Bessel polynomials · Lablypedia

y_{n} (x) = \sum_{k = 0}^{n} \frac{(n + k)!}{(n - k)! k!} {(\frac{x}{2})}^{k} .

Another definition, favored by electrical engineers, is sometimes known as the reverse Bessel polynomials

θ_{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) = 1 + 6 x + 15 x^{2} + 15 x^{3}

while the third-degree reverse Bessel polynomial is

θ_{3} (x) = x^{3} + 6 x^{2} + 15 x + 15.

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

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} θ_{n} (1 / x)

y_{n} (x) = \sqrt{\frac{2}{π x}} e^{1 / x} K_{n + \frac{1}{2}} (1 / x)

θ_{n} (x) = \sqrt{\frac{2}{π}} 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) = 15 x^{3} + 15 x^{2} + 6 x + 1 = \sqrt{\frac{2}{π 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

y_{n} (x) =_{2} F_{0} (- n, n + 1;; - x / 2) = {(\frac{2}{x})}^{- n} U (- n, - 2 n, \frac{2}{x}) = {(\frac{2}{x})}^{n + 1} U (n + 1, 2 n + 2, \frac{2}{x}) .

A similar expression holds true for the generalized Bessel polynomials (see below):

y_{n} (x; a, b) =_{2} F_{0} (- n, n + a - 1;; - x / b) = {(\frac{b}{x})}^{n + a - 1} U (n + a - 1, 2 n + a, \frac{b}{x}) .

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

θ_{n} (x) = \frac{n!}{(- 2)^{n}} L_{n}^{- 2 n - 1} (2 x)

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

θ_{n} (x) = \frac{(- 2 n)_{n}}{(- 2)^{n}}_{1} F_{1} (- n; - 2 n; 2 x)

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}{π}} x^{n + \frac{1}{2}} e^{x} K_{n - \frac{1}{2}} (x) \frac{t^{n}}{n!} = 1 + x \sum_{n = 1}^{\infty} θ_{n - 1} (x) \frac{t^{n}}{n!} = e^{x (1 - \sqrt{1 - 2 t})} .

Differentiating with respect to $t$ , cancelling $x$ , yields the generating function for the polynomials ${θ_{n}}_{n \geq 0}$

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

Similar generating function exists for the $y_{n}$ polynomials as well:

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

Upon setting $t = z - x z^{2} / 2$ , one has the following representation for the exponential function:

e^{z} = \sum_{n = 0}^{\infty} y_{n - 1} (x) \frac{(z - x z^{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) = (2 n - 1) x y_{n - 1} (x) + y_{n - 2} (x)

and

θ_{0} (x) = 1

θ_{1} (x) = x + 1

θ_{n} (x) = (2 n - 1) θ_{n - 1} (x) + x^{2} θ_{n - 2} (x)

Differential equation

The Bessel polynomial obeys the following differential equation:

x^{2} \frac{d^{2} y_{n} (x)}{d x^{2}} + 2 (x + 1) \frac{d y_{n} (x)}{d x} - n (n + 1) y_{n} (x) = 0

and

x \frac{d^{2} θ_{n} (x)}{d x^{2}} - 2 (x + n) \frac{d θ_{n} (x)}{d x} + 2 n θ_{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 $n \neq m$ ,

$\int_{0}^{2 π} y_{n} (e^{i θ}) y_{m} (e^{i θ}) i e^{i θ} d θ = 0$

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

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Generalization

Explicit form

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

y_{n} (x; α, β) := (- 1)^{n} n! {(\frac{x}{β})}^{n} L_{n}^{(1 - 2 n - α)} (\frac{β}{x}),

the corresponding reverse polynomials are

θ_{n} (x; α, β) := \frac{n!}{(- β)^{n}} L_{n}^{(1 - 2 n - α)} (β x) = x^{n} y_{n} (\frac{1}{x}; α, β) .

The explicit coefficients of the $y_{n} (x; α, β)$ polynomials are:

y_{n} (x; α, β) = \sum_{k = 0}^{n} (\binom{n}{k}) (n + k + α - 2)^{\underline{k}} {(\frac{x}{β})}^{k} .

Consequently, the $θ_{n} (x; α, β)$ polynomials can explicitly be written as follows:

θ_{n} (x; α, β) = \sum_{k = 0}^{n} (\binom{n}{k}) (2 n - k + α - 2)^{\underline{n - k}} \frac{x^{k}}{β^{n - k}} .

For the weighting function

ρ (x; α, β) :=_{1} F_{1} (1, α - 1, - \frac{β}{x})

they are orthogonal, for the relation

0 = \oint_{c} ρ (x; α, β) y_{n} (x; α, β) y_{m} (x; α, β) 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.

x^{n} = \sum_{k = 0}^{n} α (n, k, α, β) y_{n - k} (x; α, β)

where $α (n, k, α, β) = (\binom{n}{k}) \frac{(- 1)^{k} β^{n} (2 (n - k) + α - 1)}{(n - k + α - 1)_{n + 1}}$ and $n \geq 0$ .

Similarly, for the reverse generalized Bessel polynomials

x^{n} = \sum_{k = 0}^{n} α_{1} (n, k, α, β) θ_{n - k} (x; α, β)

where $α_{1} (n, k, α, β) = (- 1)^{k} \frac{(\binom{n}{k})}{β^{k}} (n + α - 1) (n + α - 2 k)_{k - 1}$ 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}^{(α, β)} (x) = \frac{a_{n}^{(α, β)}}{x^{α} e^{- \frac{β}{x}}} {(\frac{d}{d x})}^{n} (x^{α + 2 n} e^{- \frac{β}{x}})

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:

x^{2} \frac{d^{2} B_{n, m}^{(α, β)} (x)}{d x^{2}} + [(α + 2) x + β] \frac{d B_{n, m}^{(α, β)} (x)}{d x} - [n (α + n + 1) + \frac{m β}{x}] B_{n, m}^{(α, β)} (x) = 0

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

B_{n, m}^{(α, β)} (x) = \frac{a_{n, m}^{(α, β)}}{x^{α + m} e^{- \frac{β}{x}}} {(\frac{d}{d x})}^{n - m} (x^{α + 2 n} e^{- \frac{β}{x}})

Zeros

If one denotes the zeros of $y_{n} (x; α, β)$ as $α_{k}^{(n)} (α, β)$ , and that of the $θ_{n} (x; α, β)$ by $β_{k}^{(n)} (α, β)$ , then the following estimates exist:

\frac{2}{n (n + α - 1)} \leq α_{k}^{(n)} (α, 2) \leq \frac{2}{n + α - 1},

and

\frac{n + α - 1}{2} \leq β_{k}^{(n)} (α, 2) \leq \frac{n (n + α - 1)}{2},

for all $α \geq 2$ . 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). One result is the following:

\frac{2}{2 n + α - \frac{2}{3}} \leq α_{k}^{(n)} (α, 2) \leq \frac{2}{n + α - 1} .

Particular values

The Bessel polynomials $y_{n} (x)$ up to $n = 5$ are

\begin{matrix} y_{0} (x) & = 1 \\ y_{1} (x) & = x + 1 \\ y_{2} (x) & = 3 x^{2} + 3 x + 1 \\ y_{3} (x) & = 15 x^{3} + 15 x^{2} + 6 x + 1 \\ y_{4} (x) & = 105 x^{4} + 105 x^{3} + 45 x^{2} + 10 x + 1 \\ y_{5} (x) & = 945 x^{5} + 945 x^{4} + 420 x^{3} + 105 x^{2} + 15 x + 1 \end{matrix}

No Bessel polynomial can be factored into lower degree polynomials with rational coefficients. The reverse Bessel polynomials are obtained by reversing the coefficients. Equivalently, $θ_{k} (x) = x^{k} y_{k} (1 / x)$ . This results in the following:

\begin{matrix} θ_{0} (x) & = 1 \\ θ_{1} (x) & = x + 1 \\ θ_{2} (x) & = x^{2} + 3 x + 3 \\ θ_{3} (x) & = x^{3} + 6 x^{2} + 15 x + 15 \\ θ_{4} (x) & = x^{4} + 10 x^{3} + 45 x^{2} + 105 x + 105 \\ θ_{5} (x) & = x^{5} + 15 x^{4} + 105 x^{3} + 420 x^{2} + 945 x + 945 \end{matrix}