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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[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 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 integrated over the unit circle of the complex plane.[1]: 104 In other words, if ,
Generalization
Explicit form
A generalization of the Bessel polynomials have been suggested in literature, as following:
the corresponding reverse polynomials are
The explicit coefficients of the polynomials are:[1]: 108
Consequently, the 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(α, β) n are normalization coefficients.
Associated Bessel polynomials
According to this generalization we have the following generalized differential equation for associated Bessel polynomials:
where . The solutions are,
Zeros
If one denotes the zeros of as , and that of the by , 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 [6]
One result is the following:[7]
No Bessel polynomial can be factored into lower degree polynomials with rational coefficients.[9]
The reverse Bessel polynomials are obtained by reversing the coefficients.
Equivalently, .
This results in the following:
^Dita, Petre; Grama, Nicolae (May 14, 1997). "On Adomian's Decomposition Method for Solving Differential Equations". arXiv:solv-int/9705008.
^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.
^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.
^Filaseta, Michael; Trifinov, Ognian (August 2, 2002). "The Irreducibility of the Bessel Polynomials". Journal für die Reine und Angewandte Mathematik. 2002 (550): 125–140. CiteSeerX10.1.1.6.9538. doi:10.1515/crll.2002.069.