BesselPolynomials

The Bessel polynomials are an orthogonal sequence of polynomials defined by

$$y_n(x) = \sum_{k=0}^n \frac{(n+k)!}{(n-k)!k!}\left(\frac{x}{2}\right)^k$$

The reverse Bessel polynomials[^2^][^8^][^3^][^15^] are similarly defined by:

$$\theta_n(x) = x^n y_n(1/x) = \sum_{k=0}^n \frac{(n+k)!}{(n-k)!k!}\frac{x^{n-k}}{2^k}$$

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

$$y_3(x) = 15x^3 + 15x^2 + 6x + 1$$

while the third-degree reverse Bessel polynomial is

$$\theta_3(x) = x^3 + 6x^2 + 15x + 15$$

Properties

Definition in terms of Bessel functions

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

$$y_n(x) = x^n\theta_n(1/x)$$ $$y_n(x) = \sqrt{\frac{2}{\pi x}} e^{1/x} K_{n+\frac{1}{2}}(1/x)$$ $$\theta_n(x) = \sqrt{\frac{2}{\pi}} x^{n+1/2} e^x K_{n+ \frac{1}{2}}(x)$$

where $K_n(x)$ is a modified Bessel function of the second kind, $y_n(x)$ is the ordinary polynomial, and $\theta_n(x)$ is the reverse polynomial[^2^][^7^][^34^]. For example[^4^]:

$$y_3(x) = 15x^3 + 15x^2 + 6x + 1 = \sqrt{\frac{2}{\pi x}} e^{1/x} K_{3+\frac{1}{2}}(1/x)$$

Definition as a hypergeometric function

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

$$y_n(x) = {}_2F_0(-n,n+1;;-x/2) = \left(\frac{2}{x}\right)^{-n} U\left(-n,-2n,\frac{2}{x}\right) = \left(\frac{2}{x}\right)^{n+1} U\left(n+1,2n+2,\frac{2}{x}\right)$$

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

$$y_n(x;a,b) = {}_2F_0(-n,n+a-1;;-x/b) = \left(\frac{b}{x}\right)^{n+a-1} U\left(n+a-1,2n+a,\frac{b}{x}\right)$$

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

$$\theta_n(x) = \frac{n!}{(-2)^n} L_n^{-2n-1}(2x)$$

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

$$\theta_n(x) = \frac{(-2n)_n}{(-2)^n} {}_1F_1(-n;-2n;2x)$$

where $(-2n)_n$ is the Pochhammer symbol (rising factorial).

Generating function

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

$$\sum_{n=0}^\infty \sqrt{\frac{2}{\pi}} x^{n+\frac{1}{2}} e^x K_{n-\frac{1}{2}}(x) \frac{t^n}{n!} = 1 + x\sum_{n=1}^\infty \theta_{n-1}(x) \frac{t^n}{n!} = e^{x(1-\sqrt{1-2t})}$$

Differentiating with respect to $t$, cancelling $x$, yields the generating function for the polynomials ${\theta_n}_{n\ge0}$

$$\sum_{n=0}^\infty \theta_{n}(x) \frac{t^n}{n!} = \frac{1}{\sqrt{1-2t}} e^{x(1-\sqrt{1-2t})}$$

Similar generating function exists for the $y_n$ polynomials as well[^1^][^106^]:

$$\sum_{n=0}^\infty y_{n-1}(x) \frac{t^n}{n!} = \exp\left(\frac{1-\sqrt{1-2xt}}{x}\right)$$

Upon setting $t = z - xz^2/2$, one has the following representation for the exponential function[^1^][^107^]:

$$e^z = \sum_{n=0}^\infty y_{n-1}(x) \frac{(z-xz^2/2)^n}{n!}$$

Recursion

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

$$y_0(x) = 1$$ $$y_1(x) = x + 1$$ $$y_n(x) = (2n-1)x y_{n-1}(x) + y_{n-2}(x)$$

and

$$\theta_0(x) = 1$$ $$\theta_1(x) = x + 1$$ $$\theta_n(x) = (2n-1) \theta_{n-1}(x) + x^2 \theta_{n-2}(x)$$

Differential equation

The Bessel polynomial obeys the following differential equation:

$$x^2 \frac{d^2 y_n(x)}{dx^2} + 2(x+1) \frac{dy_n(x)}{dx} - n(n+1) y_n(x) = 0$$

and

$$x \frac{d^2 \theta_n(x)}{dx^2} - 2(x+n) \frac{d \theta_n(x)}{dx} + 2n \theta_n(x) = 0$$

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 $n \neq m$,

$$\int_0^{2\pi} y_n\left(e^{i\theta}\right) y_m\left(e^{i\theta}\right) ie^{i\theta} \mathrm{d}\theta = 0$$

Generalization

Explicit Form

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

$$y_n(x;\alpha,\beta) = (-1)^n n! \left(\frac{x}{\beta}\right)^n L_n^{(-1-2n-\alpha)}\left(\frac{\beta}{x}\right)$$

the corresponding reverse polynomials are

$$\theta_n(x;\alpha, \beta) = \frac{n!}{(-\beta)^n} L_n^{(-1-2n-\alpha)}(\beta x) = x^n y_n\left(\frac{1}{x};\alpha,\beta\right)$$

The explicit coefficients of the $y_n(x;\alpha, \beta)$ polynomials are[^1^][^108^]:

$$y_n(x;\alpha, \beta) = \sum_{k=0}^n \binom{n}{k} (n+k+\alpha-2)^{\underline{k}}\left(\frac{x}{\beta}\right)^k$$

Consequently, the $\theta_n(x;\alpha, \beta)$ polynomials can explicitly be written as follows:

$$\theta_n(x;\alpha, \beta) = \sum_{k=0}^n \binom{n}{k} (2n-k+\alpha-2)^{\underline{n-k}} \frac{x^k}{\beta^{n-k}}$$

For the weighting function

$$\rho(x;\alpha,\beta) = {}_1F_1(1,\alpha-1,-\frac{\beta}{x})$$

they are orthogonal, for the relation

$$0 = \oint_c \rho(x;\alpha,\beta) y_n(x;\alpha,\beta) y_m(x;\alpha,\beta) \mathrm{d} x$$

holds for $m \neq n$ and $c$ a curve surrounding the 0 point.

They specialize to the Bessel polynomials for $\alpha = \beta = 2$, in which situation $\rho(x) = \exp(-2/x)$.

Rodrigues formula for Bessel polynomials

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

$$B_n^{(\alpha,\beta)}(x) = \frac{a_n^{(\alpha,\beta)}}{x^{\alpha} e^{-\frac{\beta}{x}}} \left(\frac{d}{dx}\right)^n (x^{\alpha+2n} e^{-\frac{\beta}{x}})$$

where $a_{n}^{(\alpha,\beta)}$ are normalization coefficients.

Associated Bessel polynomials

According to this generalization we have the following generalized differential equation for associated Bessel polynomials:

$$x^2 \frac{d^2 B_{n,m}^{(\alpha,\beta)}(x)}{dx^2} + [(\alpha+2)x+\beta] \frac{dB_{n,m}^{(\alpha,\beta)}(x)}{dx} - \left[ n(\alpha+n+1) + \frac{m\beta}{x} \right] B_{n,m}^{(\alpha,\beta)}(x) = 0$$

where $0 \leq m \leq n$. The solutions are,

$$B_{n,m}^{(\alpha,\beta)}(x) = \frac{a_{n,m}^{(\alpha,\beta)}}{x^{\alpha+m} e^{-\frac{\beta}{x}}} \left(\frac{d}{dx}\right)^{n-m} (x^{\alpha+2n} e^{-\frac{\beta}{x}})$$

Zeros

If one denotes the zeros of $y_n(x;\alpha,\beta)$ as $\alpha_k^{(n)}(\alpha,\beta)$, and that of the $\theta_n(x;\alpha,\beta)$ by $\beta_k^{(n)}(\alpha,\beta)$, then the following estimates exist[^2^][^82^]:

$$\frac{2}{n(n+\alpha-1)} \leq \alpha_k^{(n)}(\alpha,2) \leq \frac{2}{n+\alpha-1}$$ $$\frac{n+\alpha-1}{2} \leq \beta_k^{(n)}(\alpha,2) \leq \frac{n(n+\alpha-1)}{2}$$

for all $\alpha \geq 2$. 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^][^6^]. One result is the following[^7^]:

$$\frac{2}{2n+\alpha-\frac{2}{3}} \leq \alpha_k^{(n)}(\alpha,2) \leq \frac{2}{n+\alpha-1}$$

Particular values

The Bessel polynomials $y_n(x)$ up to $n=5$ are[^8^]:

$$\begin{align} y_0(x) & = 1 \\\ y_1(x) & = x + 1 \\\ y_2(x) & = 3x^2 + 3x + 1 \\\ y_3(x) & = 15x^3 + 15x^2 + 6x + 1 \\\ y_4(x) & = 105x^4 + 105x^3 + 45x^2 + 10x + 1 \\\ y_5(x) & = 945x^5 + 945x^4 + 420x^3 + 105x^2 + 15x + 1 \end{align}$$

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, $\theta_k(x) = x^k y_k(1/x)$. This results in the following:

$$\begin{align} \theta_0(x) & = 1 \\\ \theta_1(x) & = x + 1 \\\ \theta_2(x) & = x^2 + 3x + 3 \\\ \theta_3(x) & = x^3 + 6x^2 + 15x + 15 \\\ \theta_4(x) & = x^4 + 10x^3 + 45x^2 + 105x + 105 \\\ \theta_5(x) & = x^5 + 15x^4 + 105x^3 + 420x^2 + 945x + 945 \\\ \end{align}$$

The Open Question of an Orthogonality Measure on the Real Line

Determining the measure of orthogonality for a given sequence of orthogonal polynomials, especially when relating to Bessel polynomials, involves a complex interplay of mathematical principles and specific properties of the polynomial sequence in question. Here's a breakdown based on the information gathered:

1. Orthogonality and Measures:

   • When a system of polynomials satisfies certain conditions, it is orthogonal with respect to some positive measure on $\mathbb{R}$ (the real line). However, this measure is not necessarily unique nor does it necessarily have a simple form like $w(x) , dx$ [NIST].

2. Bessel Polynomials:

   • Bessel polynomials are known to form an orthogonal sequence of polynomials, with various definitions based on different contexts [Wikipedia]. They are part of the classical orthogonal polynomial systems alongside Jacobi, Hermite, and Laguerre polynomials, among others, which are eigenfunctions of a second-order ordinary differential operator [Springer].

3. Real Orthogonalizing Weights:

   • For Bessel polynomials, finding a real-valued function $A(x)$ of bounded variation such that a certain orthogonality condition is satisfied is a recognized problem. The moments of the orthogonal polynomial sequence (OPS) play a crucial role in this aspect [ScienceDirect].

4. Challenges with Bessel Polynomials:

   • The orthogonality of Bessel polynomials on the real line, especially with continuously differentiable weight functions, seems to present a notable challenge. This is evident from the open problem mentioned regarding the orthogonality of Bessel polynomials on the real line, as opposed to the unit circle, where the orthogonality is well-established.

Given your description of a sequence of orthogonal polynomials related to Bessel polynomials and a weight function $WX = 1$, it's a nuanced endeavor to determine the precise measure of orthogonality. While Favard's theorem provides a framework for understanding the existence of such measures, the specific construction or determination of the measure might require a deeper investigation into the properties of the polynomial sequence in question, especially if they deviate from the standard form of Bessel polynomials.

The quest for a continuously differentiable weight function for Bessel polynomials on the real line, as opposed to the known generalized weight functions involving infinite sums of Dirac delta distributions, seems to be a complex open problem. The constructive descendants of Favard's theorem might not provide a straightforward solution due to the unique characteristics and challenges posed by Bessel polynomials in this context.

For a more precise and detailed analysis, consulting specialized literature or experts in the field of orthogonal polynomials and their orthogonality measures, especially in relation to Bessel polynomials, might be necessary.

[^1^]: 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^]: Grosswald, E. (1978). Bessel Polynomials (Lecture Notes in Mathematics). Springer: New York. ISBN: 978-0-387-09104-4

[^3^]: Berg, C. & Vignat, C. (2008). Linearization coefficients of Bessel polynomials and properties of Student-t distributions. Constructive Approximation, 27, 15–32. doi:10.1007/s00365-006-0643-6

[^4^]: Dita, P. & Grama, N. (May 14, 1997). On Adomian's Decomposition Method for Solving Differential Equations. arXiv:solv-int/9705008

[^5^]: Wolfram Alpha example

[^6^]: 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

[^7^]: 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.

[^8^]: OEIS Sequence A001498: Triangle a(n,k) (n >= 0, 0 <= k <= n) of coefficients of Bessel polynomials y_n(x) (exponents in increasing order)

[^9^]: Filaseta, M. & Trifinov, O. (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