Fact-checked by Grok 2 weeks ago

Hermite polynomials

Hermite polynomials form a classical sequence of orthogonal polynomials defined on the entire real line with respect to the Gaussian weight function e^{-x^2}, satisfying the orthogonality relation \int_{-\infty}^{\infty} H_n(x) H_m(x) e^{-x^2} \, dx = \sqrt{\pi} \, 2^n n! \, \delta_{mn}. They are named after the French mathematician Charles Hermite, who introduced and studied them in detail in 1864, although earlier work on similar polynomials dates back to in 1859. These polynomials arise prominently in as the eigenfunctions of the one-dimensional , where the wavefunctions are given by \psi_n(x) = \frac{1}{\sqrt{2^n n!}} \left( \frac{\alpha}{\pi} \right)^{1/4} e^{-\alpha x^2 / 2} H_n(\sqrt{\alpha} x) with \alpha = m \omega / \hbar, and in for expansions of functions under the normal distribution, such as in Edgeworth and Gram-Charlier series. The explicit form of the Hermite polynomials H_n(x) is provided by the Rodrigues formula: H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}, which highlights their connection to repeated of the . They satisfy the second-order y'' - 2x y' + 2n y = 0, known as Hermite's equation, making them solutions to Sturm-Liouville problems on (-\infty, \infty). A for the sequence is e^{2xt - t^2} = \sum_{n=0}^{\infty} \frac{H_n(x)}{n!} t^n, facilitating derivations of recurrence relations like H_{n+1}(x) = 2x H_n(x) - 2n H_{n-1}(x). There are two common normalizations: the "physicists'" Hermite polynomials H_n(x) as defined above, used in , and the "probabilists'" version \mathrm{He}_n(x) = 2^{-n/2} H_n(x / \sqrt{2}), which are orthogonal with respect to e^{-x^2 / 2} / \sqrt{2\pi} and scaled for unit variance in probabilistic contexts. Beyond physics and statistics, Hermite polynomials appear in via Gauss-Hermite quadrature, approximation theory, and the analysis of ensembles, such as the Gaussian Unitary Ensemble.

Definitions

Rodrigues formula

The Rodrigues formula offers a compact representation for defining the Hermite polynomials, emphasizing their connection to repeated of Gaussian functions. For the physicist's Hermite polynomials H_n(x), the formula is H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} \left( e^{-x^2} \right), where n is a non-negative . This expression generates polynomials of degree n with leading coefficient $2^n. The associated for is the Gaussian e^{-x^2} over the (-\infty, \infty). In contrast, the probabilist's Hermite polynomials \mathrm{He}_n(x) use a scaled version suited to , given by \mathrm{He}_n(x) = (-1)^n e^{x^2/2} \frac{d^n}{dx^n} \left( e^{-x^2/2} \right). These are monic polynomials of degree n with leading coefficient 1, and their orthogonality weight is e^{-x^2/2}, corresponding to the unnormalized standard normal density. The relation between the two conventions is H_n(x) = 2^{n/2} \mathrm{He}_n(\sqrt{2} x). Named after the mathematician Charles Hermite, who introduced these polynomials in 1864 while investigating solutions to certain differential equations, the highlights their role in classical analysis and physics, particularly in the . Hermite's work built on earlier ideas by Laplace and Chebyshev but provided a systematic treatment that popularized the polynomials.

Generating function

The generating function provides a compact exponential form that encapsulates all Hermite polynomials of a given type, allowing for the simultaneous generation of the sequence through expansion in the t. For the physicist's Hermite polynomials H_n(x), defined with respect to e^{-x^2}, the ordinary is e^{2xt - t^2} = \sum_{n=0}^\infty H_n(x) \frac{t^n}{n!}, valid for all complex x and t. This form arises in and other physical contexts where the polynomials appear in the wavefunctions. For the probabilist's Hermite polynomials \mathrm{He}_n(x), which are scaled versions related by \mathrm{He}_n(x) = 2^{-n/2} H_n(x / \sqrt{2}) and orthogonal with respect to e^{-x^2 / 2}, the generating function is e^{xt - t^2 / 2} = \sum_{n=0}^\infty \mathrm{He}_n(x) \frac{t^n}{n!}, commonly used in probability theory for expansions involving the Gaussian measure. This generating function can be derived from the Rodrigues formula H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} (e^{-x^2}) by considering the exponential generating series. Specifically, \sum_{n=0}^\infty H_n(x) \frac{t^n}{n!} = \sum_{n=0}^\infty (-1)^n e^{x^2} \frac{(t \frac{d}{dx})^n}{n!} e^{-x^2} = e^{x^2} \exp\left( -t \frac{d}{dx} \right) e^{-x^2}. The operator \exp\left( -t \frac{d}{dx} \right) acts as a Taylor shift, yielding e^{-(x - t)^2}, so e^{x^2} e^{-(x - t)^2} = e^{x^2} e^{-x^2 + 2xt - t^2} = e^{2xt - t^2}, confirming the exponential form via the Taylor series expansion of the shifted Gaussian. A similar derivation applies to the probabilist's version, adjusting for the scaling in the weight function. The individual polynomials are extracted as coefficients from the expansion: H_n(x) = n! [t^n] e^{2xt - t^2}, where [t^n] denotes the coefficient of t^n in the power series. Equivalently, H_n(x) = \left. \frac{\partial^n}{\partial t^n} e^{2xt - t^2} \right|_{t=0}. The same holds for \mathrm{He}_n(x) using its . Basic manipulations of the , such as partial , yield relations among the polynomials. For instance, differentiating with respect to t gives \frac{\partial}{\partial t} G(x,t) = (2x - 2t) G(x,t), where G(x,t) = e^{2xt - t^2}. Comparing coefficients of the resulting series provides the three-term recurrence H_{n+1}(x) = 2x H_n(x) - 2n H_{n-1}(x) for n \geq 1, with initial conditions H_0(x) = 1 and H_1(x) = 2x. Analogous differentiations apply to the probabilist's , producing the scaled recurrence \mathrm{He}_{n+1}(x) = x \mathrm{He}_n(x) - n \mathrm{He}_{n-1}(x).

Explicit summation formula

The explicit summation formulas provide a direct way to compute Hermite polynomials as finite sums, distinguishing between the physicist's convention H_n(x) and the probabilist's convention \mathrm{He}_n(x). For the physicist's Hermite polynomials, the formula is H_n(x) = n! \sum_{m=0}^{\lfloor n/2 \rfloor} \frac{(-1)^m (2x)^{n-2m}}{m! (n-2m)!}. This expression arises from extraction in the e^{2xt - t^2} = \sum_{n=0}^\infty H_n(x) \frac{t^n}{n!}, where the exponential terms are expanded using the and like powers of t are collected. Alternatively, the polynomials can be represented via applied to the , yielding H_n(x) = \frac{n!}{2\pi i} \oint \frac{e^{2xt - t^2}}{t^{n+1}} \, dt, from which the follows by inside the contour. For the probabilist's Hermite polynomials, scaled such that the leading coefficient is 1, the corresponding formula is \mathrm{He}_n(x) = n! \sum_{m=0}^{\lfloor n/2 \rfloor} \frac{(-1)^m x^{n-2m}}{m! (n-2m)!}. This variant relates to the physicist's by \mathrm{He}_n(x) = 2^{-n/2} H_n(x / \sqrt{2}). These formulas yield simple expressions for low degrees. For instance, H_0(x) = 1, H_1(x) = 2x, and H_2(x) = 4x^2 - 2. Similarly, \mathrm{He}_0(x) = 1, \mathrm{He}_1(x) = x, and \mathrm{He}_2(x) = x^2 - 1. The summation form is computationally efficient for small n, as it involves at most \lfloor n/2 \rfloor + 1 terms, avoiding the repeated differentiations in the . Additionally, special cases connect to double factorials; for even n = 2k, the value at is H_{2k}(0) = (-1)^k \frac{(2k)!}{k!} = (-1)^k 2^{2k} k! \cdot \frac{(2k-1)!!}{(2k)!!}, linking the constant term to products of integers.

Orthogonality and Differential Equation

Orthogonality relations

The Hermite polynomials in the physicist's convention, denoted H_n(x), satisfy the orthogonality relation \int_{-\infty}^{\infty} H_m(x) H_n(x) e^{-x^2} \, dx = \sqrt{\pi} \, 2^n n! \, \delta_{mn}, where \delta_{mn} is the Kronecker delta, equal to 1 if m = n and 0 otherwise. This relation holds over the infinite interval with weight function e^{-x^2}, establishing the polynomials as an orthogonal family in the Hilbert space L^2(\mathbb{R}, e^{-x^2} \, dx). In the probabilist's convention, denoted He_n(x), the polynomials are orthogonal with respect to the standard normal : \int_{-\infty}^{\infty} He_m(x) He_n(x) e^{-x^2/2} \, dx = \sqrt{2\pi} \, n! \, \delta_{mn}. The relation follows from the scaling He_n(x) = 2^{-n/2} H_n(x / \sqrt{2}), which adjusts the weight to e^{-x^2/2} for applications in . The squared norm in the physicist's case is \| H_n \|^2 = \int_{-\infty}^{\infty} H_n(x)^2 e^{-x^2} \, [dx](/page/DX) = \sqrt{\pi} \, 2^n n!. To prove the orthogonality, consider the Rodrigues formula H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2}. For m < n, the integral I = \int_{-\infty}^{\infty} H_m(x) H_n(x) e^{-x^2} \, dx can be expressed using the formula for H_n, yielding I = (-1)^n \int_{-\infty}^{\infty} H_m(x) \frac{d^n}{dx^n} (e^{-x^2}) \, dx. Integrating by parts n times transfers the derivatives to H_m(x), which is a polynomial of degree m < n, resulting in the boundary term vanishing at infinity and the integral becoming zero since the n-th derivative of H_m is zero. For m = n, the norm follows similarly, with the remaining term giving the factorial structure. These relations enable the Fourier-Hermite series expansion of square-integrable functions f(x) as f(x) = \sum_{n=0}^{\infty} c_n H_n(x), where coefficients c_n = \frac{1}{\| H_n \|^2} \int_{-\infty}^{\infty} f(x) H_n(x) e^{-x^2} \, dx, providing a complete orthogonal decomposition in the weighted space.

Hermite's differential equation

The Hermite polynomials arise as solutions to a specific second-order linear ordinary differential equation (ODE), known as Hermite's differential equation. There are two standard variants of these polynomials, differing by a scaling factor, which lead to slightly different forms of the ODE. The physicist's Hermite polynomials H_n(x), commonly used in quantum mechanics, satisfy \frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} + 2n y = 0, where n is a non-negative integer. In contrast, the probabilist's Hermite polynomials \mathrm{He}_n(x), often employed in probability theory and statistics, satisfy \frac{d^2 y}{dx^2} - x \frac{dy}{dx} + n y = 0. These equations are defined over the entire real line, with the physicist's version incorporating a factor of 2 in the scaling to align with applications in the quantum harmonic oscillator. The polynomial solutions of exact degree n to either equation are unique up to a multiplicative constant, which is fixed by the conventional leading coefficient of 1 for both \mathrm{He}_n(x) and H_n(x). This uniqueness follows from the theory of linear ODEs: the solution space is two-dimensional, spanned by two linearly independent solutions (one even and one odd function), but the condition of being a monic polynomial of degree exactly n selects a unique member. To verify that the Hermite polynomials satisfy the differential equation, consider the Rodrigues formula for the physicist's version: H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} \left( e^{-x^2} \right). Applying the ODE operator to this expression and using the Leibniz rule for the nth derivative of a product—specifically, \frac{d^n}{dx^n} (f g) = \sum_{k=0}^n \binom{n}{k} \frac{d^{n-k} f}{dx^{n-k}} \frac{d^k g}{dx^k}—one can show that the resulting expression equals zero, confirming satisfaction of the equation. The same approach applies to the probabilist's formula \mathrm{He}_n(x) = (-1)^n e^{x^2/2} \frac{d^n}{dx^n} \left( e^{-x^2/2} \right), with adjusted scaling. In Sturm-Liouville theory, Hermite's differential equation can be recast into self-adjoint form, highlighting its role in orthogonal polynomial systems. For the physicist's polynomials, multiplying the ODE by the weight function e^{-x^2} yields \frac{d}{dx} \left( e^{-x^2} \frac{dy}{dx} \right) + 2n e^{-x^2} y = 0, which is the standard with eigenvalue $2n and weight e^{-x^2} over (-\infty, \infty). The probabilist's version similarly takes the form \frac{d}{dx} \left( e^{-x^2/2} \frac{dy}{dx} \right) + n e^{-x^2/2} y = 0. This form underscores the orthogonality properties derived from the underlying spectral theory.

Completeness relation

The Hermite polynomials \{H_n(x)\}_{n=0}^\infty form a complete orthogonal basis for the Hilbert space L^2(\mathbb{R}, e^{-x^2}\, dx). This means that any function f \in L^2(\mathbb{R}, e^{-x^2}\, dx) admits an expansion f(x) = \sum_{n=0}^\infty c_n H_n(x), where the coefficients c_n are determined by orthogonal projection: c_n = \frac{1}{\|H_n\|^2} \int_{-\infty}^\infty f(x) H_n(x) e^{-x^2}\, dx, and the squared norms are \|H_n\|^2 = \sqrt{\pi}\, 2^n n!. Parseval's identity follows from the completeness and orthogonality: for any f \in L^2(\mathbb{R}, e^{-x^2}\, dx), \int_{-\infty}^\infty |f(x)|^2 e^{-x^2}\, dx = \sum_{n=0}^\infty |c_n|^2 \|H_n\|^2. This relation quantifies the energy preservation in the expansion. The completeness can be established by showing that polynomials are dense in L^2(\mathbb{R}, e^{-x^2}\, dx) and that the span all polynomials. Density follows from the determined nature of the for the Gaussian weight e^{-x^2}, which ensures a unique measure corresponding to the moments \mu_n = \int_{-\infty}^\infty x^n e^{-x^2}\, dx. An alternative outline uses generating functions: the \sum_{n=0}^\infty \frac{\rho^n}{n!} H_n(x) H_n(y) converges to a reproducing kernel as \rho \to 1^-, yielding the delta distribution in the weighted space upon normalization. Beyond L^2(\mathbb{R}, e^{-x^2}\, dx), the Hermite polynomials are dense in certain weighted , such as those with norms involving derivatives weighted by the Gaussian, enabling expansions for smoother functions. This completeness underpins applications to , where the Gaussian moments uniquely determine the measure via the orthogonal expansion, facilitating reconstruction without ambiguity in indeterminate cases.

Recurrence and Symmetry

Recurrence relations

The Hermite polynomials satisfy three-term recurrence relations that allow sequential computation of higher-degree polynomials from lower ones. For the physicist's Hermite polynomials H_n(x), the relation is H_{n+1}(x) = 2x H_n(x) - 2n H_{n-1}(x), valid for n \geq 1. For the probabilist's Hermite polynomials \mathrm{He}_n(x), the corresponding recurrence is \mathrm{He}_{n+1}(x) = x \mathrm{He}_n(x) - n \mathrm{He}_{n-1}(x), also for n \geq 1. These relations facilitate efficient numerical evaluation and appear in applications such as and . The recurrences require initial conditions to start the sequence. For physicist's Hermite polynomials, H_0(x) = 1 and H_1(x) = 2x. For probabilist's, \mathrm{He}_0(x) = 1 and \mathrm{He}_1(x) = x. A key derivative relation links the polynomials across degrees: for physicist's Hermite polynomials, H_n'(x) = 2n H_{n-1}(x). The probabilist's version follows analogously as \mathrm{He}_n'(x) = n \mathrm{He}_{n-1}(x). These relations can be derived by differentiating the generating functions or manipulating the differential equation satisfied by the polynomials. For the physicist's case, start with the generating function G(x,t) = e^{2xt - t^2} = \sum_{n=0}^\infty H_n(x) \frac{t^n}{n!}. Differentiating with respect to t yields \frac{\partial G}{\partial t} = (2x - 2t) G = \sum_{n=0}^\infty H_{n+1}(x) \frac{t^n}{n!}, while differentiating with respect to x gives \frac{\partial G}{\partial x} = 2t G = \sum_{n=1}^\infty H_n'(x) \frac{t^n}{n!}. Combining these and equating coefficients produces the three-term and derivative recurrences. Similar steps apply to the probabilist's generating function G(x,t) = e^{xt - t^2/2} = \sum_{n=0}^\infty \mathrm{He}_n(x) \frac{t^n}{n!}. Alternatively, the relations follow from the differential equation H_n''(x) - 2x H_n'(x) + 2n H_n(x) = 0 by differentiating and substituting lower-degree polynomials.

Symmetry properties

Hermite polynomials possess a prominent parity symmetry with respect to the origin. For the physicist's Hermite polynomials H_n(x), the relation H_n(-x) = (-1)^n H_n(x) holds for all nonnegative integers n. This implies that H_n(x) is an even function when n is even and an odd function when n is odd, a property directly derivable from the Rodrigues formula H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} or the generating function \sum_{n=0}^\infty H_n(x) \frac{t^n}{n!} = e^{2xt - t^2}. The same parity relation applies to the probabilist's Hermite polynomials \mathrm{He}_n(x). The leading term of the physicist's Hermite polynomial is $2^n x^n, so H_n(x) = 2^n x^n + lower-degree terms, reflecting the scaling inherent in their definition for applications in . In contrast, the probabilist's version has leading coefficient 1, with \mathrm{He}_n(x) = x^n + lower-degree terms, aligning with probabilistic normalizations where the weight function is e^{-x^2/2}. This difference in leading coefficients arises from the distinct conventions: the physicist's polynomials are scaled to simplify the solutions, while the probabilist's facilitate moment-generating functions for . Scaling symmetries can be explored through the generating function, which yields relations for H_n(\alpha x). Specifically, \sum_{n=0}^\infty H_n(\alpha x) \frac{t^n}{n!} = e^{2 \alpha x t - t^2}, allowing expression of scaled polynomials via expansion or connection to hypergeometric functions, such as H_n(\alpha x) = (2 \alpha x)^n \, _2F_0\left( -\frac{n}{2}, \frac{1-n}{2}; ; -\frac{1}{(\alpha x)^2} \right). These relations highlight the polynomials' behavior under linear transformations, useful in rescaling problems in analysis and physics. In the context of representation theory, univariate Hermite polynomials transform under the actions of the orthogonal group O(1) \cong \mathbb{Z}/2\mathbb{Z}, where the nontrivial element corresponds to the reflection x \to -x, preserving the polynomial up to the sign (-1)^n. This embeds into broader structures for multivariate Hermite polynomials, which furnish irreducible representations of the orthogonal group O(d) in d dimensions, relevant to isotropic multivariate Gaussians and quantum systems.

Representations and Expansions

Integral representations

One prominent integral representation of the Hermite polynomial H_n(x) is the contour integral derived from its generating function, H_n(x) = \frac{n!}{2\pi i} \oint_C \frac{e^{2xt - t^2}}{t^{n+1}} \, dt, where C is a simple closed contour encircling the origin in the positive direction, assuming the integral converges appropriately in the complex plane. This representation, known as Mehler's contour integral, allows for the extraction of coefficients from the exponential generating function e^{2xt - t^2} via Cauchy's integral formula. A Fourier-type integral representation expresses H_n(x) as H_n(x) = \frac{(-2i)^n e^{x^2}}{\pi^{1/2}} \int_{-\infty}^{\infty} e^{-t^2} t^n e^{2ixt} \, dt, which follows from the Fourier transform properties of Gaussian functions and the structure of Hermite polynomials. An equivalent real-valued form, suitable for real x, is H_n(x) = \frac{2^{n+1} e^{x^2}}{\pi^{1/2}} \int_0^{\infty} e^{-t^2} t^n \cos\left(2xt - \frac{n\pi}{2}\right) \, dt. This cosine integral is particularly useful for numerical evaluation or analysis when x > 0, as the oscillatory nature can be handled via or asymptotic techniques. For large n, these integral representations, especially the contour form, lend themselves to asymptotic analysis via the saddle-point method, yielding approximations such as H_n(x) \sim 2^{n/2} n!^{1/2} (2x)^n e^{x^2/2} in certain regimes, though full derivations appear in specialized treatments.

Asymptotic expansions

Asymptotic expansions of Hermite polynomials provide approximations for large values of the argument x with fixed degree n or for large degree n in various scaling regimes. These expansions are essential for analyzing the behavior of the polynomials in applications such as quantum mechanics and random matrix theory, where exact expressions are impractical. For fixed n and large positive x, the leading asymptotic behavior of the physicist's Hermite polynomial H_n(x) is given by H_n(x) \sim 2^n x^n as x \to +\infty, with the full expansion taking the form of a H_n(x) = 2^n x^n \sum_{k=0}^\infty (-1)^k \frac{(2n-1)!!}{(2n-2k-1)!! (2x)^{2k}}, where the double factorial denotes the product of odd numbers up to the argument, valid uniformly for x \geq \delta > 0 with relative error O(1/x^2). This expansion arises from the explicit Rodriguez formula or and holds for the region outside the oscillatory interval. For large n, the Plancherel–Rotach asymptotics describe the behavior in three distinct regimes, scaled by setting x = \sqrt{2n+1} \cos \theta with $0 < \theta < \pi, assuming the physicist's convention where the weight is e^{-x^2}. In the oscillatory regime ($0 < \theta < \pi/2 - \delta), the polynomials exhibit trigonometric-like oscillations: \frac{H_n(x)}{(2n+1)^{n/2}} \sim (-1)^n \sqrt{\frac{2}{\pi \sin \theta}} \cos\left( (n + \frac{1}{2}) \theta - \frac{\pi}{4} \right) as n \to \infty, uniformly for \theta bounded away from the endpoints. Near the turning points (\theta = \pi/2 \pm n^{-2/3} \zeta / \sqrt{2}), the transition regime uses the Airy function approximation: \frac{H_n(x)}{(2n+1)^{n/2} 2^{1/3} n^{1/6}} \sim \mathrm{Ai}(\zeta), where \zeta is a scaled variable measuring distance from the turning point x \approx \pm \sqrt{2n+1}, capturing the smooth crossover from oscillation to decay; this is uniform in bounded \zeta. In the exponential regime (\theta = \pi/2 + \phi/n^{1/2}, \phi > 0), the behavior is monotonically decaying: \frac{H_n(x)}{(2n+1)^{n/2}} \sim \frac{1}{\sqrt{2\pi}} \exp\left( n \left( \frac{1}{2} - \phi + \phi^2/2 - \log \phi \right) + \frac{1}{2} \log(2\pi n \phi) \right), uniformly for \phi \geq \delta > 0. These formulas, originally derived for the scaled case, hold with explicit error bounds of relative order O(n^{-1}) in each regime, ensuring uniform validity across overlapping transition zones. These asymptotics extend to the distribution of zeros for large n, where the largest zero satisfies x_{n,1} \sim \sqrt{2n+1} - \frac{a_1}{2^{1/2} n^{1/6}} + O(n^{-1/2}), with a_1 \approx 1.0188 the first zero of the \mathrm{Ai}(-a_1) = 0, derived from matching the Airy approximation to the oscillatory regime. Similar scalings apply to smaller zeros via the cosine mapping.

Special Values and Zeros

Special values

The Hermite polynomials H_n(x) evaluate to zero at the when n is : H_{2k+1}(0) = 0 for nonnegative integers k. This follows from their properties. For even degrees, the values are given by H_{2k}(0) = (-1)^k \frac{(2k)!}{k!}. An equivalent expression uses the : H_{2n}(0) = (-1)^n 2^n (2n-1)!!, where (2n-1)!! = 1 \cdot 3 \cdot 5 \cdots (2n-1) for n \geq 1 and (-1)!! = 1. These formulas provide explicit computational access to the central values without series expansion. At x = 1, the yield the following values for n = 0 to $10, computed via the three-term recurrence H_{n+1}(x) = 2x H_n(x) - 2n H_{n-1}(x) with initial conditions H_0(x) = 1 and H_1(x) = 2x:
nH_n(1)
01
12
22
3-4
4-20
5-8
6184
7464
8-1648
9-10720
108224
Hermite polynomials map integers to integers, as their coefficients are integers, leading to patterns in these evaluations that appear in combinatorial identities and applications. For instance, the sequence at x=1 alternates in sign for low n and grows rapidly, reflecting the factorial growth in the leading term $2^n x^n.

Zeros and their distribution

The Hermite polynomial H_n(x) of degree n has exactly n real and simple zeros, denoted x_{1,n} < x_{2,n} < \cdots < x_{n,n}, which strictly interlace those of H_{n+1}(x) in the sense that x_{k,n+1} < x_{k,n} < x_{k+1,n+1} for $1 \leq k \leq n. This interlacing property implies that the k-th zero x_{k,n} is a monotonically increasing function of the degree n for each fixed k. The zeros of H_n(x) serve as the optimal nodes for the Gauss–Hermite quadrature rule, which approximates integrals of the form \int_{-\infty}^{\infty} e^{-x^2} f(x) \, dx using n points and is exact for any polynomial f(x) of degree at most $2n-1. This choice of nodes minimizes the error in the quadrature approximation among all sets of n distinct real points, leveraging the orthogonality of the Hermite polynomials with respect to the weight e^{-x^2}. For large n, the zeros cluster densely in the interval [- \sqrt{2n}, \sqrt{2n}], with their asymptotic distribution derived via the WKB approximation applied to the Sturm–Liouville problem underlying the polynomials. After scaling by \sqrt{2n} (i.e., t = x / \sqrt{2n}), the empirical measure of the zeros converges weakly to the semicircle distribution with density \frac{2}{\pi} \sqrt{1 - t^2} on [-1, 1]. This bulk distribution captures the global spacing and concentration of the zeros away from the endpoints. Near the endpoints, finer asymptotics reveal that the largest zero satisfies x_{n,n} \sim \sqrt{2n} while the smallest zero x_{1,n} \sim -\sqrt{2n} for large n, providing tight bounds on the oscillatory interval of H_n(x). These endpoint estimates highlight the sparser distribution of zeros in the tails compared to the central clustering.

Relations to Other Functions

Connection to Laguerre polynomials

The Hermite polynomials H_n(x) exhibit a direct connection to the generalized L_n^\alpha(z) via explicit relations that distinguish even and odd degrees. For even degrees, the transformation is H_{2n}(x) = (-1)^n 2^{2n} n! L_n^{-1/2}(x^2), where \alpha = -1/2 in the generalized Laguerre polynomial. For odd degrees, it takes the form H_{2n+1}(x) = (-1)^n 2^{2n+1} n! \, x \, L_n^{1/2}(x^2), with \alpha = 1/2. These formulas establish an exact polynomial mapping, allowing properties of one family to be transferred to the other through substitution z = x^2. The relations can be derived by equating the generating functions for Hermite and Laguerre polynomials or by comparing their Rodrigues representations. The generating function for Hermite polynomials is g(t,x) = e^{2xt - t^2} = \sum_{n=0}^\infty H_n(x) \frac{t^n}{n!}, while for generalized Laguerre polynomials it is \sum_{n=0}^\infty L_n^\alpha(z) t^n = \frac{1}{(1-t)^{\alpha+1}} \exp\left( -\frac{zt}{1-t} \right). Substituting t \to it/\sqrt{2} and x \to \sqrt{z/2} aligns the expansions for even and odd terms, yielding the explicit connections after extracting coefficients. Alternatively, the Rodrigues formulas H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} and L_n^\alpha(z) = \frac{e^z z^{-\alpha}}{n!} \frac{d^n}{dz^n} (e^{-z} z^{n+\alpha}) lead to the same result upon composing derivatives under the change of variables. This connection extends to orthogonality properties through a suitable change of variables. The Hermite polynomials are orthogonal on (-\infty, \infty) with respect to the weight e^{-x^2}, satisfying \int_{-\infty}^\infty H_m(x) H_n(x) e^{-x^2} \, dx = \sqrt{\pi} \, 2^n n! \, \delta_{mn}. Substituting u = x^2 transforms the integral into one over [0, \infty) with weight e^{-u} u^{-1/2} for even functions (corresponding to \alpha = -1/2) or e^{-u} u^{1/2} for odd ( \alpha = 1/2 ), matching the Laguerre orthogonality \int_0^\infty L_m^\alpha(u) L_n^\alpha(u) e^{-u} u^\alpha \, du = \frac{\Gamma(n+\alpha+1)}{n!} \delta_{mn}. In applications, these relations are particularly useful in quantum mechanics for solving the Schrödinger equation in radial coordinates. For the three-dimensional isotropic harmonic oscillator, the wave functions in Cartesian coordinates involve products of one-dimensional Hermite functions, while in spherical coordinates, they reduce to associated Laguerre polynomials multiplied by spherical harmonics; the explicit mapping facilitates equivalence between the two approaches.

Hypergeometric function representation

The Hermite polynomials admit representations in terms of the confluent hypergeometric function of the first kind, denoted _1F_1(a; b; z) or equivalently Kummer's function M(a, b, z). This connection arises from solving the Hermite differential equation via series methods, where the solutions align with the power series expansion of the confluent hypergeometric function. For the physicist's Hermite polynomials H_n(x), the expressions separate into even and odd cases. When n = 2m is even, H_{2m}(x) = (-1)^m \frac{(2m)!}{m!} \, _1F_1\left( -m; \frac{1}{2}; x^2 \right). When n = 2m+1 is odd, H_{2m+1}(x) = (-1)^m \frac{(2m+1)!}{m!} \, 2x \, _1F_1\left( -m; \frac{3}{2}; x^2 \right). These formulas follow from the series solution of the Hermite equation and the definition of Kummer's function. The series for _1F_1(a; b; z) terminates in these cases because the parameter a = -m is a non-positive integer, limiting the hypergeometric series to a finite sum of m+1 terms and yielding a polynomial of exact degree n. This termination property ensures the expressions reproduce the polynomial nature of the Hermite functions without infinite series divergence. For the probabilist's Hermite polynomials \mathrm{He}_n(x), defined via the relation \mathrm{He}_n(x) = 2^{-n/2} H_n\left( x / \sqrt{2} \right), the hypergeometric representations are scaled accordingly. Specifically, for even n = 2m, \mathrm{He}_{2m}(x) = (-1)^m \frac{(2m)!}{m! \, 2^{m}} \, _1F_1\left( -m; \frac{1}{2}; \frac{x^2}{2} \right), and for odd n = 2m+1, \mathrm{He}_{2m+1}(x) = (-1)^m \frac{(2m+1)!}{m! \, 2^{m}} \, x \, _1F_1\left( -m; \frac{3}{2}; \frac{x^2}{2} \right). The same termination mechanism applies, confirming the polynomial character. This hypergeometric form facilitates analytic continuation of the Hermite polynomials to the complex domain, leveraging the meromorphic properties of _1F_1 defined via its series for |\arg z| < \pi and continued elsewhere by analytic means, which is useful in applications involving complex variables such as and .

Limit relations to other orthogonal polynomials

Hermite polynomials arise as limiting cases of other families of orthogonal polynomials when certain parameters tend to infinity, reflecting the transition from finite-interval orthogonality to the entire real line with Gaussian weight. A prominent example is the limit from , which are orthogonal on [-1,1] with respect to the beta distribution weight (1-z)^\alpha (1+z)^\beta. Specifically, as the parameters \alpha and \beta tend to infinity with fixed ratio or symmetrically, the appropriately scaled Jacobi polynomials converge to the Hermite polynomials. The precise relation is given by \lim_{\alpha, \beta \to \infty} \frac{P_n^{(\alpha, \beta)}\left( \frac{2x}{\alpha + \beta + 2} \right)}{\left( \frac{\alpha + \beta + 2}{2} \right)^n } = \frac{H_n(x)}{2^n}, where the convergence is uniform on compact subsets of the real line. This limit captures the degeneration of the beta weight to the Gaussian e^{-x^2} as the support effectively expands to \mathbb{R}. Since Legendre polynomials are the special case of Jacobi polynomials with \alpha = \beta = 0, a related limiting process connects Hermite polynomials to Legendre polynomials through large-degree scaling. For large n, the Hermite polynomial H_n, when scaled to the fixed interval [-1,1] by the substitution x \mapsto x \sqrt{2n}, approximates the Legendre polynomial P_n(x) in the sense of their oscillatory behavior and zero distributions. The zero density of the scaled Hermite converges to the arcsine distribution 1/(\pi \sqrt{1 - x^2}) on [-1,1], matching that of the Legendre polynomials. This relation underscores how the unbounded support of Hermite "degenerates" to the bounded [-1,1] interval in the large-n regime, with uniform convergence on compact subsets away from the turning points. A similar parameter limit obtains from ultraspherical (Gegenbauer) polynomials, which generalize as C_n^{(\lambda)}(x) = P_n^{(\lambda - 1/2, \lambda - 1/2)}(x) / constant. As \lambda \to \infty, the scaled ultraspherical polynomials converge to : \lim_{\lambda \to \infty} \frac{C_n^{(\lambda)} \left( \frac{x}{\sqrt{2\lambda - 1}} \right) }{ (2\lambda - 1)^{n/2} } = \frac{ (-1)^n 2^{n/2} }{ n! } H_n \left( \frac{x}{\sqrt{2}} \right). This follows from the by setting \alpha = \beta = \lambda - 1/2, and the convergence is again uniform on compact sets. For \lambda = 1/2, ultraspherical reduces to up to scaling, providing an indirect link where emerges from the Legendre family in the high-parameter limit. For Laguerre polynomials, orthogonal on [0, \infty) with weight x^\alpha e^{-x}, the connection to Hermite involves both exact quadratic substitutions and limiting processes. An exact relation uses the substitution to separate even and odd parts: for instance, the even-degree Hermite polynomials relate directly to generalized Laguerre via H_{2n}( \sqrt{y} ) = constant \cdot (-1)^n L_n^{(-1/2)}(y), but a limiting degeneration arises by considering a quadratic change of variables in the generating function or weight. Specifically, by setting x = i \sqrt{2 y} in the Hermite generating function and taking a limit as the imaginary part adjusts the support to [0,\infty), or more rigorously, the limit \lim_{\alpha \to \infty} \frac{ (-1)^n L_n^{(\alpha)} (x^2 / (4 \alpha)) }{ \alpha^n } = \frac{ H_{2n} (x/2) }{ 2^{2n} n! }, shows Laguerre degenerating to even Hermite polynomials, with the reverse process using quadratic scaling y = x^2 / 2 to map the half-line back, yielding Laguerre from Hermite in the low-parameter regime. Convergence holds uniformly on compact subsets of [0, \infty). This quadratic substitution highlights how the Gaussian weight e^{-x^2} restricts to e^{-y} on the positive axis in the limit. These limit relations extend to generating functions: the Hermite generating function e^{2 x t - t^2} emerges as the limit of the Jacobi generating function \sum P_n^{(\alpha,\beta)}(z) t^n under the scaling z = 2x / (\alpha + \beta + 2), \alpha, \beta \to \infty, with uniform convergence on compact sets ensuring the polynomial coefficients match in the limit. Similar transitions hold for Laguerre generating functions (1 - t)^{-\alpha - 1} e^{x t / (t - 1)} limiting to Hermite forms via quadratic rescaling. These processes preserve orthogonality and provide a unified framework for deriving properties across families.

Expansions and Applications

Series expansions in Hermite polynomials

Series expansions in Hermite polynomials provide a powerful tool for representing functions in the Hilbert space L^2(\mathbb{R}), where the basis functions are the Hermite functions \phi_n(x) = H_n(x) e^{-x^2/2}, with H_n(x) denoting the probabilist's . These functions are orthogonal with respect to the standard L^2 inner product \langle f, g \rangle = \int_{-\infty}^{\infty} f(x) g(x) , dx, and the squared norm is ||H_n e^{-x^2/2}||^2 = \sqrt{\pi} \frac{(2n)!}{2^{2n} n!}. For a function f \in L^2(\mathbb{R}), the Fourier-Hermite series is given by f(x) = \sum_{n=0}^\infty c_n H_n(x) e^{-x^2/2}, where the coefficients are c_n = \frac{\langle f, H_n e^{-x^2/2} \rangle}{||H_n e^{-x^2/2}||^2} = \frac{2^{2n} n!}{\sqrt{\pi} (2n)!} \int_{-\infty}^{\infty} f(x) H_n(x) e^{-x^2/2} \, dx.[](https://dlmf.nist.gov/18.3) This expansion leverages the completeness of the Hermite functions in L^2(\mathbb{R}), ensuring that the series converges to f in the L^2 norm for any f in the space.[](https://people.math.osu.edu/nevai.1/SZEGO/szego=szego1975=ops=OCR.pdf) The partial sum S_N(f)(x) = \sum_{n=0}^N c_n H_n(x) e^{-x^2/2} provides the best approximation to f in the L^2 norm among all linear combinations of the first N+1 basis functions, minimizing the error ||f - S_N(f)||_2. Error estimates depend on the smoothness of f; for analytic functions, the convergence is exponential, with rates determined by the distance to the nearest singularity in the complex plane. For example, if f is entire of exponential type, the L^2 error decays as O(e^{- \rho N}) for some \rho > 0 related to the growth of f. A notable example is the expansion of the Gaussian function itself, e^{-x^2/2}, which is proportional to the n=0 term since H_0(x) = 1 and higher-order terms are orthogonal to it in L^2(\mathbb{R}). Specifically, e^{-x^2/2} = c_0 H_0(x) e^{-x^2/2} with c_0 = 1, and c_n = 0 for n \geq 1. Another illustrative case is the expansion derived from the [generating function](/page/Generating_function) e^{a x - a^2/2} = \sum_{n=0}^\infty \frac{a^n}{n!} H_n(x), which, when multiplied by e^{-x^2/2}, yields the series for e^{a x - x^2/2}. This [closed-form expression](/page/Closed-form_expression) highlights how generating functions facilitate explicit [coefficient](/page/Coefficient) computation for exponential functions.[](https://people.math.osu.edu/nevai.1/SZEGO/szego=szego1975=ops=OCR.pdf) The zeros of the Hermite polynomials also play a key role in numerical applications, particularly in Gauss-Hermite [quadrature](/page/Quadrature) for approximating integrals of the form \int_{-\infty}^{\infty} g(x) e^{-x^2/2} \, dx. Using the N zeros x_{n,k} of H_N(x) and corresponding weights w_{n,k} = \frac{\sqrt{2\pi} (N-1)!}{N [H_{N-1}(x_{n,k})]^2}, the quadrature rule is \int_{-\infty}^{\infty} g(x) e^{-x^2/2} \, dx \approx \sum_{k=1}^N w_{n,k} g(x_{n,k}), exact for g a [polynomial](/page/Polynomial) of degree less than 2N. This method is especially efficient for functions with Gaussian decay.[](https://dlmf.nist.gov/18.17) ### Expected values in probability In [probability theory](/page/Probability_theory), the probabilist's Hermite polynomials $ \mathrm{He}_n(x) $ play a central role in expressing moments and cumulants of [random variable](/page/Random_variable)s, particularly those related to Gaussian distributions. For a [random variable](/page/Random_variable) $ X $ with [mean](/page/Mean) zero and unit variance, the [expectation](/page/Expectation) $ \mathbb{E}[\mathrm{He}_n(X)] = 0 $ for all $ n \geq 1 $, due to the [orthogonality](/page/Orthogonality) of the Hermite polynomials with respect to the [standard](/page/Standard) [normal](/page/Normal) [density](/page/Density) $ \phi(x) = \frac{1}{\sqrt{2\pi}} e^{-x^2/2} $. This property follows from the [integral](/page/Integral) $ \int_{-\infty}^{\infty} \mathrm{He}_n(x) \phi(x) \, dx = 0 $ for $ n > 0 $, since $ \mathrm{He}_0(x) = 1 $ is the only polynomial orthogonal to the constant [function](/page/Function) in this [space](/page/Space).[](https://arxiv.org/pdf/2409.02876) For the variance, note that $ \mathrm{He}_2(x) = x^2 - 1 $, so $ \mathbb{E}[\mathrm{He}_2(X)] = \mathbb{E}[X^2] - 1 = 0 $, confirming the unit variance assumption. Higher even-order relations similarly tie moments to the polynomials, such as $ \mathbb{E}[X^4] = 3 + \mathbb{E}[\mathrm{He}_4(X)] $, where $ \mathrm{He}_4(x) = x^4 - 6x^2 + 3 $.[](https://arxiv.org/pdf/2409.02876) The connection to generating functions further links Hermite polynomials to cumulants. The cumulant-generating function is defined as $ \log \mathbb{E}[e^{tX}] = \sum_{n=1}^{\infty} \kappa_n \frac{t^n}{n!} $, where $ \kappa_n $ are the cumulants of $ X $. For a standard normal $ X $, all cumulants vanish beyond $ \kappa_2 = 1 $, and the expansion aligns with the generating function of the probabilist's Hermite polynomials, $ e^{x t - t^2/2} = \sum_{n=0}^{\infty} \frac{\mathrm{He}_n(x) t^n}{n!} $. This relation facilitates expressing higher cumulants in terms of expectations involving Hermite polynomials, especially in approximations where deviations from Gaussianity are small. A key application is the Edgeworth expansion, which refines central limit theorem approximations by incorporating cumulants via Hermite polynomials. The density $ f(y) $ of a standardized sum is approximated as \[ f(y) \approx \phi(y) \left[ 1 + \sum_{n=3}^{\infty} \frac{\kappa_n}{n!} \mathrm{He}_n(y) \right], where the terms beyond the leading Gaussian \phi(y) capture skewness (n=3), kurtosis (n=4), and higher-order effects through the cumulants \kappa_n. This series provides asymptotic improvements in tail probabilities and quantiles, with error rates depending on the decay of cumulants, and is particularly useful for moderate sample sizes where the Berry-Esseen bound is insufficient. In the context of Gaussian processes, the Wiener chaos expansion decomposes square-integrable functionals f(\mathbf{X}) as f(\mathbf{X}) = \sum_{n=0}^{\infty} \sum_{i_1 < \cdots < i_n} c_{i_1 \dots i_n} \mathrm{He}_{i_1}(X_{j_1}) \cdots \mathrm{He}_{i_n}(X_{j_n}), where \mathbf{X} = (X_1, \dots, X_m) are jointly Gaussian with mean zero, and coefficients c_{i_1 \dots i_n} are determined by projections onto chaos spaces of order n. The expectations \mathbb{E}[f(\mathbf{X})] = c_\emptyset (the constant term), and higher moments follow from Wick's theorem applied to the orthogonal products \mathbb{E}[\mathrm{He}_{i_1}(X_{j_1}) \cdots \mathrm{He}_{i_k}(X_{j_k})] = 0 unless paired correctly. This framework enables computing moments of nonlinear functionals in stochastic differential equations and uncertainty quantification. For non-Gaussian random variables, Hermite polynomials extend to non-central moments through moment-based models that adjust the Gaussian expansion using raw moments \mu_n = \mathbb{E}[X^n]. These models, such as the cubic Hermite translation for slight non-Gaussianity, shift the argument to account for mean \mu and incorporate higher moments via polynomials like \mathrm{He}_3(x - \mu) for skewness, enabling simulation and density estimation of processes with excess kurtosis or asymmetry. The approach matches prescribed moments up to fourth order while preserving positivity, outperforming in tail accuracy for moderately non-Gaussian cases.

Kibble–Slepian formula

The Kibble–Slepian formula provides a closed-form expression for the expected value of a product of Hermite polynomials evaluated at jointly distributed zero-mean Gaussian random variables with unit variances and covariances r_{ij}. For random variables X_1, \dots, X_m with covariance matrix (r_{ij}), the formula states that E\left[ \prod_{j=1}^m \mathrm{He}_{n_j}(X_j) \right] = \sum_{\mathbf{k}} \frac{\left( \sum_{j=1}^m n_j \right)!}{\mathbf{k}!} \prod_{i=1}^m \left( \prod_{j=1}^m r_{ij}^{k_{ij}} \right), where the sum is over all multi-indices \mathbf{k} = (k_{ij}) corresponding to the possible distributions of pairings across the degrees n_j, or equivalently in permanental form involving the covariance matrix blocked according to the degrees n_j. This multivariate extension generalizes the univariate case where expectations simplify to factorial moments adjusted by the polynomial structure. The formula was first proposed by Kibble in 1945 as an extension of Mehler's bivariate result and rigorously proved by Slepian in 1972 using Fourier analysis techniques on the generating function for the polynomials. Alternative derivations include combinatorial interpretations via labeled trees and matchings, as developed by Foata, and operator methods using boson creation and annihilation operators, as shown by Louck. These approaches leverage the connection to Wick's theorem (or Isserlis' theorem for classical Gaussians), where the expectation arises from all complete contractions of the underlying power series expansion of the Hermite polynomials, with each contraction weighted by the corresponding covariance r_{ij} for cross terms and 1 for self-contractions within the same variable. In the special case m=2, the formula reduces to an extension of orthogonality relations: E[\mathrm{He}_n(X_1) \mathrm{He}_m(X_2)] = \delta_{nm} n! r^n, which holds under the unit variance assumption and captures the correlation-induced non-orthogonality for equal degrees while vanishing for unequal degrees due to parity and moment constraints. This bivariate limit aligns with and serves as a foundational check for the multivariate generalization. The finds applications in the analysis of , where it enables computation of higher-order moments for nonlinear transformations expanded in the , facilitating studies of field statistics and large deviations. In quantum optics, it supports derivations of quasiprobability distributions and coherent state overlaps in systems modeled by multivariate , aiding calculations in squeezed light and multimode quantum states.

Generalizations

Probabilist's versus physicist's variants

There are two primary conventions for Hermite polynomials: the physicist's version, denoted H_n(x), and the probabilist's version, denoted \mathrm{He}_n(x). These variants differ in their scaling and normalization to suit applications in quantum mechanics and probability theory, respectively. The scaling relation between the two is given by \mathrm{He}_n(x) = 2^{-n/2} H_n\left(x / \sqrt{2}\right), or equivalently, H_n(x) = 2^{n/2} \mathrm{He}_n\left(\sqrt{2}\, x\right). This adjustment arises from the differing variable scalings in their respective domains, where the physicist's version aligns with the coordinate x in the quantum harmonic oscillator potential, while the probabilist's version uses the standardized scale for unit-variance Gaussian distributions. The physicist's Hermite polynomials originated in the context of quantum mechanics, particularly in solving the Schrödinger equation for the harmonic oscillator, where they appear as components of the energy eigenfunctions. In contrast, the probabilist's version emerged from statistical applications, notably in Norbert Wiener's development of homogeneous chaos expansions for representing random functions in terms of Gaussian processes. Regarding orthogonality, the physicist's polynomials H_n(x) are orthogonal with respect to the weight function e^{-x^2}, satisfying \int_{-\infty}^{\infty} H_m(x) H_n(x) e^{-x^2}\, dx = \sqrt{\pi}\, 2^n n! \, \delta_{mn}, with squared norm \sqrt{\pi}\, 2^n n!. The probabilist's polynomials \mathrm{He}_n(x) are orthogonal with respect to the weight e^{-x^2/2}, satisfying \int_{-\infty}^{\infty} \mathrm{He}_m(x) \mathrm{He}_n(x) e^{-x^2/2}\, dx = n! \, \delta_{mn}, with squared norm n!. These norms reflect the unnormalized weights; when using the full standard normal density for the probabilist's case, the orthogonality integral becomes \int \mathrm{He}_m(x) \mathrm{He}_n(x) \frac{e^{-x^2/2}}{\sqrt{2\pi}}\, dx = \frac{n!}{\sqrt{2\pi}} \delta_{mn}. The recurrences also convert via the scaling. The physicist's recurrence is H_{n+1}(x) = 2x H_n(x) - 2n H_{n-1}(x), while the probabilist's is \mathrm{He}_{n+1}(x) = x \mathrm{He}_n(x) - n \mathrm{He}_{n-1}(x). Similarly, the generating functions differ: for physicists, \sum_{n=0}^{\infty} \frac{H_n(x)}{n!} t^n = e^{2xt - t^2}; for probabilists, \sum_{n=0}^{\infty} \frac{\mathrm{He}_n(x)}{n!} t^n = e^{xt - t^2/2}. These can be interchanged by substituting the scaling relation into the generating function and adjusting the parameter t. The choice of variant depends on the context: the probabilist's version is preferred in probability and statistics for expansions under the unit-variance Gaussian measure, facilitating moments and chaos decompositions, whereas the physicist's version is standard in quantum mechanics for the harmonic oscillator, where the potential scaling naturally leads to the e^{-x^2} weight.

Negative variance extensions

The negative variance extensions of Hermite polynomials generalize the standard forms by considering a Gaussian weight with negative variance parameter, resulting in a formal weight function of the form e^{\alpha x^2} for \alpha > 0, as opposed to the decaying e^{-\alpha x^2}. These polynomials, denoted H_n^{[-\alpha]}(x), have coefficients that are the values of those in the standard H_n^{[\alpha]}(x), ensuring nonnegative coefficients suitable for moment interpretations in . This construction arises in the context of expansions for moments of a with negative variance, treated analytically rather than probabilistically. The standard Hermite polynomials H_n(x) are entire functions of exponential type, permitting holomorphic to arguments, including purely imaginary values H_n(ix), where x is real. This continuation provides a scaled version aligned with the negative variance weight, as substituting the argument x \to ix in the e^{2xt - t^2} yields e^{2ixt - t^2}, facilitating expansions under the growing exponential weight e^{x^2}. The polynomials remain of n with leading coefficient (2i)^n, preserving monicity up to scaling. Orthogonality for these extensions cannot hold in the usual L^2 sense on the real line due to divergence of the weight integral, but it is recovered via contour integration in the complex plane or along the imaginary axis. Specifically, the functions H_m(ix) and H_n(ix) satisfy an orthogonality relation with respect to the standard Gaussian weight e^{-x^2} on (-\infty, \infty), derived by substitution in the classical formula. More general complex variants exhibit orthogonality over weighted complex spaces, such as \int H_m(z) \overline{H_n(z)} e^{-|z|^2} d\mu(z), where d\mu is a suitable measure on contours ensuring convergence. These extensions find applications in modeling unstable quantum systems, particularly the inverted with potential V(x) = -\frac{1}{2} x^2, whose admits solutions expressible via H_n(ix) scaled by exponentials. The eigenfunctions involve parabolic cylinder functions D_{\nu}(iz), related to Hermite polynomials by D_n(z) = 2^{-n/2} e^{-z^2/4} H_n\left( \frac{z}{\sqrt{2}} \right) for n, with the imaginary argument capturing the unstable dynamics and spectrum analysis in L^p spaces. In , they appear in asymptotic expansions of generating functions and integral representations on complex , aiding evaluations of theta-like series with growing weights.

Hermite Functions

Definition and normalization

Hermite functions are obtained by multiplying Hermite polynomials by a Gaussian , forming an for the L^2(\mathbb{R}). In the physicist's convention, the Hermite functions are defined as \psi_n(x) = (2^n n! \sqrt{\pi})^{-1/2} e^{-x^2/2} H_n(x), where H_n(x) denotes the physicist's Hermite polynomials, which satisfy with respect to the weight e^{-x^2}. This ensures that the functions are orthonormal in L^2(\mathbb{R}), satisfying \int_{-\infty}^{\infty} \psi_m(x) \psi_n(x) \, dx = \delta_{mn}. The Gaussian envelope e^{-x^2/2} provides rapid asymptotic decay, confining the functions' support effectively while preserving the polynomial oscillation structure. An analogous construction exists in the probabilist's convention, where the functions incorporate the probabilist's Hermite polynomials \mathrm{He}_n(x), orthogonal with respect to the weight e^{-x^2/2}, and are defined with the damping factor e^{-x^2/4} to achieve L^2 normalization \int_{-\infty}^{\infty} |\phi_n(x)|^2 \, dx = 1. The first few physicist's Hermite functions illustrate this structure: \psi_0(x) = \pi^{-1/4} e^{-x^2/2}, \psi_1(x) = \pi^{-1/4} \sqrt{2} \, x \, e^{-x^2/2}, \psi_2(x) = \pi^{-1/4} (2x^2 - 1) e^{-x^2/2} / \sqrt{2}. These examples highlight the increasing number of nodes and the consistent Gaussian modulation.

and combinatorial aspects

The Hermite functions \psi_n(x), defined as \psi_n(x) = \frac{1}{\sqrt{2^n n! \sqrt{\pi}}} H_n(x) e^{-x^2/2}, satisfy a relation analogous to that of the underlying Hermite polynomials but adapted to the L^2-normalized form with the Gaussian damping factor. This can be expressed using ladder operators from , where the raising operator a^\dagger = \frac{1}{\sqrt{2}} \left( x - \frac{d}{dx} \right) acts as \psi_{n+1}(x) = \frac{a^\dagger \psi_n(x)}{\sqrt{n+1}}, yielding the explicit form \psi_{n+1}(x) = \frac{1}{\sqrt{2(n+1)}} \left( x \psi_n(x) - \frac{d}{dx} \psi_n(x) \right). This relation allows iterative computation of higher functions from lower ones and is fundamental for constructing the basis in the quantum harmonic oscillator. The coefficients in the power series expansion of the Hermite polynomials H_n(x) = \sum_{k=0}^{\lfloor n/2 \rfloor} (-1)^k \frac{n!}{k! (n-2k)!} (2x)^{n-2k} admit a combinatorial interpretation in terms of partial matchings on a set of n labeled points. Specifically, the absolute value of the coefficient of x^{n-2k} is the number of ways to partition the n points into k disjoint pairs (a perfect matching on 2k points) and n-2k singletons, where the number of such matchings is \frac{n!}{k! (n-2k)! 2^k}, with the sign (-1)^k arising from the alternating structure in the Rodrigues formula or generating function derivation. This interpretation links Hermite polynomials to enumeration of pairings in graph theory, where x weights the singletons and the pairs contribute the sign. Representative examples include H_2(x) = 4x^2 - 2, where the x^2 term counts 1 way to pair 2 points (with weight 4 from 2^2), and the constant term counts 1 way to have 2 singletons (with sign -1); for H_4(x) = 16x^4 - 48x^2 + 12, the terms correspond to 0 pairs (16x^4), 1 pair ( -48x^2, 6 ways to choose and pair 2 out of 4, signed), and 2 pairs (12, 3 ways to pair all 4). The Touchard polynomials T_n(x), defined as T_n(x) = \sum_{k=0}^n S(n,k) x^k where S(n,k) are of the second kind, are related to Hermite polynomials via evaluation at imaginary arguments, specifically through the identity involving the probabilist's variant He_n, where |He_n(ix)| contributes to the positive coefficients in Touchard polynomials via connections, as the imaginary substitution transforms the alternating signs into a form amenable to enumerations. This relation highlights the combinatorial overlap, with Touchard polynomials counting set partitions weighted by x^k for k blocks. A for the Hermite functions is given by \sum_{n=0}^\infty \psi_n(x) \frac{t^n}{\sqrt{n!}} = \pi^{-1/4} e^{t^2/2 - (x - \sqrt{2} t)^2 / 2}, which can be expressed in terms of the as e^{t^2/2} e^{-t^2/2 + \sqrt{2} t x} \psi_0(x), reflecting the structure and facilitating expansions in the function basis. The connection to s arises in evaluations at specific points, such as for n=0 in the Touchard relation, where T_0(x) = 1 and the B_0 = 1, but more broadly, B_n = T_n(1) links the partition enumeration directly to the imaginary evaluation limit of Hermite polynomials at order n, providing a combinatorial bridge for n=0 as the trivial partition case.

Fourier transform eigenfunctions

The Hermite functions \psi_n(x), defined as \psi_n(x) = \frac{1}{\sqrt{2^n n! \sqrt{\pi}}} H_n(x) e^{-x^2/2} where H_n are the physicist's Hermite polynomials, serve as eigenfunctions of the Fourier transform operator \mathcal{F}, given by \mathcal{F} f(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) e^{-i \omega x} \, dx. Specifically, they satisfy \mathcal{F} \psi_n(\omega) = (-i)^n \psi_n(\omega). This relation holds under the unitary convention for the Fourier transform, ensuring preservation of the L^2 norm. The eigenvalues (-i)^n = e^{-i \pi n / 2} are pure phases that cycle every four indices: $1 for n \equiv 0 \pmod{4}, -i for n \equiv 1 \pmod{4}, -1 for n \equiv 2 \pmod{4}, and i for n \equiv 3 \pmod{4}. A proof of this eigenfunction property relies on the generating function approach. The generating function for the Hermite functions is G(x, t) = \sum_{n=0}^{\infty} \psi_n(x) \frac{t^n}{\sqrt{n!}} = \pi^{-1/4} \exp\left( -\frac{1}{2} (x^2 + t^2) + \sqrt{2} x t \right). Taking the Fourier transform yields \mathcal{F} G(\omega, t) = (-i)^t G(\omega, -i t) or an equivalent form after adjusting for the parameter shift, which, upon series expansion, produces the factor (-i)^n for each term \psi_n. More precisely, the Fourier transform of the underlying Gaussian e^{-x^2/2 + i k x} is e^{-\omega^2/2} e^{-k^2/2} e^{i k \omega} up to normalization constants, mirroring the original form with a phase rotation that induces the eigenvalues upon differentiation or expansion in powers of k. Since the is a on L^2(\mathbb{R}) and the Hermite functions form a complete therein, their images under \mathcal{F} also form a complete in the . The self-similarity of the s ensures that \{\psi_n(\omega)\}_{n=0}^{\infty} spans L^2(\mathbb{R}) equivalently. The facilitates bilateral expansions in this basis, given by K_r(x, y) = \sum_{n=0}^{\infty} r^n \psi_n(x) \psi_n(y) for |r| < 1, which equals \frac{1}{\sqrt{\pi (1 - r^2)}} \exp\left( -\frac{(x^2 + y^2) - 2 r x y}{2 (1 - r^2)} \right). This supports representations and reproducing properties in both spatial and domains due to the relation.

Applications in quantum mechanics and signal processing

Hermite functions play a central role in quantum mechanics as the stationary wave functions of the quantum harmonic oscillator, a fundamental model for systems ranging from molecular vibrations to quantum fields. The energy eigenvalues for these states are given by E_n = \hbar \omega \left( n + \frac{1}{2} \right), where n = 0, 1, 2, \dots, \hbar is the reduced Planck's constant, and \omega is the angular frequency of the oscillator. This quantized energy spectrum arises from solving the time-independent Schrödinger equation for the potential V(x) = \frac{1}{2} m \omega^2 x^2, with the Hermite functions \psi_n(x) forming a complete orthonormal basis for the Hilbert space of the system. In phase-space formulations of , the Wigner function provides a quasi-probability distribution that bridges classical and quantum descriptions. For the states, the Wigner function is expressed as W_n(x, p) = \frac{1}{\pi} \int_{-\infty}^{\infty} \psi_n(x + y) \psi_n^*(x - y) e^{2 i p y} \, dy, where units are chosen such that \hbar = 1. This representation reveals quantum interference effects, such as negative regions for higher n, highlighting the non-classical nature of the states even for the simple harmonic potential. Explicit computations for these Wigner functions often yield expressions involving , facilitating analysis of quantum correlations and revivals in time evolution. Beyond quantum mechanics, Hermite functions find applications in signal processing as a wavelet-like basis for time-frequency analysis, particularly in Gabor transforms. They form efficient frames for representing signals due to their localized Gaussian envelopes and oscillatory behavior, enabling decomposition into components with optimal time-frequency localization. For instance, linear combinations of Hermite functions generate Gabor frames on lattices in \mathbb{R}^2, which are used for adaptive filtering and in audio and image processing. This property stems from their minimal uncertainty in the time-frequency plane, making them suitable for analyzing non-stationary signals where traditional methods fall short. The of Hermite functions, \langle \psi_m | \psi_n \rangle = \delta_{mn}, ensures a complete basis for expansions, but shifted versions \psi_n(x - a) exhibit partial overlaps, quantified by non-zero inner products \langle \psi_m | \psi_n(\cdot - a) \rangle. These overlaps allow Hermite-based frames to approximate tight Gabor systems in , balancing redundancy and reconstruction stability without full . Cramér's inequality provides a uniform bound on the magnitude of normalized Hermite functions, stating that |\psi_n(x)| \leq \pi^{-1/4} for all real x and n, with the Gaussian \psi_0(x) achieving as the extremal. This bound is crucial for establishing convergence in series expansions and uncertainty principles, particularly for Gaussian extremals in optimization problems involving forms.