Incomplete Bessel K function/generalized incomplete gamma function

Some mathematicians defined this type incomplete-version of Bessel function or this type generalized-version of incomplete gamma function:

$K_v(x,y)=\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-xt-\frac{y}{t}}}{t^{v+1}}dt$

$\gamma (\alpha ,x;b)=\underset{0}{\overset{x}{\int }}{t}^{\alpha -1}{e}^{-t-\frac{b}{t}}dt$

$\mathrm{\Gamma }(\alpha ,x;b)=\underset{x}{\overset{\mathrm{\infty }}{\int }}{t}^{\alpha -1}{e}^{-t-\frac{b}{t}}dt$

Properties

$K_v(x,y)=x^v\mathrm{\Gamma }(-v,x;xy)$

$K_v(x,y)+K_{-v}(y,x)=\frac{2x^{\frac{v}{2}}}{y^{\frac{v}{2}}}{K}_v(2\sqrt{xy})$

$\gamma (\alpha ,x;0)=\gamma (\alpha ,x)$

$\mathrm{\Gamma }(\alpha ,x;0)=\mathrm{\Gamma }(\alpha ,x)$

$\gamma (\alpha ,x;b)+\mathrm{\Gamma }(\alpha ,x;b)=2b^{\frac{\alpha }{2}}{K}_{\alpha }(2\sqrt{b})$

One of the advantage of defining this type incomplete-version of Bessel function $K_v(x,y)$ is that even for example the associated Anger–Weber function defined in Digital Library of Mathematical Functions can related:

${𝐀}_{\nu }(z)=\frac{1}{\pi }\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-\nu t-z\sinh t}dt=\frac{1}{\pi }\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-(\nu +1)t-\frac{z{e}^t}{2}+\frac{z}{2e^t}}d(e^t)=\frac{1}{\pi }\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{e^{-\frac{zt}{2}+\frac{z}{2t}}}{t^{\nu +1}}dt=\frac{1}{\pi }{K}_{\nu }(\frac{z}{2},-\frac{z}{2})$

Recurrence relations

$K_v(x,y)$ satisfy this recurrence relation:

$x{K}_{v-1}(x,y)+v{K}_v(x,y)-y{K}_{v+1}(x,y)=e^{-x-y}$