IRREDUCIBILITY OF GENERALIZED HERMITE-LAGUERRE POLYNOMIALS

For a rational q = u + alpha/d with u, alpha, d is an element of Z with u >= 0, 1 <= alpha < d, gcd (alpha, d) = 1, the generalized Hermite-Laguerre polynomials G(q) (x) are defined by Gq(x) = a(n)x(n) + a(n-1) (alpha + (n - 1 + u)d)x(n-1) + .... +a(1)(Pi(n-1)(i-1) (alpha + (i + u)d)) x + a(0) (Pi(n-1)(i-0) (alpha+(i + u)d)) where a(0), a(1), . . . , a(n) are arbitrary integers. We prove some irreducibility results of G(q) (x) when q is an element of {1/3, 2/3} and extend some of the earlier irreducibility results when q of the form u + 1/2. We also prove a new improved lower bound for greatest prime factor of product of consecutive terms of an arithmetic progression whose common difference is 2 and 3.