Some properties of special functions derived from the theory of continuous transformation groups

Abstract:

The theory of continuous transformation groups is utilized in the study of some properties of special functions. On constructing the continuous transformation groups corresponding to a suitably defined infinitesimal transformation, a bilateral generating relation involving Laguerre polynomials $\{ L_n^{(\alpha )}(x)\}$ is obtained in $\S 2$. It is shown to be a generalisation of Brafman’s result. In the last section raising and lowering operators for $\{ P_n^{(\alpha ,\beta - n)}(x)\}$ and their commutator are introduced and on showing that they generate a 3-dimensional Lie algebra, the idea of c.t. groups is employed to establish a generating relation involving $\{ P_n^{(\alpha ,\beta - n)}(x)\}$ which is seen to yield a number of known results. Moreover, a bilateral generating relation involving $\{ P_n^{(\alpha ,\beta - n)}(x)\}$ is obtained; this is seen to be a generalisation of a well-known relation due to Weisner.