Bessel polynomials

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 θ**n(x) is the reverse polynomial . For example:

: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{{cite arXiv

: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): :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: :\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: :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^2y_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. 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

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

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: :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\left(1,\alpha-1,-\frac \beta x\right) 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 ≠ 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. | doi-access = free :x^n = \sum_{k=0}^n \alpha(n, k, \alpha, \beta) \ y_{n-k}(x; \alpha, \beta)

where \alpha(n, k, \alpha, \beta) = {n \choose k} \frac{(-1)^k \beta^n (2(n-k) + \alpha -1)}{(n-k+\alpha -1)_{n+1}} and n \geq 0.

Similarly, for the reverse generalized Bessel polynomials

:x^n = \sum_{k=0}^n \alpha_1(n, k, \alpha, \beta) \ \theta_{n-k}(x; \alpha, \beta)

where \alpha_1(n, k, \alpha, \beta) = (-1)^k \frac{\beta^k} (n + \alpha -1) (n +\alpha - 2k)_{k- 1} and n \geq 0.

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 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^2B_{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:

:\frac{2}{n(n+\alpha-1)}\le\alpha_k^{(n)}(\alpha,2)\le\frac{2}{n+\alpha-1}, and :\frac{n+\alpha-1}{2}\le\beta_k^{(n)}(\alpha,2)\le\frac{n(n+\alpha-1)}{2}, for all \alpha\ge2. 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: :\frac{2}{2n+\alpha-\frac23}\le\alpha_k^{(n)}(\alpha,2)\le\frac{2}{n+\alpha-1}.

Particular values

The Bessel polynomials y_n(x) up to n=5 are : \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. 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} + 3 x + 3 \ \theta_3(x) & = x^{3} + 6 x^{2} + 15 x + 15 \ \theta_4(x) & = x^{4} + 10 x^{3} + 45 x^{2} + 105 x + 105 \ \theta_5(x) & = x^{5} + 15 x^{4} + 105 x^{3} + 420 x^{2} + 945 x + 945 \ \end{align}

References

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References

1. (1948). "A New Class of Orthogonal Polynomials: The Bessel Polynomials". Trans. Amer. Math. Soc..
2. Grosswald, E.. (1978). "Bessel Polynomials (Lecture Notes in Mathematics)". Springer.
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4. (1993-01-02). "Real orthogonalizing weights for Bessel polynomials". J. Comput. Appl. Math..
5. (1976). "Zero-free parabolic regions for sequences of polynomials". SIAM J. Math. Anal..
6. (1981). "On the zeros of generalized Bessel polynomials. I". Indag. Math..
7. *{{cite OEIS
8. (August 2, 2002). "The Irreducibility of the Bessel Polynomials". Journal für die Reine und Angewandte Mathematik.