Abstract
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)(3k-i+1) Sigma(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. (c) 2012 Elsevier Ltd. All rights reserved.