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Bessel-Type Functions > BesselJ[nu,z] > Introduction to the Bessel functions

Connections within the group of Bessel functions and with other function groups

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:

Each of the Bessel functions can be represented through other Bessel functions: