This article should resolve the confusion that is caused by different definitions in various applications.
A polynomial is a finite sum of terms like akxk, where k is a positive integer or zero. There are sets of polynomials such that the product of any two different ones, multiplied by a function w(x) called a weight function and integrated over a certain interval, vanishes. The set is called a set of orthogonal polynomials. Among other things, this makes it possible to expand an arbitrary function f(x) as a sum of the polynomials, each multiplied by a coefficient c(k), which is easily and uniquely determined by integration. A Fourier series is similar, but the orthogonal functions are not polynomials. These functions can also be used to specify basis states in quantum mechanics, which must be orthogonal.
The Laguerre polynomials are orthogonal on the interval from 0 to ∞ with respect to the weight function w(x) = e-x. Surprisingly, this is sufficient to determine the polynomials up to a multiplicative factor. Let's start with the expression . When we take the derivative of the quantity on the right, we notice that it will always survive in the result, multiplied by -1, as the term with the highest power of x, namely n. Moreover, each term will contain the factor e-x that will be cancelled by the exponential in front. Therefore, the result will be a polynomial of degree n, and the coefficient of xn will be (-1)n. Moreover, the form of this definition will guarantee that the polynomials belonging to different values of n are orthogonal. It is, in fact, a Rodrigues' formula, like those used to define other kinds of orthogonal polynomials. From this formula, L0 = 1, L1 = 1 - x. The polynomials satisfy the recurrence relation Ln+1 = (2n + 1 - x)Ln - n2Ln-1, and the differential equation xL"n + (1 - x)L'n +nLn = 0.
It is now necessary to take a stand in order to define the normalization, or overall multiplicative constant, of the polynomials Ln that one will use. I have three classic references, (1) Abramowitz and Stegun, a mathematical handbook; (2) Pauling and Wilson, a quantum mechanics text; and (3) Jackson, a text on orthogonal polynomials, which I shall abbreviate AS, PW, and DJ. Each uses a different normalization! DJ makes the coefficient of xn unity, which multiplies the expression we have used by (-1)n. This, I think, is a good choice that results in the simplest expressions. PW let the coefficient alternate in sign, so they use exactly our original expression. AS want the coefficient of xn to be 1/n!, so they multiply by (-1)n/n!. The differences are slight, but they make the recurrence relations, and other formulas, have different coefficients, which can confuse the student greatly. The differential equation does not depend on the normalization, which is some comfort.
Things become more confusing when one realizes that generalized Laguerre polynomials L(α)n can be defined using a weight function xαe-x instead of the plain exponential. These functions are required for the important quantum-mechanical application of the hydrogen atom, for example. In PW, these functions arise as derivatives of the basic polynomials for α = 0, and they put Lmn = dmLn/dxm. This differs from the notations used by mathematicians in that n is no longer the order of the polynomial, which is n - m instead. The relationship between the L(α)n of AS and the Lmn of PW is not immediately obvious.
To clarify matters, we note in AS that L(α)n = (-1)αdαL(0)n+α/dxα. This relates the derivatives with the polynomials found from Rodrigues' formula satisfactorily. Taking into account the differences in the definitions, we can now assert that the PW Lmn = (-1)nL(m)n - m of AS.
Orthogonality of the generalized Laguerre polynomials is expressed by the integral , where the polynomials are defined as in AS. α does not have to be integral; in that case, the factorial should be replaced by the gamma function. m,n are, of course, always integral, and the Kronecker delta is zero if m is unequal to n, unity if m is equal to n. This integral can be evaluated by replacing the polynomial of higher order by the Rodrigues definition, and then integrating by parts successively. In many applications, such as quantum mechanics, the weight function is considered part of the function, which can then be called a Laguerre function rather than polynomial. The polynomials as defined in AS satisfy the recurrence relation (n + 1)L(α)n+1 = (2n + α + 1 - x)L(α)n - (n + α)L(α)n-1 and the differential equation xL"(α)n + (α + 1 - x)L'(α)n + nL(α)n = 0. The differential equation is not affected by changes in a multiplicative constant in the definition of the polynomial.
For a quantum-mechanical treatment of the hydrogen atom, see PW. It is a straightforward analysis, but a rather involved one. The wave function is a product of a spherical harmonic Ylm and a radial function Rnl. The integers n, l, m are the quantum numbers. n = 1, 2, 3, ..., l = 0, 1, ..., n - 1, m = l, l - 1, l - 2, ..., -l. n is the principal quantum number, on which the energy alone depends, l is the orbital quantum number giving the total angular momentum, and m is the magnetic quantum number giving the component of l along the z-axis. The radial factor of the wave function is e-x/2xlL(2l + 1)n - l - 1, where x is proportional to the radius r. The order of the polynomial, n - l - 1, gives the number of radial nodes (zeros) of the wave function. When two such functions for the same value of l are multiplied, and are integrated from 0 to ∞, the weight function becomes x2l + 1e-x, as it should, a factor x coming from the radial integration. Therefore, wave functions for the same l but different n are orthogonal. The spherical harmonics guarantee orthogonality for different l,m.
Enough information has now been presented to permit the calculation of the polynomials according to the various definitions, and to solve the differential equations occurring in applications. The Laguerre polynomials are really rather straightforward examples of orthogonal polynomials, and most of their properties can be derived from Rodrigues' formula.
Composed by J. B. Calvert
Created 14 November 2000