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^[1]: 101

${\displaystyle y_{n}(x)=\sum _{k=0}^{n}{\frac {(n+k)!}{(n-k)!k!}}\,\left({\frac {x}{2}}\right)^{k}.}$

Another definition, favored by electrical engineers, is sometimes known as the reverse Bessel polynomials^{[2]: 8 [3]: 15}

${\displaystyle \theta _{n}(x)=x^{n}\,y_{n}(1/x)=\sum _{k=0}^{n}{\frac {(n+k)!}{(n-k)!k!}}\,{\frac {x^{n-k}}{2^{k}}}.}$

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

${\displaystyle y_{3}(x)=15x^{3}+15x^{2}+6x+1}$

while the third-degree reverse Bessel polynomial is

${\displaystyle \theta _{3}(x)=x^{3}+6x^{2}+15x+15.}$

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