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 . Properties. Definition in terms of Bessel functions. The Bessel polynomial may also be defined using Bessel functions from which the polynomial draws its name. where "K"("x") is a , "y"("x") is the ordinary polynomial, and "θ"("x") is the reverse polynomial . For example: Definition as a hypergeometric function. The Bessel polynomial may also be defined as a confluent hypergeometric function (Dita, 2006) A similar expression holds true for the generalized Bessel polynomials (see below): 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 (−2"n ) is the Pochhammer symbol (rising factorial). The inversion for monomials is given by Generating function. The Bessel polynomials, with index shifted, have the generating function Differentiating with respect to formula_15, cancelling formula_16, yields the generating function for the polynomials formula_17 Similar generating function exists for the formula_19 polynomials as well: Upon setting formula_21, one has the following representation for the exponential function: Recursion. The Bessel polynomial may also be defined by a recursion formula: and Differential equation. The Bessel polynomial obeys the following differential equation: and Orthogonality. The Bessel polynomials are orthogonal with respect to the weight formula_31 integrated over 	Bessel polynomials
26876596	the unit circle of the complex plane. In other words, if formula_32, formula_33 Generalization. Explicit Form. A generalization of the Bessel polynomials have been suggested in literature (Krall, Fink), as following: the corresponding reverse polynomials are The explicit coefficients of the formula_36 polynomials are: Consequently, the formula_38 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"). Rodrigues formula for Bessel  polynomials. The Rodrigues formula for the Bessel polynomials as particular solutions of the above differential equation is : where "a" are normalization coefficients. Associated Bessel polynomials. According to this generalization we have the following generalized differential equation for associated Bessel polynomials: where formula_44. The solutions are, Zeros. If one denotes the zeros of formula_46 as formula_47, and that of the formula_48 by formula_49, then the following estimates exist: and for all formula_52. 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). One result is the following: Particular values. The first five Bessel Polynomials are expressed as: No Bessel Polynomial can be factored into lower-ordered polynomials with strictly rational coefficients. The five reverse Bessel Polynomials are obtained by reversing the coefficients. Equivalently, formula_55. This results in the following: