GENERALIZED EULERIAN POLYNOMIALS AND SOME APPLICATIONS.

We consider polynomials P_a,b,r,p (z) = ∑^rp_i=0 A_a,b,r (p, i) z^rp-i, in which the coefficients A_a,b,r (p, i) are the generalized Eulerian numbers involved in the expansion .... The numbers A_a,b,r (p, i) and their properties were studied in a previous work. The case a = 1, b = 0, r = 1, corresponds to the standard Eulerian polynomials P1,0,1,p (z). We give generalizations for the known recurrences of P1,0,1,p (z). We also show some applications of polynomials Pa,b,r,p (z), including explicit formulas for sums and alternating sums of powers of binomial coefficients. The main tool we use to obtain our results is the Z-transform of sequences. [ABSTRACT FROM AUTHOR]