In this paper the authors first
prove a theorem on bilateral generating relations for a certain sequence of functions.
It is then shown how the main result can be applied to derive a large variety of
bilateral generating functions for the Bessel, Jacobi, Hermite, Laguerre and
ultraspherical polynomials, as well as for their various generalizations. Some recent
results given by W. A. Al-Salam [1], S. K. Chatterjea [2], M. K. Das [3], S. Saran [6]
and the present authors [9] are thus observed to follow fairly easily as special cases of
the theorem proved in this paper.