Macdonald Polynomials of Type Cₙ with One-Column Diagrams and Deformed Catalan Numbers

Abstract

We present an explicit formula for the transition matrix C from the type Cₙ degeneration of the Koornwinder polynomials P₍₁ᵣ₎(x|a,−a,c,−c|q,t) with one column diagrams, to the type Cₙ monomial symmetric polynomials m₍₁ᵣ₎(x). The entries of the matrix C enjoy a set of three-term recursion relations, which can be regarded as a (a, c, t)-deformation of the one for the Catalan triangle or ballot numbers. Some transition matrices are studied associated with the type (Cₙ, Cₙ) Macdonald polynomials P⁽ᶜⁿ'ᶜⁿ⁾₍₁ᵣ₎(x|b;q,t)=P₍₁ᵣ₎(x|b¹ᐟ²,−b¹ᐟ²,q¹ᐟ²b¹ᐟ²,−q¹ᐟ²b¹ᐟ²|q,t). It is also shown that the q-ballot numbers appear as the Kostka polynomials, namely in the transition matrix from the Schur polynomials P⁽ᶜⁿ'ᶜⁿ⁾₍₁ᵣ₎(x|q;q,q) to the Hall-Littlewood polynomials PP⁽ᶜⁿ'ᶜⁿ⁾₍₁ᵣ₎(x|t;0,t).