Fibonacci-based generalizations of Bessel polynomials and their applications to Fibonacci-analogue Bessel filters

Abstract

UDC 517.584, 511.176

In the theory of orthogonal polynomials, extending classical polynomial families to multivariate and deformed settings often leads to rich and challenging generalizations. Motivated by recent developments in Fibonacci calculus and Bessel polynomial theory, this paper introduces the $F$-Bessel (Fibonacci–Bessel) polynomials, the reverse $F$-Bessel polynomials, and their two-dimensional counterparts, the 2D $F$-Bessel polynomials. Several fundamental properties of these polynomial families are established, including generating functions, identities, differential recurrence relations, and determinant representations. The reverse $F$-Bessel polynomials are further employed to construct Fibonacci analogues of classical Bessel filters. Pole distributions for the third-, fourth-, fifth-, and sixth-order $F$-Bessel filters are analyzed, while the gain, phase, phase delay, and group delay characteristics of the third-order filter are numerically investigated and compared with those of the corresponding classical Bessel filter. The results demonstrate that the proposed $F$-Bessel filters preserve the desirable features of classical Bessel filters while exhibiting several enhanced performance characteristics.