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pgb
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Can anyone PROOVE how to find out the normalisation of hermite polynomial?
Hermite polynomials are a type of orthogonal polynomial that are commonly used in mathematical physics, particularly for solving differential equations. They also have applications in probability theory and approximation theory.
The normalization condition for Hermite polynomials is that the integral of the square of the polynomial over the entire real line is equal to 1. This ensures that the polynomials are properly scaled and do not grow infinitely large.
The coefficients of Hermite polynomials can be calculated using the three-term recurrence relation, which relates the coefficients of a given polynomial to the coefficients of the previous and next polynomials in the sequence. Alternatively, they can also be calculated using the Gram-Schmidt process.
Hermite polynomials are named after the French mathematician Charles Hermite, who first studied them in the late 19th century. Hermite made significant contributions to the field of mathematical analysis and is also known for his work on elliptic and abelian functions.
Hermite polynomials are closely related to Gaussian distributions, also known as normal distributions. In fact, they are the solution to the Hermite differential equation, which arises in the study of Gaussian distributions. This connection allows for the use of Hermite polynomials in probability and statistics.