On Lagrangian distributions of the second kind

Janardan and Rao (SIAM J. Applied Math. 1983, 43, 302-313) have used the second Lagrange expansion, with f(z) and g(z) as two probability generating functions (pgfs) defined on nonnegative integers where g(0) ≠ 0, to define the class of discrete Lagrangian probability distributions of the second kind. They have also studied a number of properties of Lagrangian distributions of the second kind. Different families are generated by various choices of the pgfs f(z) and g(z). In this paper, the class of Lagrange distributions of the second kind is considerably widened to provide many more families. The convolution theorem has been modified and the central moments and cumulants have been obtained.