\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]{}
\HyPL@Entry{0<</S/D>>}
\citation{Vella}
\citation{Vella}
\citation{Woon}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{2}{section.1}}
\citation{Fuchs}
\@writefile{toc}{\contentsline {section}{\numberline {2}Background}{3}{section.2}}
\newlabel{sec1}{{2}{3}{Background}{section.2}{}}
\citation{Fuchs}
\citation{NIST}
\newlabel{eq:recursion}{{2}{4}{Background}{equation.2.2}{}}
\newlabel{CX}{{4}{4}{Background}{equation.2.4}{}}
\newlabel{ex:Bernoulli}{{2}{4}{}{theorem.2}{}}
\citation{Howard}
\newlabel{Norlund gf}{{5}{5}{}{equation.2.5}{}}
\citation{Fuchs}
\@writefile{toc}{\contentsline {section}{\numberline {3}Connection to compositions}{6}{section.3}}
\newlabel{sec2}{{3}{6}{Connection to compositions}{section.3}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Definitions}{6}{subsection.3.1}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Fuchs' result}{7}{subsection.3.2}}
\newlabel{thm:5}{{6}{7}{}{theorem.6}{}}
\newlabel{eq:compositions}{{6}{7}{}{equation.3.6}{}}
\newlabel{eq:convolutions7}{{7}{7}{}{equation.3.7}{}}
\newlabel{eq:inverse compositions}{{8}{7}{}{equation.3.8}{}}
\citation{Sills}
\citation{Hoggatt}
\citation{Gessel}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3}Invariant sequences}{8}{subsection.3.3}}
\citation{NIST}
\citation{Vella2}
\citation{Vella}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {3.3.1}Catalan numbers}{9}{subsubsection.3.3.1}}
\newlabel{eq:Catalan sum over compositions}{{10}{9}{}{equation.3.10}{}}
\newlabel{eq:inverse Catalan sum over compositions}{{11}{9}{}{equation.3.11}{}}
\citation{Vella}
\citation{Vella}
\@writefile{toc}{\contentsline {subsubsection}{\numberline {3.3.2}Hermite polynomials}{10}{subsubsection.3.3.2}}
\@writefile{toc}{\contentsline {section}{\numberline {4}A nonlinear generalization}{10}{section.4}}
\newlabel{sec3}{{4}{10}{A nonlinear generalization}{section.4}{}}
\newlabel{thm:main}{{10}{10}{}{theorem.10}{}}
\newlabel{eq:f(g(z))}{{12}{10}{}{equation.4.12}{}}
\citation{Eger}
\citation{Merlini}
\newlabel{non linear}{{13}{12}{}{theorem.13}{}}
\newlabel{eq:convolutions14}{{14}{12}{}{equation.4.14}{}}
\citation{NIST}
\citation{Vella}
\citation{Vella}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Back to the Bernoulli numbers}{13}{subsection.4.1}}
\newlabel{Stirling}{{17}{13}{Back to the Bernoulli numbers}{equation.4.17}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.2}Back to Bernoulli polynomials}{14}{subsection.4.2}}
\newlabel{Bernoulli expansion}{{16}{14}{}{theorem.16}{}}
\newlabel{gf generalized Bernoulli}{{20}{14}{}{equation.4.20}{}}
\citation{Vella2}
\citation{Vella}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.3}Back to higher-order Bernoulli polynomials}{15}{subsection.4.3}}
\citation{Srivastava}
\citation{Vella2}
\citation{Vella}
\citation{Flajolet}
\citation{Alladi}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.4}Compositions with restricted summands}{16}{subsection.4.4}}
\citation{Flajolet}
\newlabel{restricted}{{17}{17}{}{theorem.17}{}}
\newlabel{eq:comp}{{21}{17}{}{equation.4.21}{}}
\newlabel{eq:compgen}{{22}{17}{}{equation.4.22}{}}
\newlabel{eq:comptree}{{23}{17}{}{equation.4.23}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.5}Sum of digits}{18}{subsection.4.5}}
\@writefile{toc}{\contentsline {section}{\numberline {5}A general formula for composition of functions}{18}{section.5}}
\newlabel{sec4}{{5}{18}{A general formula for composition of functions}{section.5}{}}
\@writefile{toc}{\contentsline {section}{\numberline {6}Conclusion}{21}{section.6}}
\bibcite{Alladi}{1}
\bibcite{Eger}{2}
\bibcite{Flajolet}{3}
\bibcite{Fuchs}{4}
\bibcite{Gessel}{5}
\bibcite{Hoggatt}{6}
\bibcite{Howard}{7}
\bibcite{Kaneko}{8}
\bibcite{Merlini}{9}
\bibcite{NIST}{10}
\bibcite{Sills}{11}
\bibcite{Srivastava}{12}
\bibcite{Vella}{13}
\bibcite{Vella2}{14}
\bibcite{Woon}{15}
\@writefile{toc}{\contentsline {section}{\numberline {7}Acknowledgements}{22}{section.7}}
