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:

$K_{v}(x,y)=\int _{1}^{\infty }{\frac {e^{-xt-{\frac {y}{t}}}}{t^{v+1}}}~dt$

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

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

Properties

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

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

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

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

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

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:

$\mathbf {A} _{\nu }(z)={\frac {1}{\pi }}K_{\nu }\left({\frac {z}{2}},-{\frac {z}{2}}\right)$

recurrence relations

$K_{v}(x,y)$ satisfy this recurrence relation:

$xK_{v-1}(x,y)+vK_{v}(x,y)-yK_{v+1}(x,y)=e^{-x-y}$