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 Another definition, favored by electrical engineers, is sometimes known as the reverse Bessel polynomials[2]: 8 [3]: 15 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 Kn(x) is a modified Bessel function of the second kind, yn(x) is the ordinary polynomial, and θn(x) is the reverse polynomial .
[2]: 7, 34 For example:[4] The Bessel polynomial may also be defined as a confluent hypergeometric function[5]: 8 A similar expression holds true for the generalized Bessel polynomials (see below):[2]: 35 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 Bessel polynomials, with index shifted, have the generating function Differentiating with respect to
, yields the generating function for the polynomials
Similar generating function exists for the
polynomials as well:[1]: 106 Upon setting
, one has the following representation for the exponential function:[1]: 107 The Bessel polynomial may also be defined by a recursion formula: and The Bessel polynomial obeys the following differential equation: and The Bessel polynomials are orthogonal with respect to the weight
integrated over the unit circle of the complex plane.
A generalization of the Bessel polynomials have been suggested in literature, as following: the corresponding reverse polynomials are The explicit coefficients of the
( x ; α , β )
( x ; α , β )
polynomials can explicitly be written as follows: For the weighting function they are orthogonal, for the relation holds for m ≠ n and c a curve surrounding the 0 point.
They specialize to the Bessel polynomials for α = β = 2, in which situation ρ(x) = exp(−2/x).
are expressed in terms of the generalized Bessel polynomials from the inverse connection formulae which have applications in change of basis to these polynomials.
Similarly, for the reverse generalized Bessel polynomials where
( n , k , α , β ) = ( − 1
The Rodrigues formula for the Bessel polynomials as particular solutions of the above differential equation is : where a(α, β)n are normalization coefficients.
According to this generalization we have the following generalized differential equation for associated Bessel polynomials: where
The solutions are, If one denotes the zeros of
, then the following estimates exist:[2]: 82 and for all
Moreover, all these zeros have negative real part.
Sharper results can be said if one resorts to more powerful theorems regarding the estimates of zeros of polynomials (more concretely, the Parabola Theorem of Saff and Varga, or differential equations techniques).
[2]: 88 [7] One result is the following:[8] The Bessel polynomials
are[9] No Bessel polynomial can be factored into lower degree polynomials with rational coefficients.
[10] The reverse Bessel polynomials are obtained by reversing the coefficients.