Bernoulli-Euler numbers and multiboundary singularities of type $B_n^l$

TL;DR

This paper explores the properties of numbers $K_n^l$ that generalize Bernoulli-Euler numbers, establishing recurrence relations, generating functions, and differential equations to understand their structure and relations.

Contribution

It introduces a recurrence relation and generating functions for the multiboundary singularity numbers $K_n^l$, extending Bernoulli-Euler numbers and revealing new interrelations.

Findings

Derived recurrence relations for $K_n^l$

Established generating functions for fixed $l$

Identified relations between Bernoulli-Euler numbers

Abstract

In this paper we study properties of numbers $K_n^l$ of connected components of bifurcation diagrams for multiboundary singularities $B_n^l$. These numbers generalize classic Bernoulli-Euler numbers. We prove a recurrent relation on the numbers $K_n^l$. As it was known before, $K^1_n$ is $(n{+}1)$-th Bernoulli-Euler number, this gives us a necessary boundary condition to calculate $K_n^l$. We also find the generating functions for $K_n^l$ with small fixed $l$ and write partial differential equations for the general case. The recurrent relations lead to numerous relations between Bernoulli-Euler numbers. We show them in the last section of the paper.