A new extended polynomials in terms of Bessel–Struve kernel function and their representation

In this research note, we introduce two novel families of extended polynomials and numbers defined through the Bessel–Struve kernel function. These extensions serve to generalize the classical Bernoulli polynomials and numbers. By assigning specific values to the involved parameters, we recover several well-known special cases, of the classical Bernoulli polynomials. We further investigate fundamental mathematical properties of these newly defined polynomials, such as generating functions, recurrence relations, differential identities, and more. In addition, we establish several generalizations of the Stirling numbers of the second kind in connection with these extended families. Explicit expressions of the polynomials are provided for degrees up to five, and their analytical behavior is examined in detail. Moreover, we present graphical representations of the polynomials and numbers for various parameter values, offering valuable visual insights into their analytic structures.