The Extended Legendre-Stirling Numbers of the First Kind

Abstract

The Legendre-Stirling numbers of the first kind  j n Ps are defined by the coefficients of Taylor expansion of the function x(x  2)(x  6)ïŒ(x  (n 1)n) by Andrews and Littlejohn (see "A combinatorial interpretation of the Legendre-Stirling numbers", Proc. Amer. Math. Soc, 137: 2581-2590, 2009). In this paper, two new kinds of numbers   jn Ps (n  0, j  1) and  j (0 ) n Ps n j   ï‚£ ï‚£ are proposed with the coefficients of Laurent expansion of the function  1 x(x 2)(x 6) (x (n 1)n)    ïŒ ï€­  , which are called the extended Legendre-Stirling numbers of the first kind. Several properties of the two new sequences are proved, such as the recurrence relations, vertical recurrence relation, forward difference. Also, this paper shows a relational expression of the Legendre-Stirling numbers of the extended first and second kinds.