The complementary symmetric functions: Connection constants using negative sets
Let \(f(x)\) be a rational function in \(x\) of degree \(n\). As the main result of this paper, the coefficients \(c_{k}\) in the expansion \[ f(x)=\sum _{k\geq 0} ^{}c_{k}q_{n-k}(x), \quad\text{where}\quad q_i(x)=\prod _{j=1} ^{i}(x-b_j), \] and where the product has to be read as \(1/\prod _{j=1+i} ^{0}(x-b_j)\) for negative \(i\), are expressed in what are otherwise known as supersymmetric analogues of elementary symmetric functions (in \(\lambda\)-ring language, these are elementary symmetric functions in a difference of two alphabets). In several examples it is shown that special cases yield (if though well-known) identities for Stirling numbers, Lah numbers, and \(q\)-binomial coefficients.
scientific article; zbMATH DE number 1211553