Connections within the group of Bessel functions and with other function groups
Representations through more general functions
The Bessel functions , , , and are particular cases of more general functions: hypergeometric and Meijer G functions.
In particular, the functions and can be represented through the regularized hypergeometric functions (without any restrictions on the parameter ):
Similar formulas, but with restrictions on the parameter , represent and through the classical hypergeometric function :
The functions and can also be represented through the hypergeometric functions by the following formulas:
Similar formulas for other Bessel functions and always include restrictions on the parameter, namely :
In the case of integer , the right‐hand sides of the preceding six formulas evaluate to removable indeterminate expressions of the type , . The limit of the right‐hand sides exists and produces complicated series expansions including logarithmic and polygamma functions. These difficulties can be removed by using the generalized Meijer G function. The generalized Meijer G function allows represention of all four Bessel functions for all values of the parameter by the following simple formulas:
The classical Meijer G function is less convenient because it can lead to additional restrictions:
Representations through other Bessel functions
Each of the Bessel functions can be represented through other Bessel functions:
