Integral of modified Bessel function of second kind first order multiply by incomplete gamma function?

Question

Is there any possible solution or approximation for that given integral? $$\int_0^\infty {\left(v^{\frac{m}{2}-\frac{1}{4}}\right)}K_1\left[\frac{2\sqrt[4]{v}}{l}\right]\Gamma\left[m,-\frac{a+b v}{c}\right]\,\Bbb dv$$

Answer 1

Partial Answer: N.B this can be seen as a Mellin Transform, for this particular function (when $a=0$), the Mellin transform appears to be expressible in terms of the Meijer-G function.

$$ \int_0^\infty x^{s-1} K_1\left(\frac{2\sqrt[4]{x}}{l}\right)\Gamma\left(m,\frac{bx}{c}\right)\; dx = $$ $$ \frac{\left(-\frac{b}{c}\right)^{-s} G_{2,5}^{4,2}\left(-\frac{c}{16 b l^4}| \begin{array}{c} 1-s,-m-s+1 \\ -\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{3}{4},-s \\ \end{array} \right)}{4 \pi } $$ with $$ s = \frac{m}{2} + \frac{3}{4} $$ but I understand that this solution may be hard to work with.

Answer 2

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.