Classical orthogonal polynomials
Classical orthogonal polynomials are a distinguished class of orthogonal polynomials, specifically comprising the Hermite, Laguerre, and Jacobi families, which are defined as special cases or limiting cases of the more general Askey-Wilson polynomials.[1] These polynomials form a sequence \{p_n(x)\}_{n=0}^\infty where each p_n(x) is a polynomial of degree n, and they satisfy the orthogonality condition \int_a^b p_n(x) p_m(x) w(x) \, dx = 0 for n \neq m, with respect to a positive weight function w(x) over a finite or infinite interval [a, b].[2] A defining characteristic is that they satisfy a second-order linear differential equation of the Sturm-Liouville form \sigma(x) y'' + \tau(x) y' + \lambda_n y = 0, where \sigma(x) is a polynomial of degree at most 2 and \tau(x) is linear, both independent of n, and \lambda_n is the eigenvalue.[3] The classical families are distinguished by their specific intervals of orthogonality and corresponding weight functions, as summarized below: These polynomials also admit explicit representations, such as the Rodrigues formula p_n(x) = \frac{1}{a_n w(x)} \frac{d^n}{dx^n} [w(x) \sigma^n(x)], where a_n is a normalization constant, and their derivatives form another orthogonal set.[3] Key properties of classical orthogonal polynomials include a three-term recurrence relation p_{n+1}(x) = (A_n x + B_n) p_n(x) - C_n p_{n-1}(x), with coefficients A_n, B_n, C_n independent of x, and the fact that all n zeros of p_n(x) are real, simple, and lie within the interval of orthogonality, interlacing with those of p_{n+1}(x).[2] They are monic or can be normalized such that the leading coefficient is 1, and the orthogonality constants h_n = \int_a^b p_n^2(x) w(x) \, dx are positive and grow with n.[2] These features make them fundamental in the theory of orthogonal polynomials, as established in foundational works like Szegő's 1939 treatise.[5] Classical orthogonal polynomials have broad applications in mathematics and physics, including numerical integration via Gauss quadrature, approximation theory for least-squares fitting, solving Sturm-Liouville boundary value problems, and modeling quantum mechanical systems such as the harmonic oscillator (Hermite) and hydrogen atom (Laguerre). They also appear in statistics for moment problems, harmonic analysis, and random matrix theory for eigenvalue distributions.[6]General Framework
Definition of Orthogonal Polynomials
Orthogonal polynomials form a sequence \{P_n(x)\}_{n=0}^\infty of polynomials, where each P_n(x) has exact degree n, that satisfy an orthogonality condition with respect to a positive weight function w(x) > 0 on a finite or infinite interval (a, b). Specifically, the sequence is orthogonal if \int_a^b P_m(x) P_n(x) w(x) \, dx = 0 \quad \text{for all } m \neq n, and \int_a^b [P_n(x)]^2 w(x) \, dx = h_n > 0 \quad \text{for all } n, where h_n is the positive norm squared of P_n(x). The weight function w(x) must be such that the integrals exist and the moments \int_a^b x^k w(x) \, dx are finite for all k \in \mathbb{N}_0. This condition ensures the polynomials are mutually perpendicular in a weighted L^2 space, enabling efficient expansions and approximations of functions.[2] The orthogonality can be formulated using an inner product on the space of integrable functions: \langle f, g \rangle = \int_a^b f(x) g(x) w(x) \, dx. Under this inner product, the polynomials satisfy \langle P_m, P_n \rangle = 0 for m \neq n and \langle P_n, P_n \rangle = h_n > 0, making \{P_n(x)\} an orthogonal basis for the subspace of polynomials in the weighted Hilbert space. If normalized such that h_n = 1 for all n, the sequence is orthonormal.[2] The polynomials are unique up to scalar multiples, and common conventions fix the leading coefficient k_n of P_n(x) = k_n x^n + \ lower\ terms. A frequent choice is the monic normalization, where k_n = 1 for all n, simplifying computations in some contexts, though other scalings (e.g., to match specific applications) are also used.[2][7] The study of orthogonal polynomials originated in the 19th century, with early developments tied to least squares approximations; Adrien-Marie Legendre introduced related ideas in 1782 for gravitational potential expansions, while Pafnuty Chebyshev advanced the general theory in the 1850s–1870s through work on discrete approximations and continued fractions.[8][6] To contrast, orthogonality is not limited to polynomials; for example, the trigonometric functions \{\sin(nx), \cos(mx)\}_{n,m=0}^\infty form an orthogonal set on the interval [-\pi, \pi] with respect to the unweighted inner product \int_{-\pi}^\pi f(x) g(x) \, dx = 0 for distinct frequencies, as seen in Fourier series expansions.[9] Specific families of orthogonal polynomials, such as the Jacobi, Laguerre, and Hermite polynomials, exemplify the concept while satisfying additional properties like solutions to Sturm-Liouville equations.[10]Classical Orthogonal Polynomials
Classical orthogonal polynomials form a fundamental subclass of orthogonal polynomials, distinguished by their explicit solvability through Sturm-Liouville eigenvalue problems. Specifically, they satisfy the second-order linear differential equation \frac{d}{dx} \left[ \sigma(x) \frac{dy}{dx} \right] + \lambda \omega(x) y = 0, where \sigma(x) and \omega(x) are polynomials of low degree, ensuring polynomial eigenfunctions y_n(x) of degree n for eigenvalues \lambda_n = -n(n + \alpha + \beta + 1) or analogous forms depending on the family.[11] This structure arises in the context of self-adjoint operators on appropriate intervals, with \omega(x) serving as the weight function for orthogonality.[11] The classical families comprise three primary types: Jacobi, Laguerre, and Hermite polynomials, each associated with distinct intervals and weight functions. Jacobi polynomials P_n^{(\alpha,\beta)}(x) are orthogonal on the finite interval [-1, 1] with respect to the weight (1 - x)^\alpha (1 + x)^\beta, where the parameters satisfy \alpha > -1 and \beta > -1 to ensure integrability.[11] Generalized Laguerre polynomials L_n^{(\alpha)}(x) are defined on the semi-infinite interval [0, \infty) with weight x^\alpha e^{-x}, again requiring \alpha > -1.[11] Standard Hermite polynomials H_n(x) (or probabilists' He_n(x)) operate on the entire real line (-\infty, \infty) with Gaussian weight e^{-x^2} (or e^{-x^2/2}), featuring no additional parameters.[11] These families are deemed "classical" due to their weight functions satisfying Pearson's differential equation, \frac{d}{dx} \left[ \sigma(x) \omega(x) \right] = \tau(x) \omega(x), where \sigma(x) and \tau(x) are polynomials of degrees at most 2 and 1, respectively; this property enables explicit representations and differentiates them as the only such systems yielding closed-form polynomial solutions via hypergeometric series or Rodrigues formulas.[11] Bochner's classification theorem confirms that these are precisely the second-order linear differential equations with polynomial coefficients possessing infinitely many polynomial solutions.[11]Primary Families
Jacobi Polynomials
Jacobi polynomials form the most general family among the classical orthogonal polynomials, defined on the finite interval [-1, 1] with respect to the weight function w(x) = (1 - x)^\alpha (1 + x)^\beta, where \alpha > -1 and \beta > -1 ensure integrability.[11] These parameters allow flexibility in modeling singularities at the endpoints, distinguishing them from other classical families like Laguerre polynomials on [0, \infty) or Hermite polynomials on (-\infty, \infty). The orthogonality relation states that \int_{-1}^1 P_m^{(\alpha, \beta)}(x) P_n^{(\alpha, \beta)}(x) w(x) \, dx = \delta_{mn} h_n, where h_n is the normalization constant.[11] The explicit representation of the Jacobi polynomial P_n^{(\alpha, \beta)}(x) is given by the Rodrigues formula: P_n^{(\alpha, \beta)}(x) = \frac{(-1)^n}{2^n n!} (1 - x)^{-\alpha} (1 + x)^{-\beta} \frac{d^n}{dx^n} \left[ (1 - x)^{\alpha + n} (1 + x)^{\beta + n} \right]. This form highlights the role of the weight function in the differentiation process.[12] Alternatively, Jacobi polynomials connect to hypergeometric functions via P_n^{(\alpha, \beta)}(x) = \frac{(\alpha + 1)_n}{n!} \, {}_2F_1 \left( -n, n + \alpha + \beta + 1; \alpha + 1; \frac{1 - x}{2} \right), where ( \cdot )_n denotes the Pochhammer symbol and {}_2F_1 is the Gauss hypergeometric function; this expression terminates as a polynomial of degree n.[12] In the standard normalization, the leading coefficient of P_n^{(\alpha, \beta)}(x) is \frac{(n + \alpha + \beta + 1)_n}{2^n n!}, making it non-monic unless adjusted.[11] The value at the right endpoint is P_n^{(\alpha, \beta)}(1) = \frac{(\alpha + 1)_n}{n!}, independent of \beta.[12] As n \to \infty, near the endpoints x = \pm 1 (corresponding to \theta \to 0 or \theta \to \pi in x = \cos \theta), the scaled Jacobi polynomials exhibit oscillatory behavior: (\sin \frac{\theta}{2})^{\alpha + 1/2} (\cos \frac{\theta}{2})^{\beta + 1/2} P_n^{(\alpha, \beta)}(\cos \theta) \sim \pi^{-1/2} n^{-1/2} \cos \left( \frac{1}{2} (2n + \alpha + \beta + 1) \theta - \frac{1}{4} (2\alpha + 1) \pi \right), with higher-order terms O(n^{-3/2}), capturing the concentration and amplitude modulation due to the weight singularities for \alpha, \beta > -1.[13]Laguerre Polynomials
Laguerre polynomials, denoted as L_n^{(\alpha)}(x), form a family of classical orthogonal polynomials defined for nonnegative integers n and parameter \alpha > -1. They are orthogonal over the semi-infinite interval [0, \infty) with respect to the weight function w(x) = x^\alpha e^{-x}. The integral \int_0^\infty L_m^{(\alpha)}(x) L_n^{(\alpha)}(x) x^\alpha e^{-x} \, dx = 0 holds for m \neq n, with the squared norm \int_0^\infty [L_n^{(\alpha)}(x)]^2 x^\alpha e^{-x} \, dx = \frac{\Gamma(n + \alpha + 1)}{n!}.[11][14] The standard Laguerre polynomials correspond to the case \alpha = 0, denoted L_n(x) = L_n^{(0)}(x), which are orthogonal with weight e^{-x}. An explicit representation is given by the finite sum L_n^{(\alpha)}(x) = \sum_{k=0}^n (-1)^k \binom{n + \alpha}{n - k} \frac{x^k}{k!}. Alternatively, the Rodrigues formula provides L_n^{(\alpha)}(x) = \frac{e^x x^{-\alpha}}{n!} \frac{d^n}{dx^n} \left( e^{-x} x^{n + \alpha} \right). These polynomials are of degree n, with leading coefficient (-1)^n / n! in the standard normalization.[12] Laguerre polynomials are also expressible in terms of the confluent hypergeometric function of the first kind: L_n^{(\alpha)}(x) = \frac{(\alpha + 1)_n}{n!} \, {}_1F_1(-n; \alpha + 1; x), where (\alpha + 1)_n is the Pochhammer symbol and {}_1F_1 is the Kummer function.[15] In quantum mechanics, associated Laguerre polynomials (\alpha = 2l + 1, where l is the azimuthal quantum number) appear in the radial wave functions of the hydrogen atom.[16]Hermite Polynomials
Hermite polynomials constitute one of the primary families of classical orthogonal polynomials, defined over the unbounded interval (-\infty, \infty) with respect to the Gaussian weight function w(x) = e^{-x^2}.[11] They exhibit even or odd symmetry depending on the degree n, reflecting the symmetric nature of the domain and weight, which distinguishes them from polynomials on bounded or half-line intervals. This symmetry arises from their construction and makes them particularly suitable for applications involving Gaussian integrals and probability distributions.[17] There are two standard conventions for defining Hermite polynomials: the physicists' version H_n(x) and the probabilists' version \mathit{He}_n(x). The physicists' Hermite polynomials are given by the Rodrigues formula H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} \left( e^{-x^2} \right), orthogonal with respect to the weight e^{-x^2} and normalization constant h_n = \sqrt{\pi} \, 2^n n!.[11] In contrast, the probabilists' version uses \mathit{He}_n(x) = (-1)^n e^{x^2/2} \frac{d^n}{dx^n} \left( e^{-x^2/2} \right), orthogonal with respect to e^{-x^2/2} and normalization h_n = \sqrt{2\pi} \, n!, with a leading coefficient of 1 for monic polynomials.[11] These conventions differ in scaling, with the physicists' version incorporating a factor of $2^n in the leading term to align with quantum mechanical contexts.[17] An explicit expression for the physicists' Hermite polynomials is the finite sum H_n(x) = n! \sum_{m=0}^{\lfloor n/2 \rfloor} \frac{(-1)^m (2x)^{n-2m}}{m! (n-2m)!}, which highlights their polynomial nature with terms alternating in sign and powers decreasing by even steps.[17] The leading coefficient is $2^n, ensuring the highest-degree term is $2^n x^n.[11] Regarding parity, H_n(-x) = (-1)^n H_n(x), so even-degree polynomials are even functions and odd-degree ones are odd, with H_n(0) = 0 for odd n and specific nonzero values for even n, such as H_0(0) = 1 and H_2(0) = -2.[17] In physics, Hermite polynomials appear in the wavefunctions of the quantum harmonic oscillator.[11]Shared Mathematical Properties
Orthogonality Conditions
Classical orthogonal polynomials satisfy an orthogonality relation with respect to a positive weight function w(x) on a finite or infinite interval (a, b). Specifically, for the sequence of polynomials \{P_n(x)\}_{n=0}^\infty, the condition is \int_a^b P_m(x) P_n(x) w(x) \, dx = h_n \delta_{mn}, where \delta_{mn} is the Kronecker delta, and h_n > 0 is the squared norm of P_n(x). This inner product structure ensures the polynomials are orthogonal, forming a basis for the space of square-integrable functions with respect to the measure w(x) \, dx.[2] For the Jacobi polynomials P_n^{(\alpha, \beta)}(x), defined on (-1, 1) with weight w(x) = (1 - x)^\alpha (1 + x)^\beta where \alpha, \beta > -1, the squared norm is h_n = \frac{2^{\alpha + \beta + 1} \Gamma(n + \alpha + 1) \Gamma(n + \beta + 1)}{(2n + \alpha + \beta + 1) n! \, \Gamma(n + \alpha + \beta + 1)}. This explicit form arises from the hypergeometric representation and beta integral evaluations.[11] The generalized Laguerre polynomials L_n^{(\alpha)}(x), orthogonal on (0, \infty) with weight w(x) = e^{-x} x^\alpha for \alpha > -1, have squared norm h_n = \frac{\Gamma(n + \alpha + 1)}{n!}. This simplifies due to the confluent hypergeometric structure and properties of the gamma function.[11] For Hermite polynomials, two common conventions exist. The physicists' Hermite polynomials H_n(x), orthogonal on (-\infty, \infty) with weight w(x) = e^{-x^2}, satisfy h_n = \sqrt{\pi} \, 2^n n!. In contrast, the probabilists' Hermite polynomials \mathrm{He}_n(x), with weight w(x) = e^{-x^2/2}, have h_n = \sqrt{2\pi} \, n!. These norms reflect the scaling differences in the weight functions and standardization.[11] A key identity connecting these orthogonality relations is the Christoffel–Darboux formula, which provides a closed form for the sum of products of orthogonal polynomials: \sum_{k=0}^n \frac{P_k(x) P_k(y)}{h_k} = \frac{k_n}{h_n k_{n+1}} \frac{P_{n+1}(x) P_n(y) - P_n(x) P_{n+1}(y)}{x - y}, for x \neq y, where k_n is the leading coefficient of P_n(x). Here, the recurrence coefficient a_n in the three-term relation x P_n(x) = a_n P_{n+1}(x) + b_n P_n(x) + a_{n-1} P_{n-1}(x) relates to k_n / k_{n+1}, often simplifying the prefactor to a_n / (h_n a_{n-1}) in monic or orthonormal bases. This formula is fundamental for kernel representations and quadrature approximations in orthogonal polynomial theory.[2]Rodrigues' Formula
The Rodrigues formula offers a constructive and unified approach to generating the classical orthogonal polynomials by applying repeated differentiation to a product involving the weight function and a power of a specific polynomial derived from the associated Sturm-Liouville operator. In general, it takes the form P_n(x) = \frac{1}{w(x)} \frac{d^n}{dx^n} \left[ \sigma(x)^n w(x) \right], up to a normalization constant, where w(x) is the weight function over the appropriate interval, and \sigma(x) is the polynomial coefficient in the Sturm-Liouville form of the differential equation satisfied by the polynomials.[12] For Jacobi polynomials P_n^{(\alpha, \beta)}(x), defined on the interval [-1, 1] with weight w(x) = (1 - x)^\alpha (1 + x)^\beta for \alpha, \beta > -1, the polynomial \sigma(x) = 1 - x^2, and the normalized formula is P_n^{(\alpha, \beta)}(x) = \frac{(-1)^n}{2^n n! \, (1 - x)^\alpha (1 + x)^\beta} \frac{d^n}{dx^n} \left[ (1 - x)^{n + \alpha} (1 + x)^{n + \beta} \right]. This expression follows from substituting the specific \sigma(x) and w(x) into the general form, with the constant ensuring the standard leading coefficient of \frac{1}{2^n} \binom{2n}{n}^{-1} for the monic case when \alpha = \beta = 0.[12] For Laguerre polynomials L_n^{(\alpha)}(x), defined on [0, \infty) with weight w(x) = e^{-x} x^\alpha for \alpha > -1, \sigma(x) = x, and the normalized formula is L_n^{(\alpha)}(x) = \frac{1}{n! \, e^{-x} x^\alpha} \frac{d^n}{dx^n} \left[ e^{-x} x^{n + \alpha} \right]. Here, the constant n! normalizes the leading coefficient to (-1)^n / n!.[12] For Hermite polynomials H_n(x), defined on (-\infty, \infty) with weight w(x) = e^{-x^2}, \sigma(x) = 1, and the normalized formula is H_n(x) = \frac{(-1)^n}{e^{-x^2}} \frac{d^n}{dx^n} \left[ e^{-x^2} \right]. The constant (-1)^n ensures the leading coefficient is $2^n.[12] A brief proof sketch confirms that the Rodrigues formula yields a polynomial of exact degree n. The expression \sigma(x)^n w(x) is an infinitely differentiable function (smooth on the interior of the interval), and its n-th derivative is a polynomial because higher derivatives beyond the degree of \sigma^n (which is n \cdot \deg \sigma) vanish, while the leading term arises from differentiating the highest-degree part of \sigma(x)^n exactly n times, yielding a non-zero coefficient involving n! times the leading coefficient of \sigma(x) raised to the n-th power.[12][18] One key advantage of the Rodrigues formula is that it inherently encodes the orthogonality property with respect to w(x). For any polynomial Q(x) of degree less than n, the integral \int P_n(x) Q(x) w(x) \, dx = 0 follows from integration by parts applied n times, which transfers all derivatives to Q(x), resulting in zero due to Q^{(n+1)} = 0, while the boundary terms vanish owing to the specific form of w(x) \sigma(x)^k (for k < n) at the endpoints of the interval.[12][18] This directly confirms the orthogonality integrals without requiring separate verification.Sturm-Liouville Differential Equation
Classical orthogonal polynomials satisfy a shared second-order linear Sturm-Liouville differential equation in self-adjoint form \frac{d}{dx} \left[ p(x) \frac{dy}{dx} \right] + \lambda_n w(x) y = 0, where p(x) is a positive function on the interior of the interval (derived from \sigma(x) and w(x)), w(x) is the positive weight function, and \lambda_n is the eigenvalue. This equation is self-adjoint under appropriate boundary conditions, ensuring that the polynomial solutions y = p_n(x) are orthogonal with respect to the inner product defined by w(x). The structure distinguishes the classical families and facilitates explicit solutions via methods such as Rodrigues' formula.[19][20] The eigenvalue spectrum \{\lambda_n\}_{n=0}^\infty consists of simple, real values that are strictly increasing and tend to infinity as n \to \infty, guaranteeing a complete orthogonal basis in the weighted L^2 space over the interval. Boundary conditions are imposed to ensure self-adjointness; for finite or semi-infinite intervals with singular endpoints, these typically involve finiteness of the solutions or natural conditions at regular singular points.[20] For the Jacobi polynomials P_n^{(\alpha,\beta)}(x), the self-adjoint equation is \frac{d}{dx} \left[ (1 - x)^{\alpha + 1} (1 + x)^{\beta + 1} \frac{dy}{dx} \right] + \lambda_n (1 - x)^\alpha (1 + x)^\beta y = 0, with eigenvalues \lambda_n = n(n + \alpha + \beta + 1) on the interval [-1, 1] for \alpha, \beta > -1. The endpoints x = \pm 1 are regular singular points, where boundary conditions require the solutions to remain bounded or satisfy separated conditions to render the operator self-adjoint. This generalizes the Legendre case (\alpha = \beta = 0), where p(x) = 1 - x^2.[19][20] For the Laguerre polynomials L_n(x) (\alpha = 0), the self-adjoint equation is \frac{d}{dx} \left[ x e^{-x} \frac{dy}{dx} \right] + \lambda_n e^{-x} y = 0, with eigenvalues \lambda_n = n on [0, \infty). The endpoint x = 0 is a regular singular point, with boundary conditions ensuring square-integrability near the origin and decay at infinity for self-adjointness.[19][20] For the Hermite polynomials H_n(x), the self-adjoint equation is \frac{d}{dx} \left[ e^{-x^2} \frac{dy}{dx} \right] + \lambda_n e^{-x^2} y = 0, with eigenvalues \lambda_n = 2n on (-\infty, \infty). The infinite endpoints require boundary conditions that enforce rapid decay of the eigenfunctions at \pm \infty to maintain self-adjointness.[19][20]Derivations and Generating Functions
Derivation via Sturm-Liouville Theory
Classical orthogonal polynomials can be derived as eigenfunctions of specific Sturm-Liouville problems, where the weight function w(x) is prescribed on an appropriate interval, and the diffusion coefficient \sigma(x) is chosen as a low-degree polynomial to ensure the resulting coefficients are polynomials, leading to polynomial eigenfunctions. The general self-adjoint form of the Sturm-Liouville equation is \frac{d}{dx} \left[ \sigma(x) w(x) \frac{dy}{dx} \right] + \lambda w(x) y = 0, with boundary conditions such that the operator is self-adjoint. Expanding using the product rule gives the equivalent second-order linear differential equation \sigma(x) y''(x) + \tau(x) y'(x) + \lambda y(x) = 0, where \tau(x) satisfies Pearson's equation \frac{d}{dx} [\sigma(x) w(x)] = \tau(x) w(x), ensuring \tau(x) is a polynomial when \sigma(x) is. For classical families, \deg \sigma \leq 2 and \deg \tau = \deg \sigma + 1, guaranteeing a sequence of polynomial solutions y_n(x) = P_n(x) of degree n with eigenvalues \lambda_n quadratic in n. Orthogonality of distinct eigenfunctions P_m and P_n (with m \neq n) follows from the self-adjointness of the operator: integrating \int (P_n L P_m - P_m L P_n) w \, dx = 0 by parts yields (\lambda_m - \lambda_n) \int P_m P_n w \, dx = boundary term, where the boundary term vanishes due to the behavior of \sigma(x) w(x) at the endpoints of the interval.[21] For the Jacobi polynomials, begin with the weight w(x) = (1 - x)^\alpha (1 + x)^\beta on the interval (-1, 1), where \alpha > -1 and \beta > -1 ensure \int_{-1}^1 w(x) \, dx < \infty. To satisfy Pearson's equation with polynomial \tau(x), select \sigma(x) = (1 - x^2)/4; this choice yields the linear polynomial \tau(x) = [(\beta - \alpha)/4] - [(\alpha + \beta + 2)/4] x. Substituting into the differential equation gives \frac{1 - x^2}{4} y''(x) + \left[ \frac{\beta - \alpha}{4} - \frac{\alpha + \beta + 2}{4} x \right] y'(x) + \lambda y(x) = 0, with eigenvalues \lambda_n = n(n + \alpha + \beta + 1)/4. The boundary term in the integration by parts vanishes at x = \pm 1 because \sigma(x) w(x) = [(1 - x^2)/4] (1 - x)^\alpha (1 + x)^\beta \to 0 as x \to \pm 1, since both factors approach zero.[21] For the Laguerre polynomials, the weight is w(x) = x^\alpha e^{-x} on (0, \infty), with \alpha > -1. Choosing \sigma(x) = x satisfies Pearson's equation with the linear \tau(x) = \alpha + 1 - x. The differential equation becomes x y''(x) + (\alpha + 1 - x) y'(x) + \lambda y(x) = 0, with eigenvalues \lambda_n = n. Orthogonality holds because the boundary term [\sigma(x) w(x) (P_n P_m' - P_m P_n')] evaluates to zero at the limits: as x \to \infty, x \cdot x^\alpha e^{-x} = x^{\alpha + 1} e^{-x} \to 0; as x \to 0^+, x^{\alpha + 1} \to 0 since \alpha + 1 > 0.[21] For the Hermite polynomials, use w(x) = e^{-x^2} on (-\infty, \infty). With \sigma(x) = 1, Pearson's equation gives the linear \tau(x) = -2x. The differential equation is y''(x) - 2x y'(x) + \lambda y(x) = 0, with eigenvalues \lambda_n = 2n. The boundary term vanishes at \pm \infty because \sigma(x) w(x) = e^{-x^2} \to 0 exponentially fast.[21]Generating Functions
Generating functions provide a powerful analytic tool for studying classical orthogonal polynomials, allowing the derivation of recurrence relations, explicit coefficient extractions, and connections to hypergeometric functions. For these polynomials, the generating functions are typically exponential series of the form G(x, t) = \sum_{n=0}^\infty P_n(x) \frac{t^n}{n!}, where P_n(x) denotes the polynomial of degree n, though ordinary generating functions \sum_{n=0}^\infty P_n(x) t^n are also common depending on the family. These closed-form expressions enable systematic manipulation to uncover structural properties.[22] For Jacobi polynomials P_n^{(\alpha, \beta)}(x), the ordinary generating function is \sum_{n=0}^\infty P_n^{(\alpha, \beta)}(x) t^n = \frac{2^{\alpha + \beta}}{R (1 - t + R)^\alpha (1 + t + R)^\beta}, where R = \sqrt{1 - 2 x t + t^2} and \operatorname{Re}(\alpha), \operatorname{Re}(\beta) > -1, with radius of convergence |t| < \min\{1, (1 + |x| - \sqrt{1 - x^2})/|x|\} for |x| \leq 1. An equivalent form involves the Gaussian hypergeometric function: \sum_{n=0}^\infty P_n^{(\alpha, \beta)}(x) t^n = 2^{\alpha + \beta} R^{-\alpha} (1 - t + R)^{-\beta} \, {}_2F_1\left(-\frac{\alpha}{2}, \frac{1 + \alpha}{2}; \alpha + 1; \frac{4 t (1 - t)}{(1 - t + R)^2}\right). When \alpha = \beta = 0, this reduces to the Legendre case: \sum_{n=0}^\infty P_n(x) t^n = (1 - 2 x t + t^2)^{-1/2}. These expressions stem from integral representations and hypergeometric series identities.[22] The generating function for generalized Laguerre polynomials L_n^{(\alpha)}(x) is the ordinary series \sum_{n=0}^\infty L_n^{(\alpha)}(x) t^n = (1 - t)^{-\alpha - 1} \exp\left( -\frac{x t}{1 - t} \right), valid for |t| < 1 and \operatorname{Re}(\alpha) > -1. It can also be expressed using the confluent hypergeometric function: \sum_{n=0}^\infty L_n^{(\alpha)}(x) t^n = (1 - t)^{-\alpha - 1} \exp\left( -\frac{x t}{1 - t} \right) \, {}_1F_1(\alpha + 1; 1; \frac{x t}{t - 1}), highlighting the connection to Kummer's function. This form arises from the Rodrigues formula and series solutions to the Laguerre differential equation.[22] For the physicists' Hermite polynomials H_n(x), the exponential generating function is \sum_{n=0}^\infty H_n(x) \frac{t^n}{n!} = \exp(2 x t - t^2), convergent for all t \in \mathbb{C}. This simple exponential structure reflects the quadratic nature of the Hermite weight function and facilitates probabilistic interpretations in quantum mechanics.[22] The three-term recurrence relations for classical orthogonal polynomials can be derived by differentiating the generating function with respect to t, substituting the closed-form expression, and equating coefficients in the resulting series. For instance, in the Hermite case, differentiating yields \partial G / \partial t = (2x - 2t) G, which, after shifting indices and comparing terms, produces H_{n+1}(x) = 2x H_n(x) - 2n H_{n-1}(x). Similar manipulations apply to Jacobi and Laguerre families, often involving relations akin to \partial G / \partial t = x \partial G / \partial x + lower-order terms derived from the explicit form. These procedures leverage the analytic structure to obtain the recurrence coefficients without invoking orthogonality directly.[5] Coefficients P_n(x) can be extracted from G(x, t) using the Cauchy integral formula: P_n(x) = \frac{n!}{2 \pi i} \oint \frac{G(x, t)}{t^{n+1}} \, dt, where the contour encloses the origin within the radius of convergence, or via direct series expansion of the closed form, which often reduces to hypergeometric summations. This method is particularly useful for explicit computations and asymptotic analysis.[22]Special Cases and Extensions
Legendre Polynomials
Legendre polynomials arise as a special case of the Jacobi polynomials with parameters \alpha = \beta = 0, denoted P_n(x) = P_n^{(0,0)}(x). They are defined on the interval [-1, 1] with respect to the weight function w(x) = 1.[11] The explicit form is given by the Rodrigues formula: P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n. They satisfy the normalization condition P_n(1) = 1 and are orthogonal according to \int_{-1}^1 P_m(x) P_n(x) \, dx = \frac{2}{2n+1} \delta_{mn}. [12][11] The generating function for Legendre polynomials is \frac{1}{\sqrt{1 - 2xt + t^2}} = \sum_{n=0}^\infty P_n(x) t^n, \quad |t| < 1. They also obey the three-term recurrence relation (n+1) P_{n+1}(x) = (2n+1) x P_n(x) - n P_{n-1}(x), with initial conditions P_0(x) = 1 and P_1(x) = x.[22][23] Associated Legendre functions, which generalize the polynomials, are defined by P_l^m(x) = (-1)^m (1 - x^2)^{m/2} \frac{d^m}{dx^m} P_l(x), for integers l \geq m \geq 0. These functions play a central role in the theory of spherical harmonics, where the angular part of solutions to Laplace's equation in spherical coordinates is expressed as Y_{l,m}(\theta, \phi) \propto P_l^m(\cos \theta) e^{i m \phi}, enabling applications in electrostatics, gravitation, and quantum mechanics for describing angular momentum eigenstates.[24][25]Chebyshev Polynomials
Chebyshev polynomials of the first kind, denoted T_n(x), emerge as a limiting case of Jacobi polynomials when the parameters \alpha = \beta \to -1/2, specifically T_n(x) = \lim_{\alpha, \beta \to -1/2} \frac{P_n^{(\alpha, \beta)}(x)}{P_n^{(\alpha, \beta)}(1)}. This limit specializes the more general Jacobi family to the interval [-1, 1] with a weight function exhibiting singularities at the endpoints. A defining trigonometric identity is T_n(\cos \theta) = \cos(n \theta) for x = \cos \theta and \theta \in [0, \pi].[11] The Rodrigues formula provides an explicit representation: T_n(x) = \frac{(-1)^n \sqrt{\pi} \, (1 - x^2)^{1/2} }{2^n \Gamma(n + 1/2)} \frac{d^n}{dx^n} \left[ (1 - x^2)^{n - 1/2} \right]. [12]They are orthogonal over [-1, 1] with respect to the weight w(x) = 1 / \sqrt{1 - x^2}, satisfying \int_{-1}^1 T_m(x) T_n(x) w(x) \, dx = \begin{cases} \pi & m = n = 0, \\ \pi / 2 & m = n \geq 1, \\ 0 & m \neq n. \end{cases} Chebyshev polynomials of the second kind, denoted U_n(x), arise similarly as the limit \alpha = \beta \to 1/2, given by U_n(x) = (n+1) \lim_{\alpha, \beta \to 1/2} \frac{P_n^{(\alpha, \beta)}(x)}{P_n^{(\alpha, \beta)}(1)}, or equivalently as ultraspherical polynomials with parameter \lambda = 1. Their trigonometric representation is U_n(\cos \theta) = \frac{\sin((n+1) \theta)}{\sin \theta}. They are orthogonal on [-1, 1] with weight w(x) = \sqrt{1 - x^2}, with squared norms \int_{-1}^1 U_m(x) U_n(x) w(x) \, dx = \frac{\pi}{2} \delta_{mn}. The generating function for the first kind is \sum_{n=0}^\infty T_n(x) t^n = \frac{1 - t x}{1 - 2 t x + t^2}, \quad |t| < 1, \ x \in [-1, 1]. $$ Chebyshev polynomials of the first kind exhibit the minimax property, minimizing the maximum deviation from zero among monic polynomials of degree $ n $ on $[-1, 1]$. ### Gegenbauer Polynomials Gegenbauer polynomials, also known as ultraspherical polynomials, arise as the symmetric case of Jacobi polynomials where the parameters are equal, specifically $\alpha = \beta = \lambda - 1/2$ with $\lambda > 0$. They generalize several classical orthogonal polynomials and play a key role in representing solutions to PDEs in higher dimensions, such as in the [theory](/page/Theory) of hyperspherical harmonics. These polynomials are orthogonal on the interval $[-1, 1]$ with respect to the weight function $w(x) = (1 - x^2)^{\lambda - 1/2}$.[](https://dlmf.nist.gov/18.3) The explicit definition expresses the Gegenbauer polynomial $C_n^{(\lambda)}(x)$ in terms of the Jacobi polynomial $P_n^{(\alpha, \beta)}(x)$ as C_n^{(\lambda)}(x) = \frac{(2\lambda)_n}{ (\lambda + 1/2)_n} , P_n^{(\lambda - 1/2, \lambda - 1/2)}(x), where $(a)_n$ denotes the Pochhammer symbol.[](https://dlmf.nist.gov/18.7) A Rodrigues-type formula provides another representation: C_n^{(\lambda)}(x) = \frac{(-1)^n (2\lambda)_n }{ 2^n n! (\lambda + 1/2)_n } (1 - x^2)^{1/2 - \lambda} \frac{d^n}{dx^n} \left[ (1 - x^2)^{\lambda + n - 1/2} \right]. [12] The generating function for [Gegenbauer polynomials](/page/Gegenbauer_polynomials) is given by (1 - 2 x t + t^2)^{-\lambda} = \sum_{n=0}^{\infty} C_n^{(\lambda)}(x) , t^n, valid for $|t| < \min(1, |x + \sqrt{x^2 - 1}|)$.[](https://dlmf.nist.gov/18.12) Special cases include the Legendre polynomials when $\lambda = 1/2$, where $C_n^{(1/2)}(x) = P_n(x)$, and the Chebyshev polynomials of the second kind when $\lambda = 1$, where $C_n^{(1)}(x) = U_n(x)$. These connections highlight their role in unifying various orthogonal expansions in potential theory and quantum mechanics.[](https://dlmf.nist.gov/18.3) ## Characterizations and Uniqueness ### Pearson's Differential Equation Pearson's differential equation characterizes the weight functions associated with classical orthogonal polynomials by imposing a first-order linear differential equation on the weight $w(x)$: \sigma(x) w'(x) = \left[ \tau(x) - \sigma'(x) \right] w(x), where $\sigma(x)$ and $\tau(x)$ are polynomials satisfying $\deg \sigma \leq 2$ and $\deg \tau \leq 1$.[](https://opus4.kobv.de/opus4-zib/files/234/SC-96-23.pdf) This condition ensures that the orthogonal polynomials satisfy a second-order linear differential equation with polynomial coefficients of minimal degree, distinguishing classical families from more general orthogonal polynomials.[](https://opus4.kobv.de/opus4-zib/files/234/SC-96-23.pdf) The equation derives from the structure of the Sturm-Liouville problem underlying orthogonal polynomials, where the minimal-degree requirement on the coefficients leads to this specific form for the weight.[](https://link.springer.com/article/10.1007/BF01203415) Solving Pearson's equation explicitly yields the three classical weight functions: for $\deg \sigma = 2$, the Jacobi weight on $[-1, 1]$; for $\deg \sigma = 1$, the Laguerre weight on $[0, \infty)$; and for $\deg \sigma = 0$ (constant $\sigma$), the Hermite weight on $(-\infty, \infty)$.[](https://opus4.kobv.de/opus4-zib/files/234/SC-96-23.pdf) These solutions are unique up to affine transformations of the independent variable, confirming that only the Jacobi, Laguerre, and Hermite families satisfy the classical criteria.[](https://link.springer.com/article/10.1007/BF01203415) For the Jacobi case, a standard choice is $\sigma(x) = 1 - x^2$ and $\tau(x) = (\beta - \alpha) - (\alpha + \beta + 2)x$, corresponding to the weight $w(x) = (1 - x)^\alpha (1 + x)^\beta$ with parameters $\alpha, \beta > -1$.[](https://opus4.kobv.de/opus4-zib/files/234/SC-96-23.pdf) ### Classicality Criteria Classical orthogonal polynomials are distinguished from general orthogonal polynomials by specific criteria that ensure their polynomials satisfy second-order linear [differential](/page/Differential) equations with particular coefficient structures. These criteria, developed primarily in [the 1930s](/page/The_1930s) and [1940s](/page/1940s) by [Salomon Bochner](/page/Salomon_Bochner), Wolfgang Hahn, and Arthur Erdélyi, uniquely identify the continuous classical families: Jacobi, Laguerre, and [Hermite polynomials](/page/Hermite_polynomials). Bochner's theorem provides a foundational characterization: a system of monic orthogonal polynomials $\{p_n(x)\}_{n=0}^\infty$ is classical if and only if each $p_n(x)$ satisfies the Sturm-Liouville [differential equation](/page/Differential_equation) \alpha(x) p_n''(x) + \beta(x) p_n'(x) + n(n-1) \gamma + n \delta = 0, where $\alpha(x)$ is a quadratic [polynomial](/page/Polynomial), $\beta(x)$ is linear, and $\gamma, \delta$ are constants. Hahn's theorem further specifies that orthogonal polynomials are classical if and only if they satisfy a second-order [linear differential equation](/page/Linear_differential_equation) whose coefficients are rational functions independent of the degree $n$. This condition ensures the equation's form remains uniform across all degrees, a property exclusive to the classical systems.[](https://doi.org/10.1016/S0377-0427(99)00319-2) The Meixner classification, while encompassing discrete analogs such as Hahn polynomials, confirms that the continuous classical orthogonal polynomials comprise precisely the three families: Jacobi (orthogonal on $(-1,1)$ with beta weight), Laguerre (on $(0,\infty)$ with gamma weight), and Hermite (on $(-\infty,\infty)$ with Gaussian weight). These families arise as the unique solutions satisfying the above differential criteria within the continuous setting. An additional characterization involves forward and backward shift operators, defined via the three-term [recurrence relation](/page/Recurrence_relation) of the polynomials. Classical orthogonal polynomials are those for which these operators—lowering and raising the degree by one—preserve the underlying [orthogonality](/page/Orthogonality) lattice (the [support](/page/Support) of the weight measure), with coefficients that are polynomials of degree at most 1. This property links directly to Pearson's [differential equation](/page/Differential_equation) for the weight function, ensuring the measure's invariance under [differentiation](/page/Differentiation). ## Summary Table ### Table of Key Parameters and Formulas The following table provides a concise summary of the essential parameters and formulas for the principal families of classical orthogonal polynomials, including the general cases and important specializations. | Family | Interval | Weight $ w(x) $ | Rodrigues Formula | DE Coefficients ($ \sigma(x) $, $ \lambda_n $) | Norm $ h_n = \int w(x) [p_n(x)]^2 \, dx $ | Generating Function (brief) | |-------------------------|--------------|--------------------------------|-----------------------------------------------------------------------------------|----------------------------------------------------|-------------------------------------------------------------------------------------------------------------|-----------------------------------------------------| | Jacobi $ P_n^{(\alpha,\beta)}(x) $ | $(-1,1)$ | $ (1-x)^\alpha (1+x)^\beta $ | $ \frac{(-1)^n}{2^n n!} (1-x)^{-\alpha} (1+x)^{-\beta} \frac{d^n}{dx^n} [(1-x)^{n+\alpha} (1+x)^{n+\beta}] $ | $ 1-x^2 $, $ n(n+\alpha+\beta+1) $ | $ \frac{2^{\alpha+\beta+1} \Gamma(n+\alpha+1) \Gamma(n+\beta+1)}{n! (2n+\alpha+\beta+1) \Gamma(n+\alpha+\beta+1)} $ | $ \sum_{n=0}^\infty P_n^{(\alpha,\beta)}(x) z^n = \frac{2^{\alpha+\beta}}{R (1 - z + R)^\alpha (1 + z + R)^\beta} $, $ R = \sqrt{1-2xz + z^2} $ | | Laguerre $ L_n^{(\alpha)}(x) $ | $(0,\infty)$ | $ x^\alpha e^{-x} $ | $ \frac{e^x x^{-\alpha}}{n!} \frac{d^n}{dx^n} [e^{-x} x^{n+\alpha}] $ | $ x $, $ n $ | $ \frac{\Gamma(n+\alpha+1)}{n!} $ | $ \sum_{n=0}^\infty L_n^{(\alpha)}(x) z^n = (1-z)^{-\alpha-1} \exp\left( \frac{-x z}{1-z} \right) $ | | Hermite $ H_n(x) $ | $(-\infty,\infty)$ | $ e^{-x^2} $ | $ (-1)^n e^{x^2} \frac{d^n}{dx^n} [e^{-x^2}] $ | $ 1 $, $ 2n $ | $ 2^n n! \sqrt{\pi} $ | $ \sum_{n=0}^\infty \frac{H_n(x)}{n!} z^n = e^{2xz - z^2} $ | | Legendre $ P_n(x) $ | $(-1,1)$ | $ 1 $ | $ \frac{1}{2^n n!} \frac{d^n}{dx^n} [(x^2-1)^n] $ | $ 1-x^2 $, $ n(n+1) $ | $ \frac{2}{2n+1} $ | $ \sum_{n=0}^\infty P_n(x) z^n = \frac{1}{\sqrt{1-2xz + z^2}} $ | | Chebyshev $ T_n(x) $ (1st kind) | $(-1,1)$ | $ (1-x^2)^{-1/2} $ | $ \frac{(-1)^n \sqrt{\pi}}{2^n (n-1/2)!} (1-x^2)^{1/2} \frac{d^n}{dx^n} [(1-x^2)^{n-1/2}] $ | $ 1-x^2 $, $ n^2 $ | $ \begin{cases} \pi & n=0 \\ \pi/2 & n \geq 1 \end{cases} $ | $ \sum_{n=0}^\infty T_n(x) z^n = \frac{1 - x z}{1 - 2 x z + z^2} $ | | Chebyshev $ U_n(x) $ (2nd kind) | $(-1,1)$ | $ \sqrt{1-x^2} $ | $ (-1)^n 2^{-n} \frac{n+1}{(3/2)_n} (1-x^2)^{-1/2} \frac{d^n}{dx^n} [(1-x^2)^{n+1/2}] $ | $ 1-x^2 $, $ n(n+2) $ | $ \frac{\pi}{2} $ | $ \sum_{n=0}^\infty U_n(x) z^n = \frac{1}{1 - 2 x z + z^2} $ | | Gegenbauer $ C_n^{(\lambda)}(x) $ | $(-1,1)$ | $ (1-x^2)^{\lambda - 1/2} $ | $ (-1)^n \frac{(2\lambda)_n}{2^n n! (\lambda + 1/2)_n} (1-x^2)^{1/2 - \lambda} \frac{d^n}{dx^n} [(1-x^2)^{n + \lambda - 1/2}] $ | $ 1-x^2 $, $ n(n + 2\lambda) $ | $ \frac{2^{1-2\lambda} \pi \Gamma(n + 2\lambda)}{n! (n + \lambda) [\Gamma(\lambda)]^2} $ | $ \sum_{n=0}^\infty C_n^{(\lambda)}(x) z^n = (1 - 2xz + z^2)^{-\lambda} $ | *Data compiled from the NIST Digital Library of Mathematical Functions, Chapter 18.[](https://dlmf.nist.gov/18)* **Footnote on normalization and leading coefficients:** The polynomials are defined with standard conventions where the coefficient of $ x^n $ (leading coefficient $ k_n $) is positive and specific to each family: for Jacobi, $ k_n = \frac{(2n + \alpha + \beta)!}{2^n n! (n + \alpha + \beta)!} $; for Laguerre, $ k_n = (-1)^n / n! $; for Hermite $ H_n $, $ k_n = 2^n $; for Legendre, $ k_n = \frac{(2n)!}{2^n (n!)^2} $; for Chebyshev $ T_n $, $ k_n = 2^{n-1} $ ($ n \geq 1 $, $ T_0 = 1 $); for $ U_n $, $ k_n = 2^n $; for Gegenbauer, $ k_n = \frac{(2\lambda)_n}{n!} \cdot 2^n $ (adjusted for $ \lambda $). Monic versions are obtained by dividing by $ k_n $. Special cases: Legendre is Jacobi with $ \alpha = \beta = 0 $; Chebyshev $ T_n $ relates to Gegenbauer with $ \lambda = 0 $ (limit); $ U_n(x) = \frac{2}{\pi} (n+1) C_{n+1}^{(1)}(x) $ up to scaling.[](https://dlmf.nist.gov/18.3)[](https://dlmf.nist.gov/18.5) For large $ n $, these polynomials exhibit asymptotic behavior where, outside the orthogonality [interval](/page/Interval), they decay factorially as $ O( (r)^{-n} n! ) $ with $ r > 1 $ depending on the distance from the interval, while inside they oscillate with envelope determined by [the weight](/page/The_Weight).