,} 7 1 5040 ( − x 7 + 49 x 6 − 882 x 5 + 7350 x 4 − 29400 x 3 + 52920 x 2 − 35280 x + 5040 ) {\displaystyle {\tfrac {1}{5040}}(-x^{7}+49x^{6}-882x^{5}+7350x^{4}-29400x^{3}+52920x^{2}-35280x+5040)\,} 8 1 40320 ( x 8 − 64 x 7 + 1568 x 6 − 18816 x 5 + 117600 x 4 − 376320 x 3 + 564480 x 2 − 322560 x + 40320 ) {\displaystyle {\tfrac {1}{40320}}(x^{8}-64x^{7}+1568x^{6}-18816x^{5}+117600x^{4}-376320x^{3}+564480x^{2}-322560x+40320)\,} 9 1 362880 ( − x 9 + 81 x 8 − 2592 x 7 + 42336 x 6 − 381024 x 5 + 1905120 x 4 − 5080320 x 3 + 6531840 x 2 − 3265920 x + 362880 ) {\displaystyle {\tfrac {1}{362880}}(-x^{9}+81x^{8}-2592x^{7}+42336x^{6}-381024x^{5}+1905120x^{4}-5080320x^{3}+6531840x^{2}-3265920x+362880)\,} 10 1 3628800 ( x 10 − 100 x 9 + 4050 x 8 − 86400 x 7 + 1058400 x 6 − 7620480 x 5 + 31752000 x 4 − 72576000 x 3 + 81648000 x 2 − 36288000 x + 3628800 ) {\displaystyle {\tfrac {1}{3628800}}(x^{10}-100x^{9}+4050x^{8}-86400x^{7}+1058400x^{6}-7620480x^{5}+31752000x^{4}-72576000x^{3}+81648000x^{2}-36288000x+3628800)\,} n 1 n ! ( ( − x ) n + n 2 ( − x ) n − 1 + ⋯ + ( n k ) 2 k ! ( − x ) n − k + ⋯ + n ( n ! ) ( − x ) + n ! ) {\displaystyle {\tfrac {1}{n!}}((-x)^{n}+n^{2}(-x)^{n-1}+\dots +{\binom {n}{k}}^{2}{k!}(-x)^{n-k}+\dots +n({n!})(-x)+n!)\,}
The first six Laguerre polynomials. 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} generalized Laguerre polynomials , orassociated Laguerre polynomials .
One can also define the generalized Laguerre polynomials recursively, defining the first two polynomials asL 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 followingrecurrence relation for anyk ≥ 1L 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 caseα = 0L n ( 0 ) ( x ) = L n ( x ) . {\displaystyle L_{n}^{(0)}(x)=L_{n}(x).}
TheRodrigues formula for them isL 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 }.}
Thegenerating 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 ) Laguerre functions are defined byconfluent 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).} ( n + α n ) {\textstyle {n+\alpha \choose n}} binomial coefficient . Whenn is an integer the function reduces to a polynomial of degreen . 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)} Kummer's function of the second kind . The closed form for these generalized Laguerre polynomials of degreen 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!}}} 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, letD = d d x {\displaystyle D={\frac {d}{dx}}} M = x D 2 + ( α + 1 ) D {\displaystyle M=xD^{2}+(\alpha +1)D} 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: n L n ( α ) ( x ) {\displaystyle L_{n}^{(\alpha )}(x)\,} 0 1 {\displaystyle 1\,} 1 − x + α + 1 {\displaystyle -x+\alpha +1\,} 2 1 2 ( x 2 − 2 ( α + 2 ) x + ( α + 1 ) ( α + 2 ) ) {\displaystyle {\tfrac {1}{2}}(x^{2}-2\left(\alpha +2\right)x+\left(\alpha +1\right)\left(\alpha +2\right))\,} 3 1 6 ( − x 3 + 3 ( α + 3 ) x 2 − 3 ( α + 2 ) ( α + 3 ) x + ( α + 1 ) ( α + 2 ) ( α + 3 ) ) {\displaystyle {\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 1 24 ( x 4 − 4 ( α + 4 ) x 3 + 6 ( α + 3 ) ( α + 4 ) x 2 − 4 ( α + 2 ) ⋯ ( α + 4 ) x + ( α + 1 ) ⋯ ( α + 4 ) ) {\displaystyle {\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)\cdots \left(\alpha +4\right)x+\left(\alpha +1\right)\cdots \left(\alpha +4\right))\,} 5 1 120 ( − x 5 + 5 ( α + 5 ) x 4 − 10 ( α + 4 ) ( α + 5 ) x 3 + 10 ( α + 3 ) ⋯ ( α + 5 ) x 2 − 5 ( α + 2 ) ⋯ ( α + 5 ) x + ( α + 1 ) ⋯ ( α + 5 ) ) {\displaystyle {\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)\cdots \left(\alpha +5\right)x^{2}-5\left(\alpha +2\right)\cdots \left(\alpha +5\right)x+\left(\alpha +1\right)\cdots \left(\alpha +5\right))\,} 6 1 720 ( x 6 − 6 ( α + 6 ) x 5 + 15 ( α + 5 ) ( α + 6 ) x 4 − 20 ( α + 4 ) ⋯ ( α + 6 ) x 3 + 15 ( α + 3 ) ⋯ ( α + 6 ) x 2 − 6 ( α + 2 ) ⋯ ( α + 6 ) x + ( α + 1 ) ⋯ ( α + 6 ) ) {\displaystyle {\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)\cdots \left(\alpha +6\right)x^{3}+15\left(\alpha +3\right)\cdots \left(\alpha +6\right)x^{2}-6\left(\alpha +2\right)\cdots \left(\alpha +6\right)x+\left(\alpha +1\right)\cdots \left(\alpha +6\right))\,} 7 1 5040 ( − x 7 + 7 ( α + 7 ) x 6 − 21 ( α + 6 ) ( α + 7 ) x 5 + 35 ( α + 5 ) ⋯ ( α + 7 ) x 4 − 35 ( α + 4 ) ⋯ ( α + 7 ) x 3 + 21 ( α + 3 ) ⋯ ( α + 7 ) x 2 − 7 ( α + 2 ) ⋯ ( α + 7 ) x + ( α + 1 ) ⋯ ( α + 7 ) ) {\displaystyle {\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)\cdots \left(\alpha +7\right)x^{4}-35\left(\alpha +4\right)\cdots \left(\alpha +7\right)x^{3}+21\left(\alpha +3\right)\cdots \left(\alpha +7\right)x^{2}-7\left(\alpha +2\right)\cdots \left(\alpha +7\right)x+\left(\alpha +1\right)\cdots \left(\alpha +7\right))\,} 8 1 40320 ( x 8 − 8 ( α + 8 ) x 7 + 28 ( α + 7 ) ( α + 8 ) x 6 − 56 ( α + 6 ) ⋯ ( α + 8 ) x 5 + 70 ( α + 5 ) ⋯ ( α + 8 ) x 4 − 56 ( α + 4 ) ⋯ ( α + 8 ) x 3 + 28 ( α + 3 ) ⋯ ( α + 8 ) x 2 − 8 ( α + 2 ) ⋯ ( α + 8 ) x + ( α + 1 ) ⋯ ( α + 8 ) ) {\displaystyle {\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)\cdots \left(\alpha +8\right)x^{5}+70\left(\alpha +5\right)\cdots \left(\alpha +8\right)x^{4}-56\left(\alpha +4\right)\cdots \left(\alpha +8\right)x^{3}+28\left(\alpha +3\right)\cdots \left(\alpha +8\right)x^{2}-8\left(\alpha +2\right)\cdots \left(\alpha +8\right)x+\left(\alpha +1\right)\cdots \left(\alpha +8\right))\,} 9 1 362880 ( − x 9 + 9 ( α + 9 ) x 8 − 36 ( α + 8 ) ( α + 9 ) x 7 + 84 ( α + 7 ) ⋯ ( α + 9 ) x 6 − 126 ( α + 6 ) ⋯ ( α + 9 ) x 5 + 126 ( α + 5 ) ⋯ ( α + 9 ) x 4 − 84 ( α + 4 ) ⋯ ( α + 9 ) x 3 + 36 ( α + 3 ) ⋯ ( α + 9 ) x 2 − 9 ( α + 2 ) ⋯ ( α + 9 ) x + ( α + 1 ) ⋯ ( α + 9 ) ) {\displaystyle {\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)\cdots \left(\alpha +9\right)x^{6}-126\left(\alpha +6\right)\cdots \left(\alpha +9\right)x^{5}+126\left(\alpha +5\right)\cdots \left(\alpha +9\right)x^{4}-84\left(\alpha +4\right)\cdots \left(\alpha +9\right)x^{3}+36\left(\alpha +3\right)\cdots \left(\alpha +9\right)x^{2}-9\left(\alpha +2\right)\cdots \left(\alpha +9\right)x+\left(\alpha +1\right)\cdots \left(\alpha +9\right))\,} 10 1 3628800 ( x 10 − 10 ( α + 10 ) x 9 + 45 ( α + 9 ) ( α + 10 ) x 8 − 120 ( α + 8 ) ⋯ ( α + 10 ) x 7 + 210 ( α + 7 ) ⋯ ( α + 10 ) x 6 − 252 ( α + 6 ) ⋯ ( α + 10 ) x 5 + 210 ( α + 5 ) ⋯ ( α + 10 ) x 4 − 120 ( α + 4 ) ⋯ ( α + 10 ) x 3 + 45 ( α + 3 ) ⋯ ( α + 10 ) x 2 − 10 ( α + 2 ) ⋯ ( α + 10 ) x + ( α + 1 ) ⋯ ( α + 10 ) ) {\displaystyle {\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)\cdots \left(\alpha +10\right)x^{7}+210\left(\alpha +7\right)\cdots \left(\alpha +10\right)x^{6}-252\left(\alpha +6\right)\cdots \left(\alpha +10\right)x^{5}+210\left(\alpha +5\right)\cdots \left(\alpha +10\right)x^{4}-120\left(\alpha +4\right)\cdots \left(\alpha +10\right)x^{3}+45\left(\alpha +3\right)\cdots \left(\alpha +10\right)x^{2}-10\left(\alpha +2\right)\cdots \left(\alpha +10\right)x+\left(\alpha +1\right)\cdots \left(\alpha +10\right))\,}
As a contour integral [ edit ] Given the generating function specified above, the polynomials may be expressed in terms of acontour 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,}
Recurrence relations [ edit ] The addition formula for Laguerre polynomials:[ 7] L n ( α 1 + ⋯ + α r + r − 1 ) ( x 1 + ⋯ + x r ) = ∑ m 1 + ⋯ + m r = n L m 1 ( α 1 ) ( x 1 ) ⋯ L m r ( α r ) ( x r ) . {\displaystyle L_{n}^{(\alpha _{1}+\dots +\alpha _{r}+r-1)}\left(x_{1}+\dots +x_{r}\right)=\sum _{m_{1}+\dots +m_{r}=n}L_{m_{1}}^{(\alpha _{1})}\left(x_{1}\right)\cdots L_{m_{r}}^{(\alpha _{r})}\left(x_{r}\right).} 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!}},} 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)} 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),} 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);} 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-rulesL n ( α ) ( x ) = L n ( α + 1 ) ( x ) − L n − 1 ( α + 1 ) ( x ) = ∑ j = 0 k ( k j ) ( − 1 ) 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}(-1)^{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 relationsL 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}}}
SinceL n ( α ) ( x ) {\displaystyle L_{n}^{(\alpha )}(x)} n {\displaystyle n} α {\displaystyle \alpha } 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}}} i andn and immediate from the expression ofL n ( α ) ( x ) {\displaystyle L_{n}^{(\alpha )}(x)} 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).}
Differentiating thepower series representation of a generalized Laguerre polynomialk times leads tod 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 = 0α =k 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}}},} 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),} Cauchy's formula toL 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 variableα has the form,[ 8] 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}}.} 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,} 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,} 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}}}}
InSturm–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 thatL (α) n n .
The generalized Laguerre polynomials areorthogonal over[0, ∞) with respect to the measure withweighting function xα e −x [ 9]
∫ 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 ').}
IfΓ ( x , α + 1 , 1 ) {\displaystyle \Gamma (x,\alpha +1,1)}
∫ 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
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 isL 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 followingintegral is needed in thequantum mechanical treatment of thehydrogen 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).}
Let a function have the (formal) series expansionf ( x ) = ∑ i = 0 ∞ f i ( α ) L i ( α ) ( x ) . {\displaystyle f(x)=\sum _{i=0}^{\infty }f_{i}^{(\alpha )}L_{i}^{(\alpha )}(x).}
Thenf 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 associatedHilbert 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 asx 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),} 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 toe − 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}}} incomplete gamma function has the representationΓ ( 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 terms of elementary functions [ edit ] M {\displaystyle M} α {\displaystyle \alpha } [ c , d ] ⊂ ( 0 , + ∞ ) {\displaystyle [c,d]\subset (0,+\infty )} x ∈ [ c , d ] {\displaystyle x\in [c,d]} n → ∞ {\displaystyle n\to \infty } L n ( α ) ( x ) = n 1 2 α − 1 4 e 1 2 x π 1 2 x 1 2 α + 1 4 ( cos θ n ( α ) ( x ) ( ∑ m = 0 M − 1 a m ( x ) n 1 2 m + O ( 1 n 1 2 M ) ) + sin θ n ( α ) ( x ) ( ∑ m = 1 M − 1 b m ( x ) n 1 2 m + O ( 1 n 1 2 M ) ) ) {\displaystyle L_{n}^{(\alpha )}\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)} θ n ( α ) ( x ) := 2 ( n x ) 1 2 − ( 1 2 α + 1 4 ) π . {\displaystyle \theta _{n}^{(\alpha )}(x):=2(nx)^{\frac {1}{2}}-\left({\tfrac {1}{2}}\alpha +{\tfrac {1}{4}}\right)\pi .} a 0 , b 1 , a 1 , b 2 , … {\displaystyle a_{0},b_{1},a_{1},b_{2},\dots } α , x {\displaystyle \alpha ,x} n {\displaystyle n} x > 0 {\displaystyle x>0} a 0 ( x ) = 1 a 1 ( x ) = 0 b 1 ( x ) = 1 48 x 1 2 ( 4 x 2 − 24 ( α + 1 ) x + 3 − 12 α 2 ) {\displaystyle {\begin{aligned}&a_{0}(x)=1\\&a_{1}(x)=0\\&b_{1}(x)={\frac {1}{48x^{\frac {1}{2}}}}\left(4x^{2}-24(\alpha +1)x+3-12\alpha ^{2}\right)\end{aligned}}} Perron 's formula.[ 10] [ 11] : 78 x ∈ C ∖ [ 0 , ∞ ) {\displaystyle x\in \mathbb {C} \setminus [0,\infty )} [ 12] Fejér 's formula is a special case of Perron's formula withM = 1 {\displaystyle M=1} [ 13] [ 12] [ 14]
In terms of Bessel functions [ edit ] TheMehler–Heine formula states:
lim n → ∞ n − α L n ( α ) ( z 2 4 n ) = ( z 2 ) − α J α ( z ) , {\displaystyle \lim _{n\to \infty }n^{-\alpha }L_{n}^{(\alpha )}\left({\frac {z^{2}}{4n}}\right)=\left({\frac {z}{2}}\right)^{-\alpha }J_{\alpha }(z),} whereJ α {\displaystyle J_{\alpha }} Bessel function of the first kind .
See also:.[ 10]
In terms of Airy functions [ edit ] Letν = 4 n + 2 α + 2 {\displaystyle \nu =4n+2\alpha +2} Ai {\displaystyle \operatorname {Ai} } Airy function . Letα {\displaystyle \alpha } ϵ {\displaystyle \epsilon } ω {\displaystyle \omega }
ThePlancherel–Rotach asymptotics formulas:[ 15] [ 10]
e − x / 2 L n ( α ) ( x ) = ( − 1 ) n ( π sin φ ) − 1 / 2 x − α / 2 − 1 / 4 n α / 2 − 1 / 4 { sin [ ( n + α + 1 2 ) ( sin 2 φ − 2 φ ) + 3 π / 4 ] + ( n x ) − 1 / 2 O ( 1 ) } {\displaystyle e^{-x/2}L_{n}^{(\alpha )}(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 \}}} e − x / 2 L n ( α ) ( x ) = 1 2 ( − 1 ) n ( π sinh φ ) − 1 / 2 x − α / 2 − 1 / 4 n α / 2 − 1 / 4 exp [ ( n + α + 1 2 ) ( 2 φ − sinh 2 φ ) ] { 1 + O ( n − 1 ) } {\displaystyle e^{-x/2}L_{n}^{(\alpha )}(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)\}} e − x / 2 L n ( α ) ( x ) = ( − 1 ) n π − 1 2 − α − 1 / 3 3 1 / 3 n − 1 / 3 { π Ai ( − 3 − 1 / 3 t ) + O ( n − 2 / 3 ) } {\displaystyle e^{-x/2}L_{n}^{(\alpha )}(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.[ 10]
j α , m {\displaystyle j_{\alpha ,m}} m {\displaystyle m} Bessel function J α ( x ) {\displaystyle J_{\alpha }(x)}
a m {\displaystyle a_{m}} m {\displaystyle m} Ai ( x ) {\displaystyle \operatorname {Ai} (x)} 0 > a 1 > a 2 > ⋯ {\displaystyle 0>a_{1}>a_{2}>\cdots }
ν = 4 n + 2 α + 2 {\displaystyle \nu =4n+2\alpha +2}
Ifα > − 1 {\displaystyle \alpha >-1} L n ( α ) {\displaystyle L_{n}^{(\alpha )}} n {\displaystyle n} α > − 1 {\displaystyle \alpha >-1}
x 1 < ⋯ < x n {\displaystyle x_{1}<\dots <x_{n}} L n ( α ) {\displaystyle L_{n}^{(\alpha )}}
Note that( ( − 1 ) n − i L n − i ( α ) ) i = 0 n {\displaystyle \left((-1)^{n-i}L_{n-i}^{(\alpha )}\right)_{i=0}^{n}} Sturm chain .
Forα > − 1 {\displaystyle \alpha >-1} [ 16] [ 17] [ 6] [ 18]
For fixedk = 1 , … , n {\displaystyle k=1,\dots ,n} [ 16] [ 6] [ 17] ν x k > j α , k 2 x k < j α , k 2 ν / 2 + ( ν / 2 ) 2 − j α , k 2 if ν / 2 > j α , k x k < [ ν 1 / 2 + 2 − 1 / 3 ν − 1 / 6 a n − k + 1 ] 2 if | α | ⩾ 1 / 4 x k < ν + 2 2 3 a k ν 1 3 + 2 − 2 3 a k 2 ν − 1 3 {\displaystyle {\begin{aligned}\nu x_{k}&>j_{\alpha ,k}^{2}\\x_{k}&<{\frac {j_{\alpha ,k}^{2}}{\nu /2+{\sqrt {(\nu /2)^{2}-j_{\alpha ,k}^{2}}}}}\quad {\text{ if }}\nu /2>j_{\alpha ,k}\\x_{k}&<\left[\nu ^{1/2}+2^{-1/3}\nu ^{-1/6}a_{n-k+1}\right]^{2}\quad {\text{ if }}|\alpha |\geqslant 1/4\\x_{k}&<\nu +2^{\frac {2}{3}}a_{k}\nu ^{\frac {1}{3}}+2^{-{\frac {2}{3}}}a_{k}^{2}\nu ^{-{\frac {1}{3}}}\end{aligned}}} k {\displaystyle k} lim n → ∞ ν x k = j α , k 2 {\displaystyle \lim _{n\to \infty }\nu x_{k}=j_{\alpha ,k}^{2}}
See also.[ 19]
The zeroes satisfy theStieltjes relations[ 20] [ 21] ∑ 1 ≤ j ≤ n , i ≠ j 1 x i − x j = 1 2 ( 1 − α + 1 x i ) ∑ 1 ≤ j ≤ n 1 x j = n α + 1 ∑ 1 ≤ j ≤ n , i ≠ j 1 ( x i − x j ) 2 = − ( α + 1 ) ( α + 5 ) 12 x i 2 + 2 n + α + 1 6 x i − 1 12 ∑ 1 ≤ j ≤ n , i ≠ j 1 ( x i − x j ) 3 = − ( α + 1 ) ( α + 3 ) 8 x i 3 + 2 n + α + 1 8 x i 2 {\displaystyle {\begin{aligned}\sum _{1\leq j\leq n,i\neq j}{\frac {1}{x_{i}-x_{j}}}&={\frac {1}{2}}\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)}{12x_{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)}{8x_{i}^{3}}}+{\frac {2n+\alpha +1}{8x_{i}^{2}}}\\\end{aligned}}} + α + 1 2 {\displaystyle +{\frac {\alpha +1}{2}}} − 1 2 {\displaystyle -{\frac {1}{2}}} n {\displaystyle n} + 1 {\displaystyle +1} L n ( α ) {\displaystyle L_{n}^{(\alpha )}}
As the zeroes specify the polynomial up to scaling, this provides an alternative way to uniquely characterize the Laguerre polynomials.
The zeroes also satisfy[ 22] ∑ i = 1 n 1 x − x i = − ∑ k = 0 ∞ S k + 1 x k , S k := ∑ i = 1 n x i − k {\displaystyle \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}} S m − 1 / m < x 1 < S m / S m + 1 , m = 1 , 2 , … {\displaystyle S_{m}^{-1/m}<x_{1}<S_{m}/S_{m+1},\quad m=1,2,\ldots }
LetF n ( t ) := 1 n # { i : x i ≤ t } {\displaystyle F_{n}(t):={\frac {1}{n}}\#\{i:x_{i}\leq t\}} cumulative distribution function for the roots, then we have the limit law[ 23] lim n → ∞ F n ( 4 n t ) = 2 π ∫ 0 t 1 − s s d s ∀ t ∈ ( 0 , 1 ] {\displaystyle \lim _{n\to \infty }F_{n}(4nt)={\frac {2}{\pi }}\int _{0}^{t}{\sqrt {\frac {1-s}{s}}}ds\quad \forall t\in (0,1]} Wishart ensemble spectrum.
For fixedα > − 1 {\displaystyle \alpha >-1} k {\displaystyle k} n → ∞ {\displaystyle n\to \infty } [ 17] x n + 1 − k = ν + 2 2 / 3 a k ν 1 / 3 + 1 5 2 4 / 3 a k 2 ν − 1 / 3 + ( 11 35 − α 2 − 12 175 a k 3 ) ν − 1 + ( 16 1575 a k + 92 7875 a k 4 ) 2 2 / 3 ν − 5 / 3 − ( 15152 3031875 a k 5 + 1088 121275 a k 2 ) 2 1 / 3 ν − 7 / 3 + O ( ν − 3 ) , {\displaystyle {\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}}}
Forα ∈ ( − 1 , 0 ) {\displaystyle \alpha \in (-1,0)} [ 22] x 1 = α + 1 n + n − 1 2 ( α + 1 n ) 2 − n 2 + 3 n − 4 12 ( α + 1 n ) 3 + 7 n 3 + 6 n 2 + 23 n − 36 144 ( α + 1 n ) 4 − 293 n 4 + 210 n 3 + 235 n 2 + 990 n − 1728 8640 ( α + 1 n ) 5 + ⋯ {\displaystyle {\begin{aligned}x_{1}={\frac {\alpha +1}{n}}&+{\frac {n-1}{2}}\left({\frac {\alpha +1}{n}}\right)^{2}-{\frac {n^{2}+3n-4}{12}}\left({\frac {\alpha +1}{n}}\right)^{3}\\&+{\frac {7n^{3}+6n^{2}+23n-36}{144}}\left({\frac {\alpha +1}{n}}\right)^{4}\\&-{\frac {293n^{4}+210n^{3}+235n^{2}+990n-1728}{8640}}\left({\frac {\alpha +1}{n}}\right)^{5}+\cdots \end{aligned}}}
In quantum mechanics [ edit ] In quantum mechanics the Schrödinger equation for thehydrogen-like atom is exactly solvable by separation of variables in spherical coordinates. The radial part of the wave function is a (generalized) Laguerre polynomial.[ 24]
Vibronic transitions in the Franck-Condon approximation can also be described using Laguerre polynomials.[ 25]
Multiplication theorems [ edit ] Erdélyi gives the following twomultiplication theorems [ 26]
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 theHermite 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}}} H n x )Hermite polynomials based on the weighting functionexp(−x 2 ) , the so-called "physicist's version."
Because of this, the generalized Laguerre polynomials arise in the treatment of thequantum harmonic oscillator .
Applying the addition formula,( − 1 ) n 2 2 n n ! L n ( r 2 − 1 ) ( z 1 2 + ⋯ + z r 2 ) = ∑ m 1 + ⋯ + m r = n ∏ i = 1 r H 2 m i ( z i ) . {\displaystyle (-1)^{n}2^{2n}n!\,L_{n}^{\left({\frac {r}{2}}-1\right)}{\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 [ edit ] The Laguerre polynomials may be defined in terms ofhypergeometric functions , specifically theconfluent hypergeometric functions , asL 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)} ( a ) n {\displaystyle (a)_{n}} Pochhammer symbol (which in this case represents the rising factorial).
[ edit ] The generalized Laguerre polynomials satisfy theHardy –Hille formula[ 27] [ 28] ∑ 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),} α > − 1 {\displaystyle \alpha >-1} | t | < 1 {\displaystyle |t|<1} 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),} 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).} I α {\displaystyle I_{\alpha }} I α ( z ) = ∑ k = 0 ∞ 1 k ! Γ ( k + α + 1 ) ( z 2 ) 2 k + α {\displaystyle I_{\alpha }(z)=\sum _{k=0}^{\infty }{\frac {1}{k!\,\Gamma (k+\alpha +1)}}\left({\frac {z}{2}}\right)^{2k+\alpha }} Mehler kernel forHermite polynomials , which can be recovered from it by setting the Hermite polynomials as a special case of the associated Laguerre polynomials.
Substitutet ↦ − t / y {\displaystyle t\mapsto -t/y} y → ∞ {\displaystyle y\to \infty } [ 29] ∑ n = 0 ∞ t n Γ ( n + 1 + α ) L n ( α ) ( x ) = e t ( − x t ) α / 2 I α ( 2 − x t ) . {\displaystyle \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}}).} G. H. Hardy andEinar Hille .[ 30] [ 31]
The generalized Laguerre polynomials are used to describe the quantum wavefunction forhydrogen atom orbitals.[ 32] [ 33] [ 34] [ 35]
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),}
where1 F 1 ( a ; b ; x ) {\displaystyle \,_{1}F_{1}(a;b;x)} confluent hypergeometric function .In the physics literature,[ 34]
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[ 36] [ 37] [ 38]
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 toUmbral calculus by beingSheffer sequences forD / ( D − I ) {\displaystyle D/(D-I)} n ! {\displaystyle n!} [ 39] 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}} L ( n , k ) = ( n − 1 k − 1 ) n ! k ! {\textstyle L(n,k)={\binom {n-1}{k-1}}{\frac {n!}{k!}}} Lah numbers .( L n ( x ) ) n ∈ N {\textstyle ({\mathcal {L}}_{n}(x))_{n\in \mathbb {N} }} binomial type ,ie they satisfyL 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)}
^ 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 ^a b c "DLMF: §18.16 Zeros ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" .dlmf.nist.gov .^ "DLMF: §18.18 Sums ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" .dlmf.nist.gov . Retrieved2025-03-18 .^ 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" .^a b c d "DLMF: §18.15 Asymptotic Approximations ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials" .dlmf.nist.gov . Retrieved2025-07-07 .^ Perron, Oskar (1921-01-01)."Über das Verhalten einer ausgearteten hypergeometrischen Reihe bei unbegrenztem Wachstum eines Parameters" (in German).1921 (151):63– 78.doi :10.1515/crll.1921.151.63 .ISSN 1435-5345 . {{cite journal }}:Cite journal requires|journal= (help ) ^a b Szegő, p. 198. ^ Turán, Pál (1970),"Asymptotikus Értékek Meghatározásáról" Leopold Fejér Gesammelte Arbeiten I (in German), Basel: Birkhäuser Basel, pp. 445– 503,doi :10.1007/978-3-0348-5902-8_31 ,ISBN 978-3-0348-5903-5 , retrieved2025-07-07 ^ D. Borwein, J. M. Borwein, R. E. Crandall, "Effective Laguerre asymptotics",SIAM J. Numer. Anal. , vol. 46 (2008), no. 6, pp. 3285–3312doi :10.1137/07068031X ^ Szegő, pp. 200–201 ^a b Driver, K.; Jordaan, K. (January 2013)."Inequalities for Extreme Zeros of Some Classical Orthogonal andq-orthogonal Polynomials" .Mathematical Modelling of Natural Phenomena .8 (1):48– 59.doi :10.1051/mmnp/20138103 .ISSN 0973-5348 . ^a b c Gatteschi, Luigi (2002-07-01)."Asymptotics and bounds for the zeros of Laguerre polynomials: a survey" Journal of Computational and Applied Mathematics . Selected papers of the Int. Symp. on Applied Mathematics, August 2000, Dalian, China.144 (1):7– 27.doi :10.1016/S0377-0427(01)00549-0 .ISSN 0377-0427 . ^ Dimitrov, Dimitar K.; Rafaeli, Fernando R. (2009-12-01)."Monotonicity of zeros of Laguerre polynomials" Journal of Computational and Applied Mathematics . 9th OPSFA Conference.233 (3):699– 702.doi :10.1016/j.cam.2009.02.038 .ISSN 0377-0427 . ^ (Szegő 1975 , Section 6.21. Inequalities for the zeros of the classical polynomials) ^ Marcellán, F.; Martínez-Finkelshtein, A.; Martínez-González, P. (2007-10-15)."Electrostatic models for zeros of polynomials: Old, new, and some open problems" .Journal of Computational and Applied Mathematics . Proceedings of The Conference in Honour of Dr. Nico Temme on the Occasion of his 65th birthday.207 (2):258– 272.arXiv :math/0512293 doi :10.1016/j.cam.2006.10.020 .ISSN 0377-0427 . ^ (Szegő 1975 , Section 6.7. Electrostatic interpretation of the zeros of the classical polynomials) ^a b Gupta, Dharma P.; Muldoon, Martin E. (2007)."Inequalities for the smallest zeros of Laguerre polynomials and their -analogues" .JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only] .8 (1): Paper No. 24, 7 p., electronic only–Paper No. 24, 7 p., electronic only.ISSN 1443-5756 . ^ Gawronski, Wolfgang (1987-07-01)."On the asymptotic distribution of the zeros of Hermite, Laguerre, and Jonquière polynomials" Journal of Approximation Theory .50 (3):214– 231.doi :10.1016/0021-9045(87)90020-7 .ISSN 0021-9045 . ^ 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. ^ Al-Salam, W. A. (1964-03-01)."Operational representations for the Laguerre and other polynomials" Duke Mathematical Journal .31 (1).doi :10.1215/S0012-7094-64-03113-8 .ISSN 0012-7094 . ^ Szegő, page 102, Equation (5.1.16) ^ G. H. Hardy, “Summation of a series of polynomials of Laguerre,” J. London Math. Soc., v. 7, 1932, pp. 138–139; addendum, 192. ^ E. Hille, “On Laguerre’s series. I, II, III,” Proc. Nat. Acad. Sci. U.S.A., v. 12, 1926, pp. 261–269; 348–352. ^ 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 . Abramowitz, Milton ;Stegun, Irene Ann , eds. (1983) [June 1964]."Chapter 22" .Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables ISBN 978-0-486-61272-0 LCCN 64-60036 .MR 0167642 .LCCN 65-12253 .Szegő, Gábor (1975).Orthogonal Polynomials . American Mathematical Society Colloquium Publications. Vol. 23 (4th ed.). Providence, RI: American Mathematical Society.Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010),"Orthogonal Polynomials" , inOlver, Frank W. J. ; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.),NIST Handbook of Mathematical Functions ISBN 978-0-521-19225-5 MR 2723248 Spain, B.; Smith, M.G. (1970). "10. Laguerre polynomials".Functions of mathematical physics . London: Van Nostrand Reinhold Company. "Laguerre polynomials" ,Encyclopedia of Mathematics EMS Press , 2001 [1994]George Arfken and Hans Weber (2000).Mathematical Methods for Physicists . Academic Press.ISBN 978-0-12-059825-0
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