Bessel polynomials

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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.


Definition in terms of Bessel functions[edit]

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 .[1]: 7, 34  For example:[2]

Definition as a hypergeometric function[edit]

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):[1]: 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 inversion for monomials is given by

Generating function[edit]

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

Differentiating with respect to , cancelling , yields the generating function for the polynomials

Similar generating function exists for the polynomials as well:[3]: 106 

Upon setting , one has the following representation for the exponential function:


The Bessel polynomial may also be defined by a recursion formula:


Differential equation[edit]

The Bessel polynomial obeys the following differential equation:



The Bessel polynomials are orthogonal with respect to the weight integrated over the unit circle of the complex plane.[3] In other words, if ,


Explicit Form[edit]

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 polynomials are:[3]: 108 

Consequently, the polynomials can explicitly be written as follows:

For the weighting function

they are orthogonal, for the relation

holds for mn 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[edit]

The Rodrigues formula for the Bessel polynomials as particular solutions of the above differential equation is :

where a(α, β)
are normalization coefficients.

Associated Bessel polynomials[edit]

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 as , and that of the by , then the following estimates exist:[1]: 82 


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).[1]: 88 [4] One result is the following:[5]

Particular values[edit]

The first five Bessel Polynomials are expressed as:

No Bessel Polynomial can be factored into lower-ordered polynomials with strictly rational coefficients.[6] The five reverse Bessel Polynomials are obtained by reversing the coefficients. Equivalently, . This results in the following:

See also[edit]


  1. ^ a b c d Grosswald, E. (1978). Bessel Polynomials (Lecture Notes in Mathematics). New York: Springer. ISBN 978-0-387-09104-4.
  2. ^ Wolfram Alpha example
  3. ^ a b c Krall, H. L.; Frink, O. (1948). "A New Class of Orthogonal Polynomials: The Bessel Polynomials". Trans. Amer. Math. Soc. 65 (1): 100–115. doi:10.2307/1990516.
  4. ^ Saff, E. B.; Varga, R. S. (1976). "Zero-free parabolic regions for sequences of polynomials". SIAM J. Math. Anal. 7 (3): 344–357. doi:10.1137/0507028.
  5. ^ de Bruin, M. G.; Saff, E. B.; Varga, R. S. (1981). "On the zeros of generalized Bessel polynomials. I". Indag. Math. 84 (1): 1–13.
  6. ^ Filaseta, Michael; Trifinov, Ognian (August 2, 2002). "The Irreducibility of the Bessel Polynomials". Journal für die Reine und Angewandte Mathematik. 2002 (550): 125–140. CiteSeerX doi:10.1515/crll.2002.069.

External links[edit]