Generating functions of the type $\sum_0^\infty {L_n^{(\alpha + \beta x)} (x)z^n } $, $P_n^{(\alpha + \gamma n,\beta + \delta n)} (x)z^n $ have been obtained recently. In the present paper a general result of this kind is derived making use of the Lagrange expansion. A number of applications are given. These include, in addition to the Laguerre and Jacobi polynomials, the Hermite, Legendre, Bernoulli, Euler and Bell polynomials.