Bell Polynomials And Gsn On A Number Of Issues

James Stirling introduced the numbers named after him (by T. N. Thiele, according to Nielsen (1904)) in 1730. The Stirling numbers of the first kind and the second kind are the special cases of the Bell polynomial.In this paper, we discuss some recurrence relations and congruence properties of Bell polynomial, and give a class of recurrence relations satisfied by GSN pairs. Furthermore, we givethe definition of extended GSN pairs G(n,k|w,u)and G(n,k\u,\w), so we unify the binomial coefficients, Gaussian coefficients, and Stirling numbers. We provide urn ,models for G(n,k|w,0) andG(n,k|0,w). Lastly, we discuss some properties of G(n,k|w,u) withnegative integral values of n or k. In general, this paper try to extend the definition of GSN to the case where w and u are ordinary sequences, and interpretative the application in combinatorics for them.