2009 Number Theory Seminar - 11

Let a, a_0, a_1, ..., a_m be integers with a nonnegative, and define f_a(x) = sum_{j=0}^m a_j x^j / (j+a)!. Schur (in 1929) proved that f_0(x) with |a_0| = |a_n| = 1 is irreducible for all m. Schur's result has been generalized by many authors by using p-adic methods of Coleman and Filaseta. In this talk, I will give a survey of the some of these results and prove some results on the irreducibility of generalised Hermite-Laguerre polynomials by combining p-adic methods with the greatest prime factor of the product of terms of an arithmetic progression.