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Hermite polynomials are essential in mathematics, physics, and engineering, providing solutions to Hermite's differential equation and forming an orthogonal system. They are crucial in quantum mechanics as eigenfunctions of the quantum harmonic oscillator and form an orthonormal basis for Hilbert space. Their integral representations, recursion relations, and combinatorial connections highlight their versatility and importance in functional analysis and signal processing.

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## Definition and Properties of Hermite Polynomials

### Weight Function and Orthogonality

Hermite polynomials are defined by a specific weight function and have the property of orthogonality, making them useful for expanding square-integrable functions

### Differential-Operator Method

Generating Function

The differential-operator method provides a way to generate Hermite polynomials by applying the operator \( e^{-\frac{t^2}{2}} \) to the power series expansion of \( e^{xt} \)

Weierstrass Transform

The Weierstrass transform, involving the operator \( e^{t^2} \), connects Hermite polynomials to the Maclaurin series

### Integral Representations and Generalizations

Hermite polynomials can be represented through contour integrals and can be generalized to accommodate different variances while preserving their orthogonality

## Applications of Hermite Polynomials

### Role in Quantum Mechanics

Hermite functions, constructed from Hermite polynomials, serve as eigenfunctions of the quantum harmonic oscillator and are important in expanding square-integrable functions and studying quantum states

### Recursion Relations and Bounds

Hermite functions are governed by recursion relations and are subject to various inequalities, such as Cramér's inequality, which are useful for computational methods and analyzing their behavior

### Fourier Transform and Phase-Space Analysis

Hermite functions have a special behavior under the Fourier transform and are connected to the Wigner distribution, making them relevant in signal processing and quantum mechanics

## Combinatorial Connections and Completeness

### Coefficients and Combinatorial Interpretation

The coefficients of Hermite polynomials have a combinatorial interpretation, counting the number of ways to pair elements in a set, linking them to the study of combinatorics

### Completeness Relation

Christoffel-Darboux Formula

Hermite functions satisfy a completeness relation, encapsulated by the Christoffel-Darboux formula, which affirms their ability to represent any function in \( L^2(\mathbb{R}) \) space

Resolution of Identity Operator

The resolution of the identity operator also confirms the completeness of Hermite functions as a basis in functional analysis

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