Normal Ordering Associated with \(\lambda\)-Whitney Numbers of the First Kind in \(\lambda\)-Shift Algebra

It is known that the unsigned Stirling numbers of the first kind are related to normal ordering in the shift algebra. The aim of this paper is to consider the \(\lambda\)-shift algebra, which is a \(\lambda\)-analogue of the shift algebra, and to study \(\lambda\)-analogues of Whitney numbers of the first kind (called \(\lambda\)-Whitney numbers of the first kind) and those of \(r\)-Whitney numbers of the first kind arising from normal orderings in the \(\lambda\)-shift algebra. From the normal orderings in the \(\lambda\)-shift algebra, we derive some explicit expressions and recurrence relations on both of those numbers.