Bessel polynomial
The Bessel polynomials are orthogonal polynomials defined by $$y_n(x) = \displaystyle\sum_{k=0}^n \dfrac{(n+k)!}{(n-k)!k!} \left( \dfrac{x}{2} \right)^k.$$
Contents
Properties
Theorem: The following formula holds: $$y_n(x)={}_2F_0 \left( -n, 1+n;-; -\dfrac{1}{2}x \right),$$ where $y_n(x)$ denotes a Bessel polynomial and ${}_2F_0$ denotes the hypergeometric pFq.
Proof: █
Theorem
The following formula holds: $$y_n\left( \dfrac{1}{ir} \right) = \left(\dfrac{\pi r}{2} \right)^{\frac{1}{2}} e^{ir} \left[ \dfrac{J_{n +\frac{1}{2}}(r)}{i^{n+1}}+i^nJ_{-n-\frac{1}{2}}(r) \right],$$ where $y_n$ denotes a Bessel polynomial and $J_{\nu}$ denotes the Bessel J.
Proof
References
Theorem
The following formula holds: $$J_{n +\frac{1}{2}}(r) = (2\pi r)^{-\frac{1}{2}} \left[\dfrac{e^{ir}}{i^{n+1}} y_n \left( -\dfrac{1}{ir} \right) + i^{n+1}e^{-ir}y_n\left( \dfrac{1}{ir} \right) \right],$$ where $J_{n+\frac{1}{2}}$ denotes a Bessel J, $\pi$ denotes pi, $i$ denotes the imaginary number, $e^{ir}$ denotes the exponential, and $y_n$ denotes a Bessel polynomial.
Proof
References
Theorem
The following formula holds: $$J_{-n-\frac{1}{2}}(r) = (2 \pi r)^{-\frac{1}{2}} \left[ i^n e^{ir} y_n \left( -\dfrac{1}{ir} \right)+ \dfrac{e^{-ir}}{i^n} y_n\left( \dfrac{1}{ir} \right) \right],$$ where $J_{-n-\frac{1}{2}}$ denotes a Bessel function of the first kind and $y_n$ denotes a Bessel polynomial.
