Exploring Fibonacci-stirling and bivariate Fibonacci-Bell polynomials with visual and numerical realms

In recent years, Fibonacci extensions of diverse special polynomials, such as Fibonacci-Hermite, Fibonacci-Euler, Fibonacci-Bernoulli, and Fibonacci-sigmoid polynomials have been defined, and their many relations and properties have been thoroughly analyzed using the content of golden calculus. In this work, we first define the generating function of the Fibonacci-Stirling polynomials of the second kind and investigate diverse properties and relations. These involve summation formulas, addition formulas, golden derivative property, and golden integral representation for the Fibonacci-Stirling polynomials of the second kind. We then consider the generating function of the bivariate Fibonacci-Bell polynomials and investigate various useful properties and identities. These cover summation formulas, addition formulas, golden derivative properties, golden integral representation, symmetric identity, and implicit summation formulas for the bivariate Fibonacci-Bell polynomials. Moreover, we develop multifarious correlations and formulas, including the Fibonacci-Euler polynomials, the Fibonacci-Bernoulli polynomials, the Fibonacci-Stirling polynomials of the second kind, and the Fibonacci-Bell polynomials. Furthermore, we draw the zeros of the bivariate Fibonacci-Bell polynomials, forming 2D and 3D structures, and provide a table including approximate zeros of the bivariate Fibonacci-Bell polynomials. Lastly, we define the Fibonacci differential operator and then provide some Fibonacci differential operator formulas for the Fibonacci-Stirling polynomials of the second kind and the Fibonacci-Bell polynomials.