Schur Type Functions Associated with Polynomial Sequences of Binomial Type

We introduce a class of Schur type functions associated with polynomial sequences of binomial type. This can be regarded as a generalization of the ordinary Schur functions and the factorial Schur functions. This generalization satisfies some interesting expansion formulas, in which there is a curious duality. Moreover this class includes examples which are useful to describe the eigenvalues of Capelli type central elements of the universal enveloping algebras of classical Lie algebras. Introduction In this article, we introduce a class of Schur type functions associated with polynomial sequences of binomial type. Namely, suggested by the definition of the ordinary Schur function det(xi j )/ det(x N−i j ), we consider the following Schur type function: det(pλi+N−i(xj))/ det(pN−i(xj)). Here {pn(x)}n≥0 is a polynomial sequence of binomial type. This can be regarded as a generalization of the ordinary Schur functions and moreover the factorial Schur functions ([BL], [CL]). In addition to this, we also consider the following function: det(p∗λi+N−i(xj))/ det(p ∗ N−i(xj)). Here we put pn(x) = x pn+1(x) (this p ∗ n(x) is a polynomial, and satisfies good relations; see Section 1.3). The main results of this article are some expansion formulas for these functions and their mysterious duality corresponding to the exchange pn(x) ↔ p ∗ n(x) and the conjugation of partitions (Sections 3, 4, 5, and 6). Most of them are proved by elementary and straightforward calculations. Beside these results, we also give an application to representation theory of Lie algebras (Section 8). Let us briefly explain this application. The factorial Schur functions are useful to express the eigenvalues of Capelli type central elements of the universal enveloping algebras of the general linear Lie algebra (more precisely, we should say that the “shifted Schur functions” are useful; by the shift of variables, the factorial Schur functions are transformed into the shifted Schur functions ([OO1], [O])). In this article, we aim to introduce similar Schur type functions which is useful to express the eigenvalues of Capelli type central elements of the universal enveloping algebras of the orthogonal and symplectic Lie algebras. This aim is achieved in the case of the polynomial sequence corresponding to the central difference. This case associated with the central difference is also related with the analogues of the shifted Schur functions given in [OO2], which were introduced 2000 Mathematics Subject Classification. Primary 05E05, 05A40; Secondary 17B35, 15A15;