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 from the upper-incomplete-form of [modified Bessel function of the second kind](https://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions:_I%CE%B1,_K%CE%B1):

External links

• https://www.cambridge.org/core/services/aop-cambridge-core/content/view/9C572E5CE44E9E0DE8630755DF99ABAC/S0013091505000490a.pdf/incomplete-bessel-functions-i.pdf