Bessel polynomials

Short description: Mathematics concept

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

yn(x)=∑k=0n(n+k)!(n−k)!k!(x2)k.

Another definition, favored by electrical engineers, is sometimes known as the reverse Bessel polynomials[2]: 8 [3]: 15

θn(x)=xnyn(1/x)=∑k=0n(n+k)!(n−k)!k!xn−k2k.

The coefficients of the second definition are the same as the first but in reverse order. For example, the third-degree Bessel polynomial is

y3(x)=1+6x+15x2+15x3

while the third-degree reverse Bessel polynomial is

θ3(x)=x3+6x2+15x+15.

The reverse Bessel polynomial is used in the design of Bessel electronic filters.

## Contents

* 1 Properties
* 1.1 Definition in terms of Bessel functions
* 1.2 Definition as a hypergeometric function
* 1.3 Generating function
* 1.4 Recursion
* 1.5 Differential equation
* 1.6 Orthogonality
* 2 Generalization
* 2.1 Explicit form
* 2.2 Inverse Formulae
* 2.3 Rodrigues formula for Bessel polynomials
* 2.4 Associated Bessel polynomials
* 3 Zeros
* 4 Particular values
* 5 See also
* 6 References
* 7 External links

## Properties

### Definition in terms of Bessel functions

The Bessel polynomial may also be defined using Bessel functions from which the polynomial draws its name.

yn(x)=xnθn(1/x)
yn(x)=2πxe1/xKn+12(1/x)
θn(x)=2πxn+1/2exKn+12(x)

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]

y3(x)=15x3+15x2+6x+1=2πxe1/xK3+12(1/x)

### Definition as a hypergeometric function

The Bessel polynomial may also be defined as a confluent hypergeometric function[5]: 8

yn(x)=2F0(−n,n+1;;−x/2)=(2x)−nU(−n,−2n,2x)=(2x)n+1U(n+1,2n+2,2x).

A similar expression holds true for the generalized Bessel polynomials (see below):[2]: 35

yn(x;a,b)=2F0(−n,n+a−1;;−x/b)=(bx)n+a−1U(n+a−1,2n+a,bx).

The reverse Bessel polynomial may be defined as a generalized Laguerre polynomial:

θn(x)=n!(−2)nLn−2n−1(2x)

from which it follows that it may also be defined as a hypergeometric function:

θn(x)=(−2n)n(−2)n1F1(−n;−2n;2x)

where (−2n)n is the Pochhammer symbol (rising factorial).

### Generating function

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

∑n=0∞2πxn+12exKn−12(x)tnn!=1+x∑n=1∞θn−1(x)tnn!=ex(1−1−2t).

Differentiating with respect to t, cancelling x, yields the generating function for the polynomials {θn}n≥0

∑n=0∞θn(x)tnn!=11−2tex(1−1−2t).

Similar generating function exists for the yn polynomials as well:[1]: 106

∑n=0∞yn−1(x)tnn!=exp⁡(1−1−2xtx).

Upon setting t=z−xz2/2, one has the following representation for the exponential function:[1]: 107

ez=∑n=0∞yn−1(x)(z−xz2/2)nn!.

### Recursion

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

y0(x)=1
y1(x)=x+1
yn(x)=(2n−1)xyn−1(x)+yn−2(x)

and

θ0(x)=1
θ1(x)=x+1
θn(x)=(2n−1)θn−1(x)+x2θn−2(x)

### Differential equation

The Bessel polynomial obeys the following differential equation:

x2d2yn(x)dx2+2(x+1)dyn(x)dx−n(n+1)yn(x)=0

and

xd2θn(x)dx2−2(x+n)dθn(x)dx+2nθn(x)=0

### Orthogonality

The Bessel polynomials are orthogonal with respect to the weight e−2/x integrated over the unit circle of the complex plane.[1]: 104 In other words, if n≠m,

∫02πyn(eiθ)ym(eiθ)ieiθdθ=0

They are also orthogonal with respect to a real weight, provided it is a hyperfunction.[6]

## Generalization

### Explicit form

A generalization of the Bessel polynomials have been suggested in literature, as following:

yn(x;α,β):=(−1)nn!(xβ)nLn(−1−2n−α)(βx),

the corresponding reverse polynomials are

θn(x;α,β):=n!(−β)nLn(−1−2n−α)(βx)=xnyn(1x;α,β).

The explicit coefficients of the yn(x;α,β) polynomials are:[1]: 108

yn(x;α,β)=∑k=0n(nk)(n+k+α−2)k_(xβ)k.

Consequently, the θn(x;α,β) polynomials can explicitly be written as follows:

θn(x;α,β)=∑k=0n(nk)(2n−k+α−2)n−k_xkβn−k.

For the weighting function

ρ(x;α,β):=1F1(1,α−1,−βx)

they are orthogonal, for the relation

0=∮cρ(x;α,β)yn(x;α,β)ym(x;α,β)dx

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).

### Inverse Formulae

Powers of 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.[7]

xn=∑k=0nα(n,k,α,β) yn−k(x;α,β)

where α(n,k,α,β)=(nk)(−1)kβn(2(n−k)+α−1)(n−k+α−1)n+1 and n≥0.

Similarly, for the reverse generalized Bessel polynomials

xn=∑k=0nα1(n,k,α,β) θn−k(x;α,β)

where α1(n,k,α,β)=(−1)k(nk)βk(n+α−1)(n+α−2k)k−1 and n≥0.

### Rodrigues formula for Bessel polynomials

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

Bn(α,β)(x)=an(α,β)xαe−βx(ddx)n(xα+2ne−βx)

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:

x2d2Bn,m(α,β)(x)dx2+[(α+2)x+β]dBn,m(α,β)(x)dx−[n(α+n+1)+mβx]Bn,m(α,β)(x)=0

where 0≤m≤n. The solutions are,

Bn,m(α,β)(x)=an,m(α,β)xα+me−βx(ddx)n−m(xα+2ne−βx)

## Zeros

If one denotes the zeros of yn(x;α,β) as αk(n)(α,β), and that of the θn(x;α,β) by βk(n)(α,β), then the following estimates exist:[2]: 82

2n(n+α−1)≤αk(n)(α,2)≤2n+α−1,

and

n+α−12≤βk(n)(α,2)≤n(n+α−1)2,

for all α≥2. 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 [8] One result is the following:[9]

22n+α−23≤αk(n)(α,2)≤2n+α−1.

## Particular values

The Bessel polynomials yn(x) up to n=5 are[10]

y0(x)=1y1(x)=x+1y2(x)=3x2+3x+1y3(x)=15x3+15x2+6x+1y4(x)=105x4+105x3+45x2+10x+1y5(x)=945x5+945x4+420x3+105x2+15x+1

No Bessel polynomial can be factored into lower degree polynomials with rational coefficients.[11] The reverse Bessel polynomials are obtained by reversing the coefficients. Equivalently, θk(x)=xkyk(1/x). This results in the following:

θ0(x)=1θ1(x)=x+1θ2(x)=x2+3x+3θ3(x)=x3+6x2+15x+15θ4(x)=x4+10x3+45x2+105x+105θ5(x)=x5+15x4+105x3+420x2+945x+945

## See also

* Bessel function
* Neumann polynomial
* Lommel polynomial
* Hankel transform
* Fourier–Bessel series

## References

1. ↑ 1.0 1.1 1.2 1.3 1.4 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.
2. ↑ 2.0 2.1 2.2 2.3 2.4 Grosswald, E. (1978). Bessel Polynomials (Lecture Notes in Mathematics). New York: Springer. ISBN 978-0-387-09104-4.
3. ↑ Berg, Christian; Vignat, Christophe (2008). "Linearization coefficients of Bessel polynomials and properties of Student-t distributions". Constructive Approximation 27: 15–32. doi:10.1007/s00365-006-0643-6. http://www.math.ku.dk/~berg/manus/bessel.pdf. Retrieved 2006-08-16.
4. ↑ Wolfram Alpha example
5. ↑ Dita, Petre; Grama, Nicolae (May 14, 1997). "On Adomian's Decomposition Method for Solving Differential Equations". arXiv:solv-int/9705008.
6. ↑ Evans, W. D.; Everitt, W. N.; Kwon, K. H.; Littlejohn, L. L. (1993-01-02). "Real orthogonalizing weights for Bessel polynomials". J. Comput. Appl. Math. 49 (1–3): 51–57. doi:10.1016/0377-0427(93)90134-W. ISSN 0377-0427. https://doi.org/10.1016/0377-0427(93)90134-W.
7. ↑ Wolfram, D.A. (2024). "Inverse connection formulae for generalised Bessel polynomials". Bulletin of the Australian Mathematical Society 110 (3): 480-487. doi:10.1017/S0004972724000285.
8. ↑ 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.
9. ↑ 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. doi:10.1016/1385-7258(81)90013-5.
10. ↑ *Sloane, N. J. A., ed. "Sequence A001498 (Triangle a(n,k) (n >= 0, 0 <= k <= n) of coefficients of Bessel polynomials y_n(x) (exponents in increasing order).)". OEIS Foundation. https://oeis.org/A001498.
11. ↑ Filaseta, Michael; Trifinov, Ognian (August 2, 2002). "The Irreducibility of the Bessel Polynomials". Journal für die Reine und Angewandte Mathematik 2002 (550): 125–140. doi:10.1515/crll.2002.069.

* Carlitz, Leonard (1957). "A Note on the Bessel Polynomials". Duke Math. J. 24 (2): 151–162. doi:10.1215/S0012-7094-57-02421-3.
* Fakhri, H.; Chenaghlou, A. (2006). "Ladder operators and recursion relations for the associated Bessel polynomials". Physics Letters A 358 (5-6): 345-353. doi:10.1016/j.physleta.2006.05.070. Bibcode: 2006PhLA..358..345F.
* Roman, S. (1984). The Umbral Calculus (The Bessel Polynomials §4.1.7). New York: Academic Press. ISBN 978-0-486-44139-9.
* Grosswald, Emil (1978). Bessel Polynomials. Lecture Notes in Mathematics. 698. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/bfb0063135. ISBN 978-3-540-09104-2. http://link.springer.com/10.1007/BFb0063135.
* "DLMF: §18.34 Bessel Polynomials ‣ Other Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials". https://dlmf.nist.gov/18.34.

## External links

* Hazewinkel, Michiel, ed. (2001), "Bessel polynomials", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/b110410
* Weisstein, Eric W.. "Bessel Polynomial". http://mathworld.wolfram.com/BesselPolynomial.html.

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