On a certain family of generalized Laguerre polynomials

Algebraic Properties of a Family of Generalized Laguerre Polynomials

We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers r, n ≥ 0, we conjecture that L n (x) = Pn j=0 `n− j+r n− j ́ x / j! is a Q-irreducible polynomial whose Galois group contains the alternating group on n letters. That this is so for r = n was conjectured in the 1950’s by Grosswald and proven recently by Filaseta and T...

On the Genus of Generalized Laguerre Polynomials

belong to one of the three family of orthogonal polynomials, the other two being Jacobi and Legendre. In addition to their important roles in mathematical analysis, these polynomials also feature prominently in algebra and number theory. Schur ([7], [8]) pioneered the study of Galois properties of specializations of these orthogonal polynomials, and Feit [1] used them to solve the inverse Galoi...

On the Irreducibility of a Certain Class of Laguerre Polynomials

Specializations of Generalized Laguerre Polynomials

Three specializations of a set of orthogonal polynomials with “8 different q’s” are given. The polynomials are identified as q-analogues of Laguerre polynomials, and the combinatorial interpretation of the moments give infinitely many new Mahonian statistics on permutations.

A Note on Artinianess of Certain Generalized

Let ?: R0?R be a ring homomorphism and suppose that a and a0, respectively, are ideals of R and R0 such that is an Artinian ring. Let M and N be two finitely generated R-modules and suppose that (R0,m0) is a local ring. In this note we prove that the R-modules and are Artinian for all integers i and j, whenever and . Also we will show that if a is principal, then the R-modules and ...