Differential Operators and Weighted Isobaric Polynomials

where m ∈ Z. We are interested in these operators as linear operators on a special ring of polynomials, discussed in [5], namely, the ring of isobaric polynomials Λ̃, a ring isomorphic to the ring of symmetric functions Λ. The polynomials in Λ̃, or more precisely Λ̃k, are over indeterminants t1, . . . , tk and the isomorphism just mentioned is given by identifying tj with the signed elementary symmetric polynomial (−1)ej. This determines an involutory mapping whose elements in Λ̃ are called isobaric reflects. Isobaric polynomials can be defined independently of Λ as follows: for each n, let Pk,n := ∑ αAαt α where α = (α1, . . . , αk), t α = t1 1 . . . t αk k , and (1 α1 , . . . , kk) is a partition of n, n remaining constant for all monomials in the polynomial. (We call this a polynomial of degree n, even though the coefficient of the term of (maximal) degree n may be zero. In general the Aα can be taken from any commutative ring. In what follows we restrict ourselves to the ring of integers. It is our purpose in this paper to discuss the portion of the kernel of Tm lying in Λ̃, or rather, more precisely, we are interested in certain sequences of polynomials in Λ̃ all of whose entries lie in the kernel of Tm. These are the sequences of polynomials determined by a certain weighting operation whose entries lie in the module of Weighted Isobaric Polynomials (WIP-module) in Λ̃ as defined in [5]. In [4] it was shown that two such sequences are the sequence of Generalized Fibonacci Polynomials {Fn}n, for m = 2, and the sequence of Generalized Lucas Polynomials {Gn}n, for m = 1, which are, respectively, reflects of the complete symmetric polynomials (CSP) and of the power symmetric polynomials (PSP)(see also [1],[2],[3]. In this paper we shall answer the following questions. 1. Are there any other WIP sequences of solutions for these two values of m ? 2. Are there any other WIP sequences of solutions for other integer values of m ?