Explicit formulae for sums of products of Bernoulli polynomials, including poly-Bernoulli polynomials

We study sums of products of Bernoulli polynomials, including poly-Bernoulli polynomials. As a main result, for any positive integer m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$m$\end{document}, explicit expressions of sums of m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$m$\end{document} products are given. This result extends that of the first author, as well as the famous Euler formula about sums of two products of Bernoulli numbers.