Special Functions from Power Series

Many of the most important functions in mathematics and physics arise as power series solutions to second-order linear ODEs with specific singularity structures.

Legendre Functions

Definition9.4Legendre equation

The Legendre equation is:

$(1 - x^2)y'' - 2xy' + \ell(\ell + 1)y = 0,$

where $\ell$ is a parameter. It has regular singular points at $x = \pm 1$ and an ordinary point at $x = 0$ . In standard form: $P(x) = -2x/(1-x^2)$ , $Q(x) = \ell(\ell+1)/(1-x^2)$ .

ExampleLegendre polynomials

At $x_0 = 0$ (ordinary point), the power series solution yields the recurrence:

$a_{n+2} = \frac{n(n+1) - \ell(\ell+1)}{(n+2)(n+1)} a_n.$

When $\ell$ is a non-negative integer, one series terminates, producing the Legendre polynomial $P_\ell(x)$ :

$P_0(x) = 1, \quad P_1(x) = x, \quad P_2(x) = \frac{3x^2 - 1}{2}, \quad P_3(x) = \frac{5x^3 - 3x}{2}.$

These are orthogonal on $[-1,1]$ : $\int_{-1}^{1} P_m(x)P_n(x)\,dx = \frac{2}{2n+1}\delta_{mn}$ .

The Rodrigues formula gives $P_n(x) = \frac{1}{2^n n!}\frac{d^n}{dx^n}(x^2-1)^n$ .

Hypergeometric Functions

Definition9.5Hypergeometric equation

The Gauss hypergeometric equation is:

$x(1-x)y'' + [c - (a+b+1)x]y' - aby = 0,$

with regular singular points at $x = 0, 1, \infty$ . The solution analytic at $x = 0$ is the hypergeometric function:

${}_{2}F_{1}(a, b; c; x) = \sum_{n=0}^{\infty} \frac{(a)_n(b)_n}{(c)_n \, n!} x^n, \quad |x| < 1,$

where $(a)_n = a(a+1)\cdots(a+n-1)$ is the Pochhammer symbol (rising factorial).

RemarkUnifying role of hypergeometric functions

Many special functions are special cases of ${}_{2}F_{1}$ :

Legendre: $P_\ell(x) = {}_{2}F_{1}(-\ell, \ell+1; 1; (1-x)/2)$ .
Chebyshev: $T_n(\cos\theta) = {}_{2}F_{1}(-n, n; 1/2; (1-\cos\theta)/2)$ .
Jacobi polynomials: expressible via ${}_{2}F_{1}$ with negative integer parameter.

The hypergeometric equation is the most general second-order Fuchsian equation with exactly three regular singular points (by Riemann's theorem, any such equation reduces to the hypergeometric form via a Mobius transformation).

Sturm-Liouville Theory

Theorem9.3Sturm-Liouville eigenvalue problem

The Sturm-Liouville problem consists of the ODE:

$\frac{d}{dx}\left[p(x)\frac{dy}{dx}\right] + [q(x) + \lambda w(x)]y = 0, \quad a < x < b,$

with boundary conditions at $a$ and $b$ , where $p > 0$ , $w > 0$ , and $p, q, w$ are continuous. Then:

There exist infinitely many eigenvalues $\lambda_1 < \lambda_2 < \cdots \to \infty$ .
The eigenfunctions $\{y_n\}$ form a complete orthogonal set in $L^2([a,b], w)$ .
Any $f \in L^2([a,b], w)$ has the eigenfunction expansion $f = \sum c_n y_n$ with $c_n = \langle f, y_n \rangle / \|y_n\|^2$ .

ExampleClassical Sturm-Liouville problems

Fourier-Legendre: $p = 1-x^2$ , $q = 0$ , $w = 1$ on $(-1,1)$ . Eigenvalues $\lambda_n = n(n+1)$ , eigenfunctions $P_n(x)$ .
Fourier-Bessel: $p = x$ , $q = -\nu^2/x$ , $w = x$ on $(0,b)$ . Eigenvalues $\lambda_n = j_{\nu,n}^2/b^2$ (where $j_{\nu,n}$ are zeros of $J_\nu$ ), eigenfunctions $J_\nu(\sqrt{\lambda_n}x)$ .
Hermite: $p = e^{-x^2}$ , $q = 0$ , $w = e^{-x^2}$ on $(-\infty,\infty)$ (singular Sturm-Liouville). Eigenvalues $\lambda_n = 2n$ , eigenfunctions $H_n(x)$ .