Fully degenerate Bernoulli numbers and polynomials

The aim of this article is to study the fully degenerate Bernoulli polynomials and numbers, which are a degenerate version of Bernoulli polynomials and numbers and arise naturally from the Volkenborn integral of the degenerate exponential functions on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi mathvariant="double-struck">Z</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> </m:math> {{Z}}ₚ . We find some explicit expressions for the fully degenerate Bernoulli polynomials and numbers in terms of the degenerate Stirling numbers of the second kind, the degenerate <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>r</m:mi> </m:math> r -Stirling numbers of the second kind, and the degenerate Stirling polynomials. We also consider the degenerate poly-Bernoulli polynomials and derive explicit representations for them in terms of the same degenerate Stirling numbers and polynomials.