Euler-Frobenius numbers

These numbers are defined as the coefficients of the Euler-Frobenius polynomials P-n,P-lambda(Z) = Sigma(n)(l-0) A(n,l)(lambda)Z(l), which usually are introduced via the rational function expansion Sigma(infinity)(nu-0) (nu + lambda)(n)Z(nu) = P-n,P-lambda(Z)/(1-Z)(n+1) n being a nonnegative integer and lambda is an element of [0, 1). The special case A(n,l)(0) is known from combinatorics (Eulerian numbers) and the general one A(n,l)(lambda) occurs, for example, in approximation theory, summability, and rounding error analysis. By supplementing and extending known results on Eulerian numbers, various theorems for the Euler-Frobenius numbers A(n,l)(lambda) and related quantities are established including unimodality, monotonicity properties, and asymptotic expansions given by a local central limit theorem.