Bessel polynomials

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 (Krall & Frink, 1948)

Another definition, favored by electrical engineers, is sometimes known as the reverse Bessel polynomials (See Grosswald 1978, Berg 2000).

The coefficients of the second definition are the same as the first but in reverse order. For example, the third-degree Bessel polynomial is

while the third-degree reverse Bessel polynomial is

The reverse Bessel polynomial is used in the design of Bessel electronic filters.

The Bessel polynomial may also be defined using Bessel functions from which the polynomial draws its name.

where K_n(x) is a modified Bessel function of the second kind, y_n(x) is the ordinary polynomial, and θ_n(x) is the reverse polynomial (pg 7 and 34 Grosswald 1978). For example:

The Bessel polynomial may also be defined as a confluent hypergeometric function (Dita, 2006)

The reverse Bessel polynomial may be defined as a generalized Laguerre polynomial:

from which it follows that it may also be defined as a hypergeometric function:

where (−2n)_n is the Pochhammer symbol (rising factorial).

The inversion for monomials is given by

The Bessel polynomials, with index shifted, have the generating function

Differentiating with respect to ${\displaystyle t}$, cancelling ${\displaystyle x}$, yields the generating function for the polynomials ${\displaystyle \{\theta _{n}\}_{n\geq 0}}$

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