Four families of orthogonal polynomials of C2 and symmetric and antisymmetric generalizations of sine and cosine functions

Abstract: Four families of generalizations of trigonometric functions were recently introduced. In the paper the functions are transformed into four families of orthogonal polynomials depending on two variables. Recurrence relations for construction of the polynomials are presented. Orthogonality relations of the four families of polynomials are found together with the appropriate weight fuctions. Tables of the lowest degree polynomials are shown. Numerous trigonometric-like identities are found. Two of the four families of functions are identified as the functions encountered in the Weyl character formula for the finite dimensional irreducible representations of the compact Lie group Sp(4). The other two families of functions seem to play no role in Lie theory so far in spite of their analogous `good' properties.