The associated Bessel functions and recurrence formulas

The associated Bessel function and its recurrence formulas are shown in this article. The associated Bessel functions are expressed by the first kind of Hankel and spherical Hankel functions. It is shown that associated Bessel functions (both real and imaginary part) can be expressed as a finite combination of exponential and trigonometric functions. Moreover, cylindrical Bessel functions can be defined by associated spherical Bessel functions as given in this article. Higher orders of functions can be obtained from recurrence formulas.