Jacobi–Stirling polynomials and -partitions

We investigate the diagonal generating function of the Jacobi–Stirling numbers of the second kind JS ( n + k , n ; z ) by generalizing the analogous results for the Stirling and Legendre–Stirling numbers. More precisely, letting JS ( n + k , n ; z ) = p k , 0 ( n ) + p k , 1 ( n ) z + ⋯ + p k , k ( n ) z k , we show that ( 1 − t ) 3 k − i + 1 ∑ n ≥ 0 p k , i ( n ) t n is a polynomial in t with nonnegative integral coefficients and provide combinatorial interpretations of the coefficients by using Stanley’s theory of P -partitions.