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Laguerre Differential Equation


The Laguerre differential equation is given by

 xy^('')+(1-x)y^'+lambday=0.
(1)

Equation (1) is a special case of the more general associated Laguerre differential equation, defined by

 xy^('')+(nu+1-x)y^'+lambday=0
(2)

where lambda and nu are real numbers (Iyanaga and Kawada 1980, p. 1481; Zwillinger 1997, p. 124) with nu=0.

The general solution to the associated equation (2) is

 t=C_1U(-lambda,1+nu,x)+C_2L_lambda^nu(x),
(3)

where U(a,b,x) is a confluent hypergeometric function of the first kind and L_lambda^nu(x) is a generalized Laguerre polynomial.

Note that in the special case lambda=0, the associated Laguerre differential equation is of the form

 y^('')(x)+P(x)y^'(x)=0,
(4)

so the solution can be found using an integrating factor

mu=exp(intP(x)dx)
(5)
=exp(int(nu+1-x)/xdx)
(6)
=exp[(nu+1)lnx-x]
(7)
=x^(nu+1)e^(-x),
(8)

as

y=C_1int(dx)/mu+C_2
(9)
=C_1int(e^x)/(x^(nu+1))dx+C_2
(10)
=C_2-C_1x^(-nu)E_(1+nu)(-x),
(11)

where E_n(x) is the En-function.

The associated Laguerre differential equation has a regular singular point at 0 and an irregular singularity at infty. It can be solved using a series expansion,

xsum_(n=2)^(infty)n(n-1)a_nx^(n-2)+(nu+1)sum_(n=1)^(infty)na_nx^(n-1)-xsum_(n=1)^(infty)na_nx^(n-1)+lambdasum_(n=0)^(infty)a_nx^n=0
(12)
sum_(n=2)^(infty)n(n-1)a_nx^(n-1)+(nu+1)sum_(n=1)^(infty)na_nx^(n-1)-sum_(n=1)^(infty)na_nx^n+lambdasum_(n=0)^(infty)a_nx^n=0
(13)
sum_(n=1)^(infty)(n+1)na_(n+1)x^n+(nu+1)sum_(n=0)^(infty)(n+1)a_(n+1)x^n-sum_(n=1)^(infty)na_nx^n+lambdasum_(n=0)^(infty)a_nx^n=0
(14)
[(nu+1)a_1+lambdaa_0]+sum_(n=1)^(infty){[(n+1)n+(nu+1)(n+1)]a_(n+1)-na_n+lambdaa_n}x^n=0
(15)
[(nu+1)a_1+lambdaa_0]+sum_(n=1)^(infty)[(n+1)(n+nu+1)a_(n+1)+(lambda-n)a_n]x^n=0.
(16)

This requires

a_1=-lambda/(nu+1)a_0
(17)
a_(n+1)=(n-lambda)/((n+1)(n+nu+1))a_n
(18)

for n>1. Therefore,

 a_(n+1)=(n-lambda)/((n+1)(n+nu+1))a_n
(19)

for n=1, 2, ..., so

y=sum_(n=0)^(infty)a_nx^n
(20)
=a_0_1F_1(-lambda,nu+1,x)
(21)
=a_0[1-lambda/(nu+1)x-(lambda(1-lambda))/(2(nu+1)(nu+2))x^2-(lambda(1-lambda)(2-lambda))/(2·3(nu+1)(nu+2)(nu+3))x^3+...].
(22)

If lambda is a nonnegative integer, then the series terminates and the solution is given by

 y=a_0(lambda!L_lambda^nu(x))/((nu+1)_lambda),
(23)

where L_lambda^nu(x) is an associated Laguerre polynomial and (a)_n is a Pochhammer symbol. In the special case nu=0, the associated Laguerre polynomial collapses to a usual Laguerre polynomial and the solution collapses to

 y=a_0L_lambda(x).
(24)

See also

Associated Laguerre Polynomial, Laguerre Polynomial

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References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1481, 1980.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 120, 1997.

Referenced on Wolfram|Alpha

Laguerre Differential Equation

Cite this as:

Weisstein, Eric W. "Laguerre Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LaguerreDifferentialEquation.html

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