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mathematical physics > special function > hypergeometric function > Bessel polynomial

mathematical analysis > functional analysis > special function > hypergeometric function > Bessel polynomial

... > function > elementary function > polynomial > orthogonal polynomials > Bessel polynomial

algebra > polynomial > orthogonal polynomials > Bessel polynomial

Terme préférentiel

Bessel polynomial

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

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

Another definition, favored by electrical engineers, is sometimes known as the reverse Bessel polynomials

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

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

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

The reverse Bessel polynomial is used in the design of Bessel electronic filters.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Bessel_polynomials)

polynôme de Bessel

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