Bessel polynomials

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Published on November 21, 2024 12:00:00 AM ● Video Link: https://www.youtube.com/watch?v=eLfBhwIhESI

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In mathematics, the Bessel polynomials are an orthogonal sequence of polynomials. There are a number of different but closely related definitions. The definition favored by mathematicians is given by the series
$$y_n(x)=\sum_{k=0}^n\frac{(n+k)!}{(n-k)!k!}\,\left(\frac{x}{2}\right)^k.$$
Another definition, favored by electrical engineers, is sometimes known as the reverse Bessel polynomials
$$\theta_n(x)=x^n\,y_n(1/x)=\sum_{k=0}^n\frac{(n+k)!}{(n-k)!k!}\,\frac{x^{n-k}}{2^{k}}.$$
The coefficients of the second definition are the same as the first but in reverse order. For example, the third-degree Bessel polynomial is
y₃(x) = 15x³ + 15x² + 6x + 1
while the third-degree reverse Bessel polynomial is
θ₃(x) = x³ + 6x² + 15x + 15.
The reverse Bessel polynomial is used in the design of Bessel electronic filters.

Source: https://en.wikipedia.org/wiki/Bessel_polynomials
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