Journal of the
Korean Mathematical Society JKMS

In this paper, we introduce $w$-Catalan polynomials as a generalization of Catalan polynomials and derive fourteen basic identities of symmetry in three variables related to $w$-Catalan polynomials and analogues of alternating power sums. In addition, specializations of one of the variables as one give us new and interesting identities of symmetry even for two variables. The derivations of identities are based on the $p$-adic integral expression for the generating function of the $w$-Catalan polynomials and the quotient of $p$-adic integrals for that of the analogues of the alternating power sums.

Keywords: Catalan polynomial, $w$-Catalan polynomial, fermionic $p$-adic integral, identities of symmetry

MSC numbers: 11B83, 11S80, 05A19