Special functions · Orthogonal polynomials
The third classical family completes the set: Legendre polynomials live on \([-1, 1]\), Laguerre polynomials on \([0, \infty)\), and Hermite polynomials on the whole real line, with the Gaussian bell \(e^{-x^2}\) as weight. They are the polynomials of probability theory (the normal distribution) and of quantum mechanics, where they describe the stationary states of the harmonic oscillator. As before, everything flows from a generating function.
(The book numbers this point as 2° as well; it is the first point, 1°.) The generating function is an entire function of \(t\), so the series converges for all \(t\) and \(x\). Writing \( e^{2tx - t^2} = e^{x^2}e^{-(t - x)^2} \) shows where it comes from: shifting a Gaussian bell. Expanding:
\[ H_0 = 1,\quad H_1 = 2x,\quad H_2 = 4x^2 - 2,\quad H_3 = 8x^3 - 12x,\quad H_4 = 16x^4 - 48x^2 + 12. \]The leading coefficient of \(H_n\) is \(2^n\), and since \( e^{2t(-x) - t^2} = e^{2(-t)x - (-t)^2} \), \( H_n(-x) = (-1)^nH_n(x) \): \(H_n\) is even or odd with \(n\).
Why: multiply the series \( e^{2tx} = \sum_j\frac{(2x)^j}{j!}t^j \) and \( e^{-t^2} = \sum_k\frac{(-1)^k}{k!}t^{2k} \). The coefficient of \(t^n\) collects \(j + 2k = n\), i.e. \( \sum_k\frac{(-1)^k(2x)^{n-2k}}{k!(n - 2k)!} \); multiplying by \(n!\) gives the formula.
Not in the book, but the most compact description, is Rodrigues' formula:
\[ H_n(x) = (-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2}. \]It follows from the generating function: \( H_n = \frac{\partial^n}{\partial t^n}\Big[e^{x^2}e^{-(x - t)^2}\Big]_{t=0} \), and differentiating \(e^{-(x - t)^2}\) in \(t\) is the same as differentiating in \(x\) with a minus sign. Check: \( (e^{-x^2})' = -2xe^{-x^2} \), so \( H_1 = 2x \). ✓
Derivation: let \(G = e^{2tx - t^2}\). Then \( \frac{\partial G}{\partial t} = (2x - 2t)G \). Comparing coefficients of \(\frac{t^n}{n!}\): \( H_{n+1} = 2xH_n - 2nH_{n-1} \).
Differentiating \(G\) in \(x\) instead, \( \frac{\partial G}{\partial x} = 2tG \), gives the derivative rule
\[ H_n'(x) = 2n\,H_{n-1}(x), \]so, like Bernoulli polynomials, Hermite polynomials form an Appell-type sequence.
On the whole line, the Gaussian weight \(e^{-x^2}\) makes all integrals of polynomials finite, with \( \int_{-\infty}^\infty e^{-x^2}dx = \sqrt\pi \) (the Gaussian integral).
Why (the same pattern as for Legendre and Laguerre): for \(m < n\), use Rodrigues' formula, \( e^{-x^2}H_n = (-1)^n\frac{d^n}{dx^n}e^{-x^2} \), and integrate by parts \(n\) times against \(H_m\). The boundary terms vanish because \(e^{-x^2}\) decays faster than any polynomial grows, and \( H_m^{(n)} = 0 \). For \(m = n\), only the leading term \(2^nx^n\) contributes after \(n\) differentiations: \( 2^n\,n!\int e^{-x^2}dx = 2^nn!\sqrt\pi \).
Why: from the two rules of Section 3, \( H_n' = 2nH_{n-1} \) and \( H_n'' = 2nH_{n-1}' = 2n\cdot2(n - 1)H_{n-2} \). Rewrite the recurrence with \(n - 1\): \( H_n - 2xH_{n-1} + 2(n - 1)H_{n-2} = 0 \). Multiply by \(2n\) and substitute: \( 2nH_n - 2xH_n' + H_n'' = 0 \).
For non-integer \(n\), the equation still has solutions, but none of them is a polynomial, and they grow like \(e^{x^2}\) at infinity. Only for \(n = 0, 1, 2, \dots\) is there a solution (the polynomial \(H_n\)) with \( e^{-x^2/2}y \) decaying at \(\pm\infty\). This is exactly the mechanism of energy quantization in the next section.
A particle in the potential \( V = \frac12m\omega^2x^2 \) (a mass on a spring, a vibrating molecule) obeys the Schrödinger equation. In suitable units it reads
\[ -\psi'' + x^2\psi = 2\varepsilon\,\psi. \]Substituting \( \psi = e^{-x^2/2}y(x) \) turns it into Hermite's equation \( y'' - 2xy' + (2\varepsilon - 1)y = 0 \). A physically acceptable solution must be normalizable (\(\int|\psi|^2 < \infty\)), which by Section 5 forces \( 2\varepsilon - 1 = 2n \). Hence:
\[ \psi_n(x) = \frac{1}{\sqrt{2^nn!\sqrt\pi}}\,H_n(x)\,e^{-x^2/2}, \qquad \varepsilon_n = n + \frac12. \]The energy comes in equal steps, \(E_n = \hbar\omega\big(n + \frac12\big)\), and the lowest energy is not zero (the zero-point energy). The normalization constant is exactly the square root of the orthogonality constant, so the \(\psi_n\) form an orthonormal family.
| Legendre \(P_n\) | Laguerre \(L_n\) (book) | Hermite \(H_n\) | |
|---|---|---|---|
| Interval | \([-1, 1]\) | \([0, \infty)\) | \((-\infty, \infty)\) |
| Weight | \(1\) | \(e^{-x}\) | \(e^{-x^2}\) |
| Generating function | \( (1 - 2xt + t^2)^{-1/2} \) | \( \frac{1}{1 - t}e^{-\frac{xt}{1-t}} \) (\(\frac{t^n}{n!}\)) | \( e^{2tx - t^2} \) (\(\frac{t^n}{n!}\)) |
| Rodrigues | \( \frac{1}{2^nn!}\frac{d^n}{dx^n}(x^2 - 1)^n \) | \( e^x\frac{d^n}{dx^n}(x^ne^{-x}) \) | \( (-1)^ne^{x^2}\frac{d^n}{dx^n}e^{-x^2} \) |
| Recurrence | \( (n + 1)P_{n+1} = (2n + 1)xP_n - nP_{n-1} \) | \( L_{n+1} = (2n + 1 - x)L_n - n^2L_{n-1} \) | \( H_{n+1} = 2xH_n - 2nH_{n-1} \) |
| Norm \(\int w\,p_n^2\) | \( \frac{2}{2n + 1} \) | \( (n!)^2 \) | \( 2^nn!\sqrt\pi \) |
| Equation | \( (1 - x^2)y'' - 2xy' + n(n + 1)y = 0 \) | \( xy'' + (1 - x)y' + ny = 0 \) | \( y'' - 2xy' + 2ny = 0 \) |
| Physics | angular part (spheres) | radial part (hydrogen) | harmonic oscillator |
Try each one before opening the solution. More solved problems from this topic are in Series.
\( H_5 = 2xH_4 - 8H_3 = 32x^5 - 96x^3 + 24x - 64x^3 + 96x = 32x^5 - 160x^3 + 120x \).
From the explicit formula only the term with \(n - 2k = 0\) survives: \( H_{2k}(0) = \frac{(-1)^k(2k)!}{k!} \): \( 1, -2, 12, -120, \dots \) (and \(H_{2k+1}(0) = 0\)).
\( y = 4x^2 - 2 \), \( y' = 8x \), \( y'' = 8 \): \( 8 - 16x^2 + 4(4x^2 - 2) = 0 \). ✓
\( H_2 = 4x^2 - 2 \Rightarrow x^2 = \frac14H_2 + \frac12H_0 \).
\( 2^3\cdot3!\cdot\sqrt\pi = 48\sqrt\pi \).
Zeros of \(H_2 = 4x^2 - 2\): \( x = \pm\frac{1}{\sqrt2} \). Energy \( \varepsilon_2 = \frac52 \) (in units of \(\hbar\omega\)).
Hermite polynomials complete the classical trio of orthogonal polynomials, each tied to its interval and weight: Legendre with \(1\) on \([-1, 1]\), Laguerre with \(e^{-x}\) on \([0, \infty)\), Hermite with \(e^{-x^2}\) on the whole line. The structure is the same each time: a generating function gives the explicit formula and the recurrence; a Rodrigues formula gives orthogonality through repeated integration by parts; and a second-order differential equation has the polynomial as its only well-behaved solution. In the harmonic oscillator, that last fact is exactly why energy is quantized.
Related posts: Legendre polynomials, Laguerre polynomials, and the Gamma function (\(\int e^{-x^2}dx = \sqrt\pi = \Gamma(\frac12)\)).