Exploring Algebraic and Probabilistic Properties for Degenerate Bivariate Appell Polynomials

The purpose of this paper is to carry out a comprehensive analysis of the properties of degenerate bivariate Appell polynomials by making extensive use of algebraic and probabilistic techniques. The determinant expressions for the degenerate cosine-Appell and sine-Appell polynomials and for their conjugate forms are established by employing several recursive formulas. Further, we focus on exploring probabilistic properties, in which there associated a family of degenerate bivariate Appell polynomials to every random variable χ with some exponential moments eχ u , which provides some powerful tools for a deeper understanding of the behavior and properties of degenerate bivariate Appell polynomials. Examples are framed which introduces a random variable whose mass function is given in terms of degenerate bivariate Bernoulli polynomials. Some expressions for the expectation E[χ] of the random variable associated with the degenerate bivariate Bernoulli polynomials are derived. It is demonstrated that the probabilistic approach offers interesting new results and developments for these polynomials.