Degenerate Algorithms for degenerate Bernoulli and Euler numbers

TL;DR

This paper develops degenerate versions of classical algorithms for Bernoulli and Euler numbers by introducing a parameter lambda, deriving explicit formulas, and establishing functional relationships between generating functions.

Contribution

It presents novel degenerate algorithms for Bernoulli and Euler numbers, including explicit formulas and functional relationships involving degenerate Stirling numbers.

Findings

Derived explicit formulas for degenerate Bernoulli and Euler numbers.

Established functional relationships between generating functions.

Showed how specific initial conditions produce degenerate Bernoulli and Euler numbers.

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.