SIMAI 2025

On mixed-type Fibonacci-Mittag-Leffler and Lucas-Mittag-Leffler polynomials: some properties

  • Ramírez, William (AY339179)

Please login to view abstract download link

In this talk, i will present recent results on a new class of special functions defined by merging classical Fibonacci and Lucas structures with Mittag-Leffler-type functions. This approach gives rise to two novel families of polynomials: the {Fibonacci--Mittag-Leffler (FML)} and {Lucas--Mittag-Leffler (LML)} polynomials. \medskip These families are introduced via the following generating functions: \begin{equation}\label{e11} \left(\frac{z}{1 - xz - z^2}\right)\left(\frac{1 + z}{1 - z}\right)^x = \sum_{n=0}^{\infty} {}_F M_n(x) \frac{z^n}{n!}, \end{equation} \begin{equation}\label{deffff2} \left(\frac{2 - xz}{1 - xz - z^2}\right)\left(\frac{1 + z}{1 - z}\right)^x = \sum_{n=0}^{\infty} {}_L M_n(x) \frac{z^n}{n!}, \end{equation} \medskip where \( |z| < \min\left\{ \frac{2}{x + \sqrt{x^2 + 4}}, 1 \right\} \). \medskip In the rest of the talk, I will focus on: \begin{itemize} \item Deriving recurrence relations and determinantal representations for these polynomials. \item Establishing key {algebraic identities}. \item Analyzing the {location and asymptotic behavior of their zeros}. \item And applying a {generalized Hurwitz theorem} in two variables to further explore their analytic structure. \end{itemize}