X iv : q - a lg / 9 60 90 16 v 1 1 7 Se p 19 96 Two parameter Deformed Multimode Oscillators and q - Symmetric States

ar X iv : q - a lg / 9 60 10 10 v 1 1 1 Ja n 19 96 UAHEP 956 November 1995 TWO - PARAMETER DEFORMATION OF THE POINCARÉ ALGEBRA

We examine a two-parameter (, λ) deformation of the Poincarè algebra which is covariant under the action of SLq(2, C). When λ → 0 it yields the Poincarè algebra, while in the → 0 limit we recover the classical quadratic algebra discussed previously in [1], [2]. The analogues of the Pauli-Lubanski vector w and Casimirs p 2 and w 2 are found and a set of mutually commuting operators is constructed.

ar X iv : q - a lg / 9 60 60 16 v 1 2 3 Ju n 19 96 LIE ALGEBRAS AND THE FOUR COLOR THEOREM DROR

We present a “reasonable” statement about Lie algebras that is equivalent to the Four Color Theorem.

ar X iv : q - a lg / 9 70 90 01 v 2 2 6 Se p 19 97 Bilinear identity for q - hypergeometric integrals

ar X iv : q - a lg / 9 60 20 33 v 1 2 7 Fe b 19 96 The multiple gamma function and its q - analogue ∗

We give an asymptotic expansion (the higher Stirling formula) and an infinite product representation (the Weierstrass product formula) of the Vignéras multiple gamma function by considering the classical limit of the multiple q-gamma function.

ar X iv : q - a lg / 9 60 80 20 v 1 2 3 A ug 1 99 6 SHIFTED JACK POLYNOMIALS , BINOMIAL FORMULA , AND APPLICATIONS

In this note we prove an explicit binomial formula for Jack polynomials and discuss some applications of it. 1. Jack polynomials ([M,St]). In this note we use the parameter θ = 1/α inverse to the standard parameter α for Jack polynomials. Jack symmetric polynomials Pλ(x1, . . . , xn; θ) are eigenfunctions of Sekiguchi differential operators D(u; θ) = V (x) det [