New Families of Certain Special Polynomials: A Kaniadakis Calculus Viewpoint

In this paper, we introduce a new family of Stirling polynomials of the second kind, Bell polynomials, bivariate Bell polynomials, Bernoulli polynomials of higher order, and Euler polynomials of higher order arising from the Kaniadakis calculus viewpoint. We refer to each of them as κ-polynomials. Through the defined concepts of Kaniadakis calculus, we derive explicit formulas, summation formulas, and addition formulas for the polynomials discussed in the present paper. We also present the Volkenborn integral and the fermionic p-adic integral representations in terms of the κ-Stirling polynomials of the second kind, bivariate κ-Bell polynomials, κ-Bernoulli polynomials of higher order, and κ-Euler polynomials of higher order. We establish some formulae, including old and new polynomials. Finally, we investigate determinantal representations for the κ-Euler polynomials and the κ-Bernoulli polynomials.