On combinatorial numbers and polynomials and their applications.

The aim of this study is to define a higher-order extension of the numbers arising from the finite sums containing higher powers of binomial coefficients. By using functional equations of the generating functions for these numbers, we derive some identities and relations between these numbers and λ-array polynomials. Finally, we give some remarks and observations on relations among the aforementioned numbers, λ-array polynomials and Eulerian type splines with their applications. [ABSTRACT FROM AUTHOR]