Degenerate Algorithms for degenerate Bernoulli and Euler numbers
Taekyun Kim, Dae san Kim
TL;DR
The paper develops degenerate (\lambda-parametrized) versions of the A- and B-algorithms, linking their final sequences to initial sequences via degenerate Stirling numbers of the second kind ${n \brace k}_{\lambda}$. It proves key generating-function identities, notably $\overline{F}_{\lambda}(t)=F_{\lambda}(1-e_{\lambda}(t))$ and $\overline{G}_{\lambda}(t)=e_{\lambda}(t)G_{\lambda}(1-e_{\lambda}(t))$, and provides explicit formulas for final sequences such as $a_{n,0}(\lambda)=\sum_{k=0}^{n}(-1)^{k}k!{n \brace k}_{\lambda}a_{0,k}(\lambda)$ and $b_{n,0}(\lambda)=\sum_{k=0}^{n}(-1)^{k}k!({n+1 \brace k+1}_{\lambda}+n\lambda {n \brace k+1}_{\lambda})b_{0,k}(\lambda)$. By choosing suitable initial sequences, the authors recover degenerate Bernoulli numbers $\beta_{n,\lambda}$ and degenerate Euler numbers $\mathcal{E}_{n,\lambda}$, as well as their variants at 1, thereby providing a unified combinatorial framework for degenerate numbers via $\lambda$-deformed recursions and generating functions. These results extend classical Seidel-type methods to the degenerate setting and illuminate connections between ordinary and exponential generating functions under $\lambda$-deformation.
Abstract
This paper introduces and investigates degenerate versions of the A-algorithm and B-algorithm by incorporating a parameter lambda into their respective recurrence relations. We derive explicit formulas for the final sequences of these algorithms in terms of the initial sequences and the degenerate Stirling numbers of the second kind. Furthermore, we establish functional relationships between the ordinary generating functions of the initial sequences and the exponential generating functions of the final sequences. Specifically, we demonstrate that these degenerate algorithms yield degenerate Bernoulli and Euler numbers under specific initial conditions.