Is there any formula to represent Laguerre functions with fractional index (in this case only divided by 2) in terms of Bessel functions $I_0(x)$ and $I_1(x)$?
I found this formula in Wolfram documentation:
$$L_{n-\frac{1}{2}}(x)=e^{\frac{x}{2}}\sum_{k=0}^{n}(-2x)^{k}\binom{n}{k}\frac{k!}{(2k)!}\ \sum_{p=0}^{k}\left(-\frac{1}{2}\right)^{p}\binom{k}{p}\sum_{j=0}^{p}\binom{p}{j}I_{p-2j}\left(\frac{x}{2}\right)$$
But going to calculate them explicitly I have this form
$$L_{\frac{1}{2}}(x)=-e^{x/2}\left[(x-1)I_0\left(\frac{x}{2}\right)-x I_1\left(\frac{x}{2}\right)\right]\\
L_{\frac{3}{2}}(x)=\frac{e^{x/2}}{3}\left[(2x^2-6x+3)I_0\left(\frac{x}{2}\right)-x(2x-4)I_1\left(\frac{x}{2}\right)\right]\\
L_{\frac{5}{2}}(x)=-\frac{e^{x/2}}{15}\left[{(4x^3-28x^2+45x-15)I_0\left(\frac{x}{2}\right)\atop-x(4x^2-24x+23) I_1\left(\frac{x}{2}\right)}\right]\\
L_{\frac{7}{2}}(x)=\frac{e^{x/2}}{105}\left[{(8x^4-104x^3+376x^2-420x+105)I_0\left(\frac{x}{2}\right)\atop-4x(2x^3-24x^2+71x-44)I_1\left(\frac{x}{2}\right)}\right]\\
L_{\frac{9}{2}}(x)=-\frac{e^{x/2}}{945}\left[{(16x^5-336x^4+2220x^3-5484x^2+4725x-945) I_0\left(\frac{x}{2}\right)\atop-x(16x^4-320x^3+1908x^2-3720x+1689) I_1\left(\frac{x}{2}\right)}\right]$$
Here appear only in terms of the subscript $0$ and $1$.
So I think this can be represented as $$L_{n-\frac{1}{2}}(x)=e^{x/2}\left[I_0\left(\frac{x}{2}\right)\sum_{i}P_i(x)+I_1\left(\frac{x}{2}\right)x\sum_{j}Q_j(x)\right]$$
This is the definition of Laguerre function
$$L_n(x):=\frac{e^x}{n!}\frac{\mathrm{d}^n}{\mathrm{d}x^n}\left(e^{-x}x^n\right)$$
Do you think it is appropriate to apply the formulas of the fractional calculation?
