A Broad Study on Some New Families of q‐Polynomials

ABSTRACT In this paper, we introduce diverse new central special polynomials and numbers utilizing two types of ‐exponential functions. We first consider ‐central factorial numbers and polynomials of the second kind and investigate some of their properties and formulas, such as addition formulas, summation formulas, ‐derivative properties, and Jackson integral representations. As part of our main content, we define trivariate central ‐Bell polynomials and acquire several identities and relations, such as some summation and addition formulas, three ‐derivative properties, two Jackson integral representations, two implicit summation formulas, and a symmetric identity. We investigate an important correlation between these two new ‐polynomials and provide some of its consequences. Then, we provide many correlations between new and old ‐polynomials and ‐numbers, such as the ‐Stirling numbers and polynomials of the second kind, the ‐combinatorial Simsek polynomials and numbers of the first kind, the newly defined ‐polynomials and ‐numbers, ‐Euler polynomials, and ‐Bernoulli polynomials. Also, we consider a new ‐extension of Stirling numbers of the second kind and type 2 ‐Bernoulli polynomials and derive some mixed correlations related to the other new and old ‐polynomials and ‐numbers. Furthermore, we compute two ‐operator formulas for trivariate central ‐Bell polynomials and two ‐operator formulas for bivariate and one‐variable ‐central factorial polynomials of the second kind. In the end, we present graphical illustrations and zero distribution patterns of these newly introduced ‐polynomials, which exhibit a striking and structured scattering in the real and complex planes, offering both aesthetic appeal and deep analytical significance.