How to calculate integral involving Bessel function?

Question

$$\int_0^{\infty}y^{1+\frac{m+c}{2}}K_{c-m}(2b\sqrt{y})dy,$$

How to calculate this integral? Note that, $K_{c-m}$ is the modified Bessel function of the second kind.

Answer 1

Let $t = 2b\sqrt{y}$, then $\frac{t}{2b^2} dt = dy$, so $$ \int_0^\infty \left(\frac{t}{2b}\right)^{2+m+c} K_{c-m}(t) \frac{t}{2b^2} dt $$ $$ \frac{(2b)^{-3-m-c}}{b} \int_0^\infty t^{3+m+c} K_{c-m}(t) dt $$ Now applying the integral identity here, $$ \frac{(2b)^{-3-m-c}}{b} 2^{2+m+c}\Gamma(m+2)\Gamma(c+2) $$ $$ \frac{b^{-4-m-c}}{2} \Gamma(m+2)\Gamma(c+2) $$