Incomplete Bessel functions

In mathematics, the incomplete Bessel functions are types of special functions which act as a type of extension from the complete-type of Bessel functions.

Definition
The incomplete Bessel functions are defined as the same delay differential equations of the complete-type Bessel functions:
 * $$J_{v-1}(z,w)-J_{v+1}(z,w)=2\dfrac{\partial}{\partial z}J_v(z,w)$$
 * $$Y_{v-1}(z,w)-Y_{v+1}(z,w)=2\dfrac{\partial}{\partial z}Y_v(z,w)$$
 * $$I_{v-1}(z,w)+I_{v+1}(z,w)=2\dfrac{\partial}{\partial z}I_v(z,w)$$
 * $$K_{v-1}(z,w)+K_{v+1}(z,w)=-2\dfrac{\partial}{\partial z}K_v(z,w)$$
 * $$H_{v-1}^{(1)}(z,w)-H_{v+1}^{(1)}(z,w)=2\dfrac{\partial}{\partial z}H_v^{(1)}(z,w)$$
 * $$H_{v-1}^{(2)}(z,w)-H_{v+1}^{(2)}(z,w)=2\dfrac{\partial}{\partial z}H_v^{(2)}(z,w)$$

And the following suitable extension forms of delay differential equations from that of the complete-type Bessel functions:
 * $$J_{v-1}(z,w)+J_{v+1}(z,w)=\dfrac{2v}{z}J_v(z,w)-\dfrac{2\tanh vw}{z}\dfrac{\partial}{\partial w}J_v(z,w)$$
 * $$Y_{v-1}(z,w)+Y_{v+1}(z,w)=\dfrac{2v}{z}Y_v(z,w)-\dfrac{2\tanh vw}{z}\dfrac{\partial}{\partial w}Y_v(z,w)$$
 * $$I_{v-1}(z,w)-I_{v+1}(z,w)=\dfrac{2v}{z}I_v(z,w)-\dfrac{2\tanh vw}{z}\dfrac{\partial}{\partial w}I_v(z,w)$$
 * $$K_{v-1}(z,w)-K_{v+1}(z,w)=-\dfrac{2v}{z}K_v(z,w)+\dfrac{2\tanh vw}{z}\dfrac{\partial}{\partial w}K_v(z,w)$$
 * $$H_{v-1}^{(1)}(z,w)+H_{v+1}^{(1)}(z,w)=\dfrac{2v}{z}H_v^{(1)}(z,w)-\dfrac{2\tanh vw}{z}\dfrac{\partial}{\partial w}H_v^{(1)}(z,w)$$
 * $$H_{v-1}^{(2)}(z,w)+H_{v+1}^{(2)}(z,w)=\dfrac{2v}{z}H_v^{(2)}(z,w)-\dfrac{2\tanh vw}{z}\dfrac{\partial}{\partial w}H_v^{(2)}(z,w)$$

Where the new parameter $$w$$ defines the integral bound of the upper-incomplete form and lower-incomplete form of the modified Bessel function of the second kind:
 * $$K_v(z,w)=\int_w^\infty e^{-z\cosh t}\cosh vt~dt$$
 * $$J_v(z,w)=\int_0^we^{-z\cosh t}\cosh vt~dt$$

Properties

 * $$J_v(z,w)=J_v(z)+\dfrac{e^\frac{v\pi i}{2}J(iz,v,w)-e^{-\frac{v\pi i}{2}}J(-iz,v,w)}{i\pi}$$
 * $$Y_v(z,w)=Y_v(z)+\dfrac{e^\frac{v\pi i}{2}J(iz,v,w)+e^{-\frac{v\pi i}{2}}J(-iz,v,w)}{\pi}$$
 * $$I_{-v}(z,w)=I_v(z,w)$$ for integer $$v$$
 * $$I_{-v}(z,w)-I_v(z,w)=I_{-v}(z)-I_v(z)-\dfrac{2\sin v\pi}{\pi}J(z,v,w)$$
 * $$I_v(z,w)=I_v(z)+\dfrac{J(-z,v,w)-e^{-v\pi i}J(z,v,w)}{i\pi}$$
 * $$I_v(z,w)=e^{-\frac{v\pi i}{2}}J_v(iz,w)$$
 * $$K_{-v}(z,w)=K_v(z,w)$$
 * $$K_v(z,w)=\dfrac{\pi}{2}\dfrac{I_{-v}(z,w)-I_v(z,w)}{\sin v\pi}$$ for non-integer $$v$$
 * $$H_v^{(1)}(z,w)=J_v(z,w)+iY_v(z,w)$$
 * $$H_v^{(2)}(z,w)=J_v(z,w)-iY_v(z,w)$$
 * $$H_{-v}^{(1)}(z,w)=e^{v\pi i}H_v^{(1)}(z,w)$$
 * $$H_{-v}^{(2)}(z,w)=e^{-v\pi i}H_v^{(2)}(z,w)$$
 * $$H_v^{(1)}(z,w)=\dfrac{J_{-v}(z,w)-e^{-v\pi i}J_v(z,w)}{i\sin v\pi}=\dfrac{Y_{-v}(z,w)-e^{-v\pi i}Y_v(z,w)}{\sin v\pi}$$ for non-integer $$v$$
 * $$H_v^{(2)}(z,w)=\dfrac{e^{v\pi i}J_v(z,w)-J_{-v}(z,w)}{i\sin v\pi}=\dfrac{Y_{-v}(z,w)-e^{v\pi i}Y_v(z,w)}{\sin v\pi}$$ for non-integer $$v$$

Differential equations
$$K_v(z,w)$$ satisfies the inhomogeneous Bessel's differential equation
 * $$z^2\dfrac{d^2y}{dz^2}+z\dfrac{dy}{dz}-(x^2+v^2)y=(v\sinh vw+z\cosh vw\sinh w)e^{-z\cosh w}$$

Both $$J_v(z,w)$$, $$Y_v(z,w)$$ , $$H_v^{(1)}(z,w)$$ and $$H_v^{(2)}(z,w)$$ satisfy the partial differential equation
 * $$z^2\dfrac{\partial^2y}{\partial z^2}+z\dfrac{\partial y}{\partial z}+(z^2-v^2)y-\dfrac{\partial^2y}{\partial w^2}+2v\tanh vw\dfrac{\partial y}{\partial w}=0$$

Both $$I_v(z,w)$$ and $$K_v(z,w)$$ satisfy the partial differential equation
 * $$z^2\dfrac{\partial^2y}{\partial z^2}+z\dfrac{\partial y}{\partial z}-(z^2+v^2)y-\dfrac{\partial^2y}{\partial w^2}+2v\tanh vw\dfrac{\partial y}{\partial w}=0$$

Integral representations
Base on the preliminary definitions above, one would derive directly the following integral forms of $$J_v(z,w)$$, $$Y_v(z,w)$$:
 * $$\begin{align}

J_v(z,w)&=J_v(z)+\dfrac{1}{\pi i}\left(\int_0^we^{\frac{v\pi i}{2}-iz\cosh t}\cosh vt~dt-\int_0^we^{iz\cosh t-\frac{v\pi i}{2}}\cosh vt~dt\right) \\&=J_v(z)+\dfrac{1}{\pi i}\left(\int_0^w\cos\left(z\cosh t-\dfrac{v\pi}{2}\right)\cosh vt~dt-i\int_0^w\sin\left(z\cosh t-\dfrac{v\pi}{2}\right)\cosh vt~dt\right.\\ &\quad\quad\quad\quad\quad\quad\left.-\int_0^w\cos\left(z\cosh t-\dfrac{v\pi}{2}\right)\cosh vt~dt-i\int_0^w\sin\left(z\cosh t-\dfrac{v\pi}{2}\right)\cosh vt~dt\right) \\&=J_v(z)+\dfrac{1}{\pi i}\left(-2i\int_0^w\sin\left(z\cosh t-\dfrac{v\pi}{2}\right)\cosh vt~dt\right) \\&=J_v(z)-\dfrac{2}{\pi}\int_0^w\sin\left(z\cosh t-\dfrac{v\pi}{2}\right)\cosh vt~dt\end{align}$$
 * $$\begin{align}

Y_v(z,w)&=Y_v(z)+\dfrac{1}{\pi}\left(\int_0^we^{\frac{v\pi i}{2}-iz\cosh t}\cosh vt~dt+\int_0^we^{iz\cosh t-\frac{v\pi i}{2}}\cosh vt~dt\right) \\&=Y_v(z)+\dfrac{1}{\pi}\left(\int_0^w\cos\left(z\cosh t-\dfrac{v\pi}{2}\right)\cosh vt~dt-i\int_0^w\sin\left(z\cosh t-\dfrac{v\pi}{2}\right)\cosh vt~dt\right.\\ &\quad\quad\quad\quad\quad\quad\left.+\int_0^w\cos\left(z\cosh t-\dfrac{v\pi}{2}\right)\cosh vt~dt+i\int_0^w\sin\left(z\cosh t-\dfrac{v\pi}{2}\right)\cosh vt~dt\right) \\&=Y_v(z)+\dfrac{2}{\pi}\int_0^w\cos\left(z\cosh t-\dfrac{v\pi}{2}\right)\cosh vt~dt\end{align}$$ With the Mehler–Sonine integral expressions of $$J_v(z)=\dfrac{2}{\pi}\int_0^\infty\sin\left(z\cosh t-\dfrac{v\pi}{2}\right)\cosh vt~dt$$ and $$Y_v(z)=-\dfrac{2}{\pi}\int_0^\infty\cos\left(z\cosh t-\dfrac{v\pi}{2}\right)\cosh vt~dt$$ mentioned in Digital Library of Mathematical Functions,

we can further simplify to $$J_v(z,w)=\dfrac{2}{\pi}\int_w^\infty\sin\left(z\cosh t-\dfrac{v\pi}{2}\right)\cosh vt~dt$$ and $$Y_v(z,w)=-\dfrac{2}{\pi}\int_w^\infty\cos\left(z\cosh t-\dfrac{v\pi}{2}\right)\cosh vt~dt$$, but the issue is not quite good since the convergence range will reduce greatly to $$|v|<1$$.