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A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers

Kim Taekyun, Kim Hye Kyung

Demonstratio Mathematica

2024

Vol. 57, nr 1

art. no. 20240065

In this study, we introduce the λ-analogue of Lah numbers and λ-analogue of r-Lah numbers in the view of degenerate version, respectively. We investigate their properties including recurrence relation and several identities of λ-analogue of Lah numbers arising from degenerate differential operators. Using these new identities, we study the normal ordering of degenerate integral power of the number operator in terms of boson operators, which is represented by means of λ-analogue of Lah numbers and λ-analogue of r-Lah numbers, respectively.

Normal ordering associated with λ-Stirling numbers in λ-shift algebra

Kim Taekyun, Kim Dae San, Kim Hye Kyung

Demonstratio Mathematica

2023

Vol. 56, nr 1

art. no. 20220250

It is known that the Stirling numbers of the second kind are related to normal ordering in the Weyl algebra, while the unsigned Stirling numbers of the first kind are related to normal ordering in the shift algebra. Recently, Kim-Kim introduced a λ-analogue of the unsigned Stirling numbers of the first kind and that of the r-Stirling numbers of the first kind. In this article, we introduce a λ-analogue of the shift algebra (called λ-shift algebra) and investigate normal ordering in the λ-shift algebra. From the normal ordering in the λ-shift algebra, we derive some identities about the λ-analogue of the unsigned Stirling numbers of the first kind.