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

An extension of the Bernoulli polynomials inspired by the Tsallis statistics