A κ -extension of Hermite polynomials in the framework of Kaniadakis calculus

Abstract In this paper, we introduce a new family of κ -Hermite polynomials as a natural κ -extension of the classical Hermite polynomials within the framework of Kaniadakis calculus. The construction is based on a κ -modified generating function obtained by replacing the classical exponential function with the κ -exponential function together with the corresponding κ -factorial structure. This approach allows us to derive explicit representations of the κ -Hermite polynomials and the associated κ -Hermite numbers and to establish several fundamental properties that generalize those of the classical Hermite family. In particular, recurrence relations, κ -derivative formulas, and operational identities in the κ -calculus framework are obtained. Furthermore, a two-variable extension H n , κ x , y ${H}_{n,\kappa }\left(x,y\right)$ is introduced and its main structural properties are investigated. Mixed-type κ -Hermite–Bernoulli, κ -Hermite–Euler, and κ -Hermite–Stirling polynomials are also constructed, and explicit connection formulas are established. It is further shown that, for 0 < | κ | < 1, no nontrivial nonnegative scalar weight with finite moments through order four yields orthogonality in the natural κ -integral setting. Graphical comparisons for the first three degrees exhibiting nontrivial κ -dependence, namely n = 3, 4, and 5, illustrate how the polynomials depart from their classical counterparts as | κ | increases. It is shown that all formulae and constructions that admit classical counterparts consistently reduce to those counterparts in the limit κ → 0, confirming the mathematical consistency of the proposed κ -extension. The present work complements recent studies on κ -generalized polynomial families, such as κ -Bernoulli, κ -Euler, and κ -Stirling polynomials, and provides a systematic κ -extension of the Hermite polynomials.