0.7 Some special functions and their q-analogues

The classical exponential function and the trigonometric functions and can be expressed in terms of hypergeometric functions as

 

 

and

 

Further we have the well-known Bessel function which can be defined by

 

Applying this formula to the generating function (1.11.11) of the Laguerre polynomials we obtain :

These functions all have several q-analogues. The exponential function for instance has two different natural q-extensions, denoted by and defined by

 

and

 

These q-analogues of the exponential function are related by

They are q-extensions of the exponential function since

If we set a = 0 in the q-binomial theorem we find for the q-exponential functions :

 

Further we have

 

For instance, these formulas can be used to obtain other versions of a generating function for several sets of orthogonal polynomials mentioned in this report.

If we assume that |z| < 1 we may define

 

and

 

These are q-analogues of the trigonometric functions and . On the other hand we may define

 

and

 

Then it is not very difficult to verify that

Further we have

The q-analogues of the trigonometric functions can be used to find different forms of formulas appearing in this report, although we will not use them.

Some q-analogues of the Bessel functions are given by

 

and

 

These q-Bessel functions are connected by

They are q-analogues of the Bessel function since

These q-Bessel functions were introduced by F.H. Jackson in 1905. They are therefore referred to as Jackson q-Bessel functions. Another q-analogue of the Bessel function is the so-called Hahn-Exton q-Bessel function which can be defined by

 

As an example we note that

where denotes the q-Laguerre polynomial defined by (3.21.1). We also have

where denotes the little q-Laguerre (or Wall) polynomial defined by (3.20.1).

Finally we remark that the generating function (3.20.11) for the little q-Laguerre (or Wall) polynomials can also be written as

or as




Last modified on July 15, 1998