On power series distributions associated with LAGRANGE expansion

The generalized logarithmic series distribution

“…It is also possible to get (1.1) as a limiting form of the zero truncated generalized negative binomial distribution, see Jain (1975). Famoye (1987) showed in a recent "letter" that the GLSD is unimodal but not strongly unimodal, or equivalently, not log-concave.…”

Generalized distributions of order k associated with success runs in Bernoulli trials

“…For k = 1, Proposition 2.4(i) reduces to the PGF of the generalized negative binomial distribution (see Jain [5]) and (ii) reduces to its mean and variance (see Jain and Consul [6]). …”

“…Table 4.1 shows the expected frequencies of the counts of bacteria in leucocytes estimated by the generalized Poisson distribution of order 2, type I (GP 2,I (·)), using the following moment estimators of the parameters θ and λ: Poisson distribution (GP(·)) (see, e.g., Jain [5]) have been fitted for comparison. We observe that the generalized Poisson distribution of order 2, type I, gives a good fit to the data.…”

“…Jain and Consul [6] introduced and studied the generalized negative binomial distribution (see also Mohanty [13]). The generalized logarithmic series distribution and the Borel-Tanner or generalized Poisson distribution were obtained as limiting cases of the generalized negative binomial distribution by Jain and Consul [6], and Jain and Singh [8], respectively (see also Jain [5], Jain and Gupta [7], and Haight and Breuer [4]). Recently, Sen and Mishra [23] obtained the generalized Polya distribution, which unifies the usual Polya and inverse Polya distributions.…”