Multifarious applications and generalizations of some special polynomials

Multifarious explicit formulas for not only the p-adic integral of the p-adic gamma function and its derivatives but also p-adic Euler constant have been provided. Then, several p-adic integral representations of the p-adic gamma function and its derivative through the Daehee, Changhee and Boole polynomials with their some extensions have been given. Besides, a lot of p-adic integral representations of the p-adic Euler constant utilizing the Daehee, Changhee and Boole polynomials with their some generalizations have been developed. Two variables truncated Fubini polynomials and fully degenerate central Bell polynomials are considered and some identities and properties for these polynomials, involving summation formulas, recurrence relations, and derivative property are acquired. Multifarious correlations including old and new polynomials are discovered. Degenerate truncated forms of the various special polynomials and numbers, including both Stirling numbers of the second kind and Fubini, Bernoulli, Euler and Bell polynomials are considered and their several properties, identities, and interesting relationships are analyzed extensively by using the series manipulation method and some special proof techniques. Several interesting surface plots of the aforementioned polynomials in the special cases are given. Lastly, degenerate Poisson distribution is introduced. Then, the first few raw moments and the moment generating function are developed. Using this moment generating function, a new degenerate form of Bell polynomials is defined and some of its properties are provided.