Laguerre Polynomial

Solutions to the Laguerre Differential Equation with $\nu=0$ are called Laguerre polynomials. The Laguerre polynomials $L_n(x)$ are illustrated above for $x\in[0,1]$ and $n=1$, 2, ..., 5.

The Rodrigues formula for the Laguerre polynomials is

$$L_n(x) = {e^x\over n!}{d^n\over dx^n}(x^ne^{-x})$$ (1)

and the Generating Function for Laguerre polynomials is

$$g(x,z) = {\mathop{\rm exp}\nolimits \left({-{xz\over 1-z}}\right)\over 1-z}$$

$$= 1+(-x+1)z+({\textstyle{1\over 2}}x^2-2x+1)z^2+(-{\textstyle{1\over 6}}x^3+{\textstyle{3\over 2}}x^2-3x+1)z^3+\ldots.$$ (2)

A Contour Integral is given by

$$L_n(x) = {1\over 2\pi i}\int{e^{-xz/(1-z)}\over (1-z)z^{n+1}}\, dz.$$ (3)

The Laguerre polynomials satisfy the Recurrence Relations

$$(n+1)L_{n+1}(x)=(2n+1-x)L_n(x)-nL_{n-1}(x)$$ (4)

(Petkovsek et al. 1996) and

$$xL_n'(x)=nL_n(x)-nL_{n-1}(x).$$ (5)

The first few Laguerre polynomials are

$$\begin{eqnarray*} L_0(x) &=& 1\\ L_1(x) &=& -x+1\\ L_2(x) &=& {\textstyle{1... ...^2-4x+2)\\ L_3(x) &=& {\textstyle{1\over 6}}(-x^3+9x^2-18x+6). \end{eqnarray*}$$

Solutions to the associated Laguerre Differential Equation with $\nu\not=0$ are called associated Laguerre polynomials $L_n^k(x)$. In terms of the normal Laguerre polynomials,

$$L_n(x)=L_n^0(x).$$ (6)

The Rodrigues formula for the associated Laguerre polynomials is

$$L_n^k(x) = {e^xx^{-k}\over n!}{d^n\over dx^n}(e^{-x}x^{n+k})$$

$$= (-1)^n {d^n\over dx^n}[L_{n+k}(x)]$$ (7)

$$= \sum_{m=0}^\infty (-1)^m {(n+k)!\over (n-m)!(k+m)!m!}x^m$$ (8)

and the Generating Function is

$$g(x,z)={\mathop{\rm exp}\nolimits \left({-{xz\over 1-z}}\rig... ...-x)z+{\textstyle{1\over 2}}[x^2-2(k+2)x+(k+1)(k+2)]z^2+\ldots.$$ (9)

The associated Laguerre polynomials are orthogonal over $[0, \infty)$ with respect to the Weighting Function $x^ne^{-x}$.

$$\int_0^\infty e^{-x}x^k L_n^k(x)L_m^k(x)\,dx = {(n+k)!\over n!}\delta_{mn},$$ (10)

where $\delta_{mn}$ is the Kronecker Delta. They also satisfy

$$\int_0^\infty e^{-x}x^{k+1}[L_n^k(x)]^2\,dx = {(n+k)!\over n!}(2n+k+1).$$ (11)

Recurrence Relations include

$$\sum_{\nu=0}^n L_\nu^{(\alpha)}(x)=L_n^{(\alpha+1)}(x)$$ (12)

and

$$L_n^{(\alpha)}(x)=L_n^{(\alpha+1)}(x)-L_{n-1}^{(\alpha+1)}(x).$$ (13)

The Derivative is given by

$${d\over dx}L_n^{(\alpha)}(x) = -L_{n-1}^{(\alpha+1)}(x)$$

$$= x^{-1}[nL_n^{(\alpha)}(x)-(n+\alpha)L_{n-1}^{(\alpha)}(x).$$ (14)

In terms of the Confluent Hypergeometric Function,

$$L_n^k(x) = {(k+1)_n\over n!}\,{}_1F_1(-b;k+1;x).$$ (15)

An interesting identity is

$$\sum_{n=0}^\infty {L_n^{(\alpha)}(x)\over \Gamma(n+\alpha+1)} w^n =e^w (xw)^{-\alpha/2}J_\alpha(2\sqrt{xw}\,),$$ (16)

where $\Gamma(z)$ is the Gamma Function and $J_\alpha(z)$ is the Bessel Function of the First Kind (Szegö 1975, p. 102). An integral representation is

$$e^{-x}x^{\alpha/2}L_n^{(\alpha)}(x)={1\over n!} \int_0^\infty e^{-t}t^{n+\alpha/2} J_\alpha(2\sqrt{tx}\,)\,dt$$ (17)

for $n=0$, 1, ...and $\alpha>-1$. The Discriminant is

$$D_n^{(\alpha)}=\prod_{\nu=1}^n \nu^{\nu-2n+2} (\nu+\alpha)^{\nu-1}$$ (18)

(Szegö 1975, p. 143). The Kernel Polynomial is

$$K_n^{(k)}(x,y)={n+1\over\Gamma(k+1)} {n+k\choose n}^{-1} {L_n^{(k)}(x)L_{n+1}^{(k)}(y)-L_{n+1}^{(k)}(x)L_n{(k)}(y)\over x-y},$$ (19)

where ${n\choose k}$ is a Binomial Coefficient (Szegö 1975, p. 101).

The first few associated Laguerre polynomials are

$$\begin{eqnarray*} L_0^k(x) &=& 1\\ L_1^k(x) &=& -x+k+1\\ L_2^k(x) &=& {\tex... ...xtstyle{1\over 6}}[-x^3+3(k+3)x^2-3(k+2)(k+3)x+(k+1)(k+2)(k+3)]. \end{eqnarray*}$$

See also Sonine Polynomial

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Orthogonal Polynomials.'' Ch. 22 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 771-802, 1972.

Arfken, G. ``Laguerre Functions.'' §13.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 721-731, 1985.

Chebyshev, P. L. ``Sur le développement des fonctions à une seule variable.'' Bull. Ph.-Math., Acad. Imp. Sc. St. Pétersbourg 1, 193-200, 1859.

Chebyshev, P. L. Oeuvres, Vol. 1. New York: Chelsea, pp. 499-508, 1987.

Iyanaga, S. and Kawada, Y. (Eds.). ``Laguerre Functions.'' Appendix A, Table 20.VI in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1481, 1980.

Laguerre, E. de. ``Sur l'intégrale $\int_x^{+\infty} x^{-1}e^{-x}\,dx$.'' Bull. Soc. math. France 7, 72-81, 1879. Reprinted in Oeuvres, Vol. 1. New York: Chelsea, pp. 428-437, 1971.

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, pp. 61-62, 1996.

Sansone, G. ``Expansions in Laguerre and Hermite Series.'' Ch. 4 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 295-385, 1991.

Spanier, J. and Oldham, K. B. ``The Laguerre Polynomials $L_n(x)$.'' Ch. 23 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 209-216, 1987.

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., 1975.