Superbinomial coefficients (Q2137074)

In this paper, several families of polynomials that are related to certain Euler type summation operators are investigated. The authors are interested in a family of polynomials \(p(n, x)\) that, for particular polynomials \(a(n, x)\), satisfy recursion formulas of the form \(p(n, x + 1) + 2p(n, x) + p(n, x- 1) = a(n, x)^{2}\). The authors consider polynomials \(p(n, x)\) of degree \(\leq 2(n - 1)\) with \(p(0, x) = 0\), such that the values \(p(n, m)+(-1)^{m+n}n\) are symmetric for all \(n, m \geq 0\). Imposing the additional conditions \(p(n, x) = p(n, -x)\), this uniquely determines the polynomials \(p(n, x)\). For \(m \geq n\), the authors further define \(P(n, m) = p(n, x)|_{x=m}\). Extended by symmetry \(P(m, n) = P(n, m)\), these numbers (up to a simultaneous index shift of \(n, m\) by 1) are called the superbinomial coefficients. They arise from the representation theory of the superlinear groups SL(\(n|n\)). Being integer-valued at integral points, they satisfy combinatorial properties and nearby symmetries, due to triangle recursion relations involving squares of polynomials. Some of these interpolate the Delannoy numbers. The results are motivated by and strongly related to the study of irreducible Lie supermodules, where dimension polynomials in many cases show similar features.