Probabilistic identities involving fully degenerate Bernoulli polynomials and degenerate Euler polynomials

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Assume that X is the Bernoulli random variable with parameter 1 2 , and that random variables X 1 , X 2 , … are a sequence of mutually independent copies of X. We also assume that Y is the uniform random variable on the interval [ 0 , 1 ] , and that random variables Y 1 , Y 2 , … are a sequence of mutually independent copies of Y. We consider the fully degenerate Bernoulli polynomials and their higher-order analogues. We also consider the degenerate Euler polynomials and their higher-order analogues. The aim of this paper is to compute the expectations of some random variables associated with those polynomials and random variables explicitly, and to derive certain identities between such expectations.