The $r$-Stirling Genocchi Numbers

This paper introduces the \( r \)-Stirling Genocchi numbers, a new sequence derived by combining Broder's \( r \)-Stirling numbers with the classical Genocchi numbers, which are closely related to Bernoulli numbers and have notable applications in algebraic combinatorics. While the original \( r \)-Stirling numbers were developed using combinatorial methods to count set partitions, this study adopts an algebraic approach to explore the properties of the new sequence. Through tools like generating functions, recurrence relations, and algebraic transformations, the paper uncovers deeper structural insights and highlights the broader mathematical connections between partition theory, number theory, and combinatorial analysis.