Generalized λ-Riordan arrays with their applications to λ-Sheffer sequences

In this paper, the authors introduced the generalized [Formula: see text]-Riordan arrays as an extension of Riordan arrays. These arrays offer a versatile framework for studying combinatorial structures and generating functions. The primary focus of this research is to establish a profound connection between [Formula: see text]-Riordan arrays and [Formula: see text]-Sheffer sequences. This study delves into the intricate relationship between [Formula: see text]-Riordan arrays and specific members of the [Formula: see text]-Sheffer class, shedding light on their underlying mathematical properties. By exploring the structural features of [Formula: see text]-Riordan arrays, we uncover novel insights into their connection with well-known members of [Formula: see text]-Sheffer family. Furthermore, the authors use the generic elements of [Formula: see text]-Riordan arrays to derive determinant expressions as the applications. These expressions serve as powerful tools for understanding the underlying mathematics of [Formula: see text]-Sheffer sequences. By providing explicit determinant formulas, this research contributes to a deeper understanding of the relationships between [Formula: see text]-Riordan arrays and [Formula: see text]-Sheffer sequences, offering valuable insights into combinatorial mathematics and its broader applications.