概率退化Bernstein多项式
Probabilistic degenerate Bernstein polynomials
AI总结：
本文研究与原点邻域矩生成函数存在的随机变量Y相关的概率退化Bernstein多项式，推导其显式表达式与恒等式，并讨论三类常见随机变量的特殊情形。
摘要：
近年来，许多特殊数与多项式的退化形式及概率推广均得到了研究，例如退化Bernstein多项式与概率Bernstein多项式此前已被探讨。假设Y是一个原点邻域内矩生成函数存在的随机变量。本文旨在研究与Y相关的概率退化Bernstein多项式，这类多项式既是退化Bernstein多项式的概率推广，也是与$Y$相关的概率Bernstein多项式的退化形式。我们推导了这类多项式的若干显式表达式及相关恒等式，此外还讨论了Poisson随机变量、Bernoulli随机变量和二项随机变量的特殊情形。
英文摘要：
In recent years, both degenerate versions and probabilistic extensions of many special numbers and polynomials have been explored. For instance, degenerate Bernstein polynomials and probabilistic Bernstein polynomials were investigated earlier. Assume that Y is a random variable whose moment generating function exists in a neighborhood of the origin. The aim of this paper is to study probabilistic degenerate Bernstein polynomials associated with Y which are both probabilistic extension of the degenerate Bernstein polynomials and degenerate version of the probabilistic Bernstein polynomials associated with $Y$. We derive several explicit expressions and certain related identities for those polynomials. In addition, we treat the special cases of the Poisson random variable, the Bernoulli random variable and of the binomial random variable.
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