\relax 
\providecommand\hyper@newdestlabel[2]{}
\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument}
\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined
\global\let\oldcontentsline\contentsline
\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}}
\global\let\oldnewlabel\newlabel
\gdef\newlabel#1#2{\newlabelxx{#1}#2}
\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}}
\AtEndDocument{\ifx\hyper@anchor\@undefined
\let\contentsline\oldcontentsline
\let\newlabel\oldnewlabel
\fi}
\fi}
\global\let\hyper@last\relax 
\gdef\HyperFirstAtBeginDocument#1{#1}
\providecommand*\HyPL@Entry[1]{}
\citation{Kaneko}
\HyPL@Entry{0<</S/D>>}
\citation{BH1}
\citation{BH2}
\citation{GKP}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{2}{section.1}}
\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces The poly-Bernoulli numbers $B_n^{(-k)}$}}{2}{table.1}}
\newlabel{tabel}{{1}{2}{The poly-Bernoulli numbers $B_n^{(-k)}$}{table.1}{}}
\newlabel{combform}{{2}{2}{Introduction}{equation.1.2}{}}
\citation{Brewbaker}
\citation{Bayad}
\citation{GKP}
\citation{Bayad}
\citation{Bayad}
\citation{Callan}
\newlabel{lieu}{{4}{3}{Introduction}{equation.1.4}{}}
\newlabel{eulerszita}{{5}{3}{Introduction}{equation.1.5}{}}
\newlabel{pBpol}{{6}{3}{Introduction}{equation.1.6}{}}
\newlabel{BayHam}{{1}{3}{}{theorem.1}{}}
\newlabel{pBeu2}{{7}{3}{}{equation.1.7}{}}
\citation{Callan}
\@writefile{toc}{\contentsline {section}{\numberline {2}Main results}{4}{section.2}}
\citation{Bayad}
\newlabel{theo0}{{4}{5}{}{theorem.4}{}}
\newlabel{pBeupar}{{8}{5}{}{equation.2.8}{}}
\newlabel{theo3}{{5}{5}{}{theorem.5}{}}
\newlabel{pBeu3}{{9}{5}{}{equation.2.9}{}}
\newlabel{theo1}{{6}{5}{}{theorem.6}{}}
\newlabel{pBeu1}{{10}{5}{}{equation.2.10}{}}
\newlabel{theo4}{{7}{5}{}{theorem.7}{}}
\newlabel{pBeu4}{{11}{5}{}{equation.2.11}{}}
\newlabel{theo2}{{8}{5}{}{theorem.8}{}}
\newlabel{pBeu5}{{12}{5}{}{equation.2.12}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3}Proofs of the main results}{5}{section.3}}
\citation{Bona}
\citation{GKP}
\citation{Bona}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Eulerian numbers}{6}{subsection.3.1}}
\newlabel{opeu}{{13}{6}{Eulerian numbers}{equation.3.13}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Combinatorial proofs of the theorems}{7}{subsection.3.2}}
\bibcite{AIK}{1}
\bibcite{Bayad}{2}
\bibcite{BH1}{3}
\@writefile{toc}{\contentsline {section}{\numberline {4}Acknowledgments}{9}{section.4}}
\bibcite{BH2}{4}
\bibcite{BN}{5}
\bibcite{Bona}{6}
\bibcite{Brewbaker}{7}
\bibcite{Callan}{8}
\citation{OEIS}
\bibcite{GKP}{9}
\bibcite{Kaneko}{10}
\bibcite{OEIS}{11}