<s>[INST] Definite integral of the product of modified Bessel function of the first kind, exponential, and a power term Is there a solution to the following integral?
$$
\int_0^{\infty} t^{-0.5}e^{-at}I_{l}\left(kt\right)dt,\;\;\;a,k>0
$$
Here, $I_{l}$ is the modified Bessel function of the first kind, and $a,k$ are constants. I have found solutions of similar integrals without the $t^{-0.5}$ term, and with $l=0$ (e.g. here and here). The closest question and explanation I could find was this. However, the subscript $l$ is important for my question because the entire integral sits inside a summation over integer values of $l\in[-\infty,\infty]$.
Any kind of help/suggestions will be greatly appreciated. Thanks!
 [/INST] $$
\int_0^{\infty}{t^{-\frac{1}{2}}I_l\left( kt \right) e^{-at}dt}
\\
=\sum_{n=0}^{\infty}{\frac{1}{n!\Gamma \left( n+l+1 \right)}\left( \frac{k}{2} \right) ^{2n+l}}\int_0^{\infty}{t^{-\frac{1}{2}+2n+l}e^{-at}dt}
\\
=\sum_{n=0}^{\infty}{\frac{1}{n!\Gamma \left( n+l+1 \right)}\left( \frac{k}{2} \right) ^{2n+l}\frac{\Gamma \left( 2n+l+\frac{1}{2} \right)}{a^{2n+l+\frac{1}{2}}}}
\\
=\frac{1}{\sqrt{2a\pi}}\left( \frac{k}{a} \right) ^l\sum_{n=0}^{\infty}{\frac{\Gamma \left( n+\frac{l}{2}+\frac{1}{4} \right) \Gamma \left( n+\frac{l}{2}+\frac{3}{4} \right)}{n!\Gamma \left( n+l+1 \right)}\left( \frac{k^2}{a^2} \right) ^n}
\\
=\frac{1}{\sqrt{2a\pi}}\left( \frac{k}{a} \right) ^l\frac{\Gamma \left( \frac{l}{2}+\frac{1}{4} \right) \Gamma \left( \frac{l}{2}+\frac{3}{4} \right)}{\Gamma \left( l+1 \right)}\,\,_2F_1\left( \frac{l}{2}+\frac{1}{4},\frac{l}{2}+\frac{3}{4};l+1;\frac{k^2}{a^2} \right) 
\\
Q_{\nu}^{\mu}(z)=\frac{e^{\mu \pi i}}{2^{\nu +1}}\frac{\Gamma (\nu +\mu +1)}{\Gamma \left( \nu +\frac{3}{2} \right)}\frac{\Gamma \left( \frac{1}{2} \right) \left( z^2-1 \right) ^{\frac{\mu}{2}}}{z^{\nu +\mu +1}}F\left( \frac{\nu +\mu}{2}+1,\frac{\nu +\mu +1}{2};\nu +\frac{3}{2};\frac{1}{z^2} \right) 
\\
Q_{l-\frac{1}{2}}(z)=\frac{1}{2^{l+\frac{1}{2}}}\frac{\Gamma (l+\frac{1}{2})}{\Gamma \left( l+1 \right)}\frac{\Gamma \left( \frac{1}{2} \right)}{z^{l+\frac{1}{2}}}F\left( \frac{l}{2}+\frac{3}{4},\frac{l}{2}+\frac{1}{4};l+1;\frac{1}{z^2} \right) 
\\
\int_0^{\infty}{t^{-\frac{1}{2}}I_l\left( kt \right) e^{-at}dt}
\\
=\frac{1}{\sqrt{2a\pi}}\left( \frac{k}{a} \right) ^l\frac{\Gamma \left( \frac{l}{2}+\frac{1}{4} \right) \Gamma \left( \frac{l}{2}+\frac{3}{4} \right)}{\Gamma \left( l+1 \right)}\,\,_2F_1\left( \frac{l}{2}+\frac{1}{4},\frac{l}{2}+\frac{3}{4};l+1;\frac{k^2}{a^2} \right) 
\\
=\frac{2}{\sqrt{2k\pi}}Q_{l-\frac{1}{2}}(\frac{a}{k})
$$
</s>