On Bivariate Fubini-Fibonacci Polynomials and Numbers

In this paper, we introduce a new type of Fubini polynomials called bivariate Fubini-Fibonacci polynomials, denoted by \(F_{n}^{f}(x, y)\) using golden exponential function, via generating function\[\frac{e_{f}^{xt}}{1-y(e_{f}^{t}-1)}=\sum_{n=0}^{\infty} F_{n}^{f}(x,y) \frac{t^{n}}{f_{n}!}. \nonumber\]We then derive some fundamental properties of these polynomials including addition formula, explicit formula, recurrence relations, and derivative and integral formulas. Moreover, we establish the relationships between Fubini-Fibonacci polynomials and other Fibonacci polynomials and obtain the harmonic-based \(f\)-exponential generating functions of Fubini-Fibonacci polynomials and numbers.