Some identities of the probabilistic degenerate Genocchi polynomials and numbers

초록

At 1979, L. Carlitz introduced a degenerate version of the Stirling, Bernoulli, and Eulerian numbers. Recently, some researchers have studied various degenerate special polynomials and numbers. Let Y be a random variable whose moment generating function, E [e (Yt)] = Sigma(infinity) (n=0) E [Y-n] tn/nl, (|t| < r). The purpose of this paper is to introduce the probabilistic extension of degenerate Genocchi polynomials, namely the probabilistic degenerate Genocchi polynomials associated with random variable Y, 2t/E[e(lambda)(Y)(t)]+1 (L [e(psi)Y(t)(x)])(x) = Sigma(infinity)(n=0)G(n,lambda)(Y)(x)t(n)/n!. We obtain some properties, explicit expressions, recurrence relations and certain identities for those polynomials and numbers. As the various special cases of random variable Y, we deal with the gamma random variable with parameters alpha, beta > 0, the Bernoulli random variable with probability of success p and the Poisson random variable with parameter alpha > 0. Also, we investigate the properties of the G(n,lambda)(Y)(x) and illustrate our results with some examples.

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