Other approaches for generalized Bernoulli–Euler polynomials and beyond

In this paper we develop two approaches for studying a large family of generalized Bernoulli–Euler polynomials. For the determinental approach, using Little Fermat’s Theorem, we establish a congruence identity and we give an explicit formulas of the generalized Bernoulli–Euler polynomials in terms of the Stirling numbers. The linear recursive approach allows us to formulate some properties of the generalized Bernoulli–Euler numbers and the generalized Bernoulli–Euler polynomials. Moreover, combinatorial formulas for these polynomials are provided.