# Bessel function K convergence to Laguerre polynomials

1. Aug 29, 2013

### kd6ac

I am interested to know whether the modified Bessel functions of the second kind, also known as BesselK, defined as
$$K_\alpha(x) = \frac{\pi}{2} \frac{I_{-\alpha} (x) - I_\alpha (x)}{\sin (\alpha \pi)}$$
where
$$I_\alpha(x) = i^{-\alpha} J_\alpha(ix) =\sum_{m=0}^\infty \frac{1}{m! \Gamma(m+\alpha+1)}\left(\frac{x}{2}\right)^{2m+\alpha}$$
and
$$\frac{J_\alpha(x)}{\left( \frac{x}{2}\right)^\alpha}= \frac{e^{-t}}{\Gamma(\alpha+1)} \sum_{k=0} \frac{L_k^{(\alpha)}\left( \frac{x^2}{4 t}\right)}{{k+ \alpha \choose k}} \frac{t^k}{k!}$$
can be shown to converge to the Laguerre polynomials written on the above equation with the symbol
$$L$$
for
$$\alpha \rightarrow \infty \;.$$
Note that the above equations make use of the GAMMA function
$$\Gamma(n) = (n-1)!$$