<s>[INST] Integral of modified Bessel function of second kind first order multiply by incomplete gamma function? Is there any possible solution or approximation for that given integral? $$\int_0^\infty {\big(v^{\frac{m}{2}-\frac{1}{4}}\big)}K_1\Bigg[\frac{2\sqrt[4]{v}}{l}\Bigg]\Gamma\left[m,-\frac{a+b v}{c}\right]\text{d}v.$$
 [/INST] I suppose that the approach could be the same as for your previous post.
If $a=0$, we have
$$\frac{1}{4 \pi }\left(-\frac{b}{c}\right)^{-\frac{2m+3}{4}}\,\,G_{2,5}^{4,2}\left(-\frac{c}{16\, b\, l^4}|
\begin{array}{c}
 \frac{1-6 m}{4} ,\frac{1-2m}{4}  \\
 -\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{3}{4},-\frac{2m+3}{4} )
\end{array}
\right)$$
Now, write
$$\Gamma\left[m,-\frac{a+b v}{c}\right]=\sum_{p=0}^\infty d_n\, a^n$$ with
$$d_0=\Gamma \left(m,-\frac{b v}{c}\right)\qquad d_1=\frac 1c\left(-\frac{b}{c}\right)^{m-1} e^{\frac{b }{c}v}$$
$$d_{n+2}=\frac{(n+1)  (b v+c (m-n-1))\,d_{n+1}+n\, d_n}{b c (n+1) (n+2) v}$$
The problem is that I am unable to compute even the first integral.
</s>