An extension of the Bernoulli polynomials inspired by the Tsallis statistics

In [Arch. Math. 7, 28 (1956), Utilitas Math. 15, 51 (1979)] Carlitz introduced the degenerate Bernoulli numbers and polynomials by replacing the exponential factors in the corresponding classical generating functions with their deformed analogs: $\exp(t) \rightarrow (1+\lambda t)^{1/\lambda}$, and $\exp(tx) \rightarrow (1+\lambda t)^{x/\lambda}$. The deformed exponentials reduce to their ordinary counterparts in the $\lambda \rightarrow 0$ limit. In the present work we study the extension of the Bernoulli polynomials obtained via an alternate deformation $\exp(tx) \rightarrow (1+\lambda tx)^{1/\lambda}$ that is inspired by the concepts of $q$-exponential function and $q$-logarithm used in the nonextensive Tsallis statistics.