Research Projects

Generalised Laguerre Polynomials (GLP) denoted by $L_n^{(\alpha)}(x)$ is a family of orthogonal polynomials that have been extensively studied primarily because of the important roles they play in various branches of mathematical analysis and mathematical physics. GLP also enjoys algebraic properties of great interest and in this project we aim to study the irreducibility of GLP over rationals. Till date, irreducibility results are explicitly known only for some classes of rational sequences $(\alpha_n)$. In general for a real number $\alpha,$ it is not known whether $L_n^{(\alpha)}(x)$ is irreducible. For example, the irreducibility of $L_n^{(\alpha)}(x)$ for an arbitrary given rational number $\alpha $ with denominator greater than $4$ is not known. Specifically, we are interested to characterize sequences $(\alpha_n)$ such that $L_n^{(\alpha_n)}(x)$ is irreducible for all $n$. To achieve this we need to tackle some related Diophantine problems. These are problems related to refinements and extensions on Sylvester's theorem on greatest prime factor of product of consecutive terms in an arithmetic progression, improving the range of values of $n$ for which Grimm's conjecture hold, extending the tables of Lehmer, Luca and Najman on greatest prime factor of product of two consecutive integers.