Jacobi-Stirling Numbers

The Jacobi-Stirling numbers of the first kind are the polynomial sequence defined by

(1)

for , with and for positive integers and . The companion Jacobi-Stirling numbers of the second kind satisfy

(2)

with the same boundary conditions. At , the two families specialize to the Legendre-Stirling numbers of the first and second kinds, respectively; this simply sets in the two recurrences.

The two kinds are inverse polynomial connection coefficients. More explicitly, if

(3)

then

(4)

and

(5)

Here a polynomial basis is a set of polynomials in which every polynomial in the relevant vector space has a unique expression as a finite linear combination of basis elements. Thus the first-kind numbers convert the polynomial basis to the monomial basis , while the second-kind numbers convert powers back to the basis .

The name "Jacobi" refers to the classical second-order Jacobi differential operator

(6)

whose polynomial eigenfunctions are the Jacobi polynomials, with eigenvalues . Taking explains the factors above: the Jacobi-Stirling numbers arise when powers of this differential operator are expanded in a natural differential-operator basis (Andrews et al. 2013).