\relax 
\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{\let \contentsline\oldcontentsline
\let \newlabel\oldnewlabel}
\else
\global \let \hyper@last\relax 
\fi

\citation{GF}
\citation{S}
\citation{C}
\citation{M}
\citation{AS}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}}
\citation{AS}
\@writefile{toc}{\contentsline {section}{\numberline {2}Notation and definitions}{2}{section.2}}
\newlabel{D.2.1}{{2.1}{2}{Notation and definitions\relax }{Item.3}{}}
\newlabel{D.2.2}{{2.2}{2}{Notation and definitions\relax }{equation.2.2}{}}
\newlabel{D.2.3}{{2.3}{2}{Notation and definitions\relax }{equation.2.2}{}}
\newlabel{D.5.1a}{{2.4}{3}{Notation and definitions\relax }{equation.2.4}{}}
\newlabel{D.5.1b}{{2.5}{3}{Notation and definitions\relax }{equation.2.4}{}}
\newlabel{D.5.1c}{{2.6}{3}{Notation and definitions\relax }{equation.2.4}{}}
\newlabel{D.5.2a}{{2.7}{3}{Notation and definitions\relax }{equation.2.7}{}}
\newlabel{D.5.2b}{{2.8}{3}{Notation and definitions\relax }{equation.2.7}{}}
\newlabel{D.5.2c}{{2.9}{3}{Notation and definitions\relax }{equation.2.7}{}}
\newlabel{D.6.1}{{2.10}{3}{Notation and definitions\relax }{equation.2.10}{}}
\newlabel{D.6.2}{{2.11}{3}{Notation and definitions\relax }{equation.2.10}{}}
\newlabel{D.6.4}{{2.12}{3}{Notation and definitions\relax }{equation.2.12}{}}
\newlabel{D.6.5}{{2.13}{3}{Notation and definitions\relax }{equation.2.12}{}}
\newlabel{D.6.6}{{2.14}{3}{Notation and definitions\relax }{equation.2.14}{}}
\newlabel{D.6.7}{{2.15}{3}{Notation and definitions\relax }{equation.2.14}{}}
\@writefile{toc}{\contentsline {section}{\numberline {3}The generating function $B^{c}(x,y,t)$}{4}{section.3}}
\newlabel{1.1}{{3.1}{4}{The generating function $B^{c}(x,y,t)$\relax }{equation.3.1}{}}
\newlabel{P.1}{{3.1}{4}{The generating function $B^{c}(x,y,t)$\relax }{proposition.3.1}{}}
\newlabel{1.2}{{3.2}{4}{The generating function $B^{c}(x,y,t)$\relax }{equation.3.2}{}}
\newlabel{1.3}{{3.3}{4}{The generating function $B^{c}(x,y,t)$\relax }{equation.3.3}{}}
\newlabel{1.4}{{3.4}{4}{The generating function $B^{c}(x,y,t)$\relax }{equation.3.4}{}}
\newlabel{P.1.1}{{3.5}{4}{The generating function $B^{c}(x,y,t)$\relax }{equation.3.5}{}}
\newlabel{P.1.2}{{3.6}{4}{The generating function $B^{c}(x,y,t)$\relax }{equation.3.6}{}}
\newlabel{P.1.3}{{3.7}{4}{The generating function $B^{c}(x,y,t)$\relax }{equation.3.7}{}}
\newlabel{P.1.4}{{3.8}{5}{The generating function $B^{c}(x,y,t)$\relax }{equation.3.8}{}}
\newlabel{P.1.5}{{3.9}{5}{The generating function $B^{c}(x,y,t)$\relax }{equation.3.9}{}}
\newlabel{P.1.6}{{3.10}{5}{The generating function $B^{c}(x,y,t)$\relax }{equation.3.10}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Special cases}{5}{subsection.3.1}}
\newlabel{1.8b}{{3.11}{5}{Special cases\relax }{equation.3.11}{}}
\newlabel{1.9b}{{3.12}{5}{Special cases\relax }{equation.3.12}{}}
\newlabel{1.10b}{{3.13}{5}{Special cases\relax }{equation.3.13}{}}
\newlabel{1.5}{{3.14}{5}{Special cases\relax }{equation.3.14}{}}
\newlabel{1.6}{{3.15}{5}{Special cases\relax }{equation.3.14}{}}
\newlabel{1.7}{{3.16}{5}{Special cases\relax }{equation.3.16}{}}
\newlabel{1.8}{{3.17}{5}{Special cases\relax }{equation.3.16}{}}
\newlabel{1.9}{{3.18}{5}{Special cases\relax }{equation.3.18}{}}
\newlabel{1.10}{{3.19}{5}{Special cases\relax }{equation.3.18}{}}
\citation{M}
\citation{AS}
\citation{C}
\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces The number triangle $B_{n,k}$}}{6}{table.1}}
\newlabel{B}{{1}{6}{The numbers $B_{n,k}(1)$ and $B_{n,k}(2)$\relax }{table.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}The numbers $B_{n,k}(1)$ and $B_{n,k}(2)$}{6}{subsection.3.2}}
\newlabel{1.11}{{3.20}{6}{The numbers $B_{n,k}(1)$ and $B_{n,k}(2)$\relax }{equation.3.20}{}}
\newlabel{1.12'}{{3.21}{6}{The numbers $B_{n,k}(1)$ and $B_{n,k}(2)$\relax }{equation.3.21}{}}
\newlabel{1.13'}{{3.22}{6}{The numbers $B_{n,k}(1)$ and $B_{n,k}(2)$\relax }{equation.3.21}{}}
\newlabel{1.19a'}{{3.23}{6}{The numbers $B_{n,k}(1)$ and $B_{n,k}(2)$\relax }{equation.3.23}{}}
\newlabel{1.19b'}{{3.24}{6}{The numbers $B_{n,k}(1)$ and $B_{n,k}(2)$\relax }{equation.3.24}{}}
\citation{AS}
\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces The number triangle $A_{n,k}$}}{7}{table.2}}
\newlabel{A}{{2}{7}{The numbers $B_{n,k}(1)$ and $B_{n,k}(2)$\relax }{table.2}{}}
\newlabel{1.14}{{3.25}{7}{The numbers $B_{n,k}(1)$ and $B_{n,k}(2)$\relax }{equation.3.25}{}}
\newlabel{1.15}{{3.26}{7}{The numbers $B_{n,k}(1)$ and $B_{n,k}(2)$\relax }{equation.3.26}{}}
\newlabel{1.16}{{3.27}{7}{The numbers $B_{n,k}(1)$ and $B_{n,k}(2)$\relax }{equation.3.26}{}}
\newlabel{1.17}{{3.28}{7}{The numbers $B_{n,k}(1)$ and $B_{n,k}(2)$\relax }{equation.3.28}{}}
\newlabel{1.18}{{3.29}{7}{The numbers $B_{n,k}(1)$ and $B_{n,k}(2)$\relax }{equation.3.29}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.3}Examples of some $B_{n}(x,y;c)$}{7}{subsection.3.3}}
\@writefile{toc}{\contentsline {section}{\numberline {4}Properties of the $B_{n}(x,y;c)$}{8}{section.4}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Additive property with respect to the parameter $c$}{8}{subsection.4.1}}
\newlabel{1.20}{{4.1}{8}{Additive property with respect to the parameter $c$\relax }{equation.4.1}{}}
\newlabel{1.21}{{4.2}{8}{Additive property with respect to the parameter $c$\relax }{equation.4.2}{}}
\newlabel{1.22}{{4.3}{8}{Additive property with respect to the parameter $c$\relax }{equation.4.3}{}}
\newlabel{1.23}{{4.4}{8}{Additive property with respect to the parameter $c$\relax }{equation.4.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.2}Infinite series}{8}{subsection.4.2}}
\newlabel{P.2}{{4.1}{8}{Infinite series\relax }{proposition.4.1}{}}
\newlabel{2.1}{{4.5}{8}{Infinite series\relax }{equation.4.5}{}}
\citation{Ru}
\newlabel{2.3}{{4.6}{9}{Infinite series\relax }{equation.4.6}{}}
\newlabel{2.4}{{4.7}{9}{Infinite series\relax }{equation.4.7}{}}
\newlabel{2.5}{{4.8}{9}{Infinite series\relax }{equation.4.8}{}}
\newlabel{2.6}{{4.9}{9}{Infinite series\relax }{equation.4.8}{}}
\citation{L}
\citation{W.St2}
\newlabel{2.7}{{4.10}{10}{Infinite series\relax }{equation.4.10}{}}
\newlabel{2.8}{{4.11}{10}{Infinite series\relax }{equation.4.10}{}}
\newlabel{2.11a}{{4.12}{10}{Infinite series\relax }{equation.4.12}{}}
\newlabel{2.9}{{4.13}{10}{Infinite series\relax }{equation.4.13}{}}
\newlabel{2.10}{{4.14}{10}{Infinite series\relax }{equation.4.13}{}}
\newlabel{2.11b}{{4.15}{10}{Infinite series\relax }{equation.4.15}{}}
\newlabel{2.14}{{4.16}{10}{Infinite series\relax }{equation.4.16}{}}
\newlabel{2.12}{{4.17}{10}{Infinite series\relax }{equation.4.17}{}}
\newlabel{2.13}{{4.18}{10}{Infinite series\relax }{equation.4.18}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.3}Generating expression}{11}{subsection.4.3}}
\newlabel{P.5}{{4.2}{11}{Generating expression\relax }{proposition.4.2}{}}
\newlabel{5.1}{{4.19}{11}{Generating expression\relax }{equation.4.19}{}}
\newlabel{5.2}{{4.20}{11}{Generating expression\relax }{equation.4.20}{}}
\newlabel{5.3}{{4.21}{11}{Generating expression\relax }{equation.4.21}{}}
\citation{M}
\@writefile{toc}{\contentsline {section}{\numberline {5}Properties of the $B_{n,k}(c)$}{12}{section.5}}
\newlabel{P.3}{{5.1}{12}{Properties of the $B_{n,k}(c)$\relax }{proposition.5.1}{}}
\newlabel{3.1}{{5.1}{12}{Properties of the $B_{n,k}(c)$\relax }{equation.5.1}{}}
\newlabel{3.2}{{5.2}{12}{Properties of the $B_{n,k}(c)$\relax }{equation.5.2}{}}
\newlabel{3.3}{{5.3}{12}{Properties of the $B_{n,k}(c)$\relax }{equation.5.3}{}}
\citation{M}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Expressions}{13}{subsection.5.1}}
\newlabel{P.4}{{5.2}{13}{Expressions\relax }{proposition.5.2}{}}
\newlabel{4.1a}{{5.4}{13}{Expressions\relax }{equation.5.4}{}}
\newlabel{4.1b}{{5.5}{13}{Expressions\relax }{equation.5.4}{}}
\newlabel{4.5a}{{5.6}{13}{Expressions\relax }{equation.5.6}{}}
\newlabel{4.5b}{{5.7}{13}{Expressions\relax }{equation.5.7}{}}
\newlabel{4.1}{{5.8}{13}{Expressions\relax }{equation.5.8}{}}
\newlabel{4.2}{{5.9}{14}{Expressions\relax }{equation.5.9}{}}
\newlabel{4.3}{{5.10}{14}{Expressions\relax }{equation.5.10}{}}
\newlabel{4.3b}{{5.11}{14}{Expressions\relax }{equation.5.11}{}}
\newlabel{P.15}{{5.3}{14}{Expressions\relax }{proposition.5.3}{}}
\newlabel{15.1}{{5.12}{14}{Expressions\relax }{equation.5.12}{}}
\citation{AS}
\newlabel{15.2}{{5.13}{15}{Expressions\relax }{equation.5.13}{}}
\newlabel{15.3}{{5.14}{15}{Expressions\relax }{equation.5.14}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.2}Polynomial expression}{15}{subsection.5.2}}
\newlabel{P.6}{{5.4}{15}{Polynomial expression\relax }{proposition.5.4}{}}
\newlabel{6.1}{{5.15}{15}{Polynomial expression\relax }{equation.5.15}{}}
\newlabel{6.2}{{5.16}{15}{Polynomial expression\relax }{equation.5.16}{}}
\citation{Ro1984}
\newlabel{6.3a}{{5.17}{18}{Polynomial expression\relax }{equation.5.17}{}}
\newlabel{6.3b}{{5.18}{18}{Polynomial expression\relax }{equation.5.17}{}}
\newlabel{6.3c}{{5.19}{18}{Polynomial expression\relax }{equation.5.17}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3}Symmetry}{19}{subsection.5.3}}
\newlabel{4.4}{{5.20}{19}{Symmetry\relax }{equation.5.20}{}}
\newlabel{4.4a}{{5.21}{19}{Symmetry\relax }{equation.5.21}{}}
\newlabel{4.4b}{{5.22}{19}{Symmetry\relax }{equation.5.22}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4}A result of Ruiz}{19}{subsection.5.4}}
\citation{R}
\newlabel{4.5}{{5.23}{20}{A result of Ruiz\relax }{equation.5.23}{}}
\newlabel{4.5c}{{5.24}{20}{A result of Ruiz\relax }{equation.5.24}{}}
\newlabel{4.5f}{{5.25}{20}{A result of Ruiz\relax }{equation.5.25}{}}
\@writefile{toc}{\contentsline {section}{\numberline {6}Properties of the $b_{n,k,l}$}{20}{section.6}}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.1}Recursion relation for the $b_{n,k,l}$}{20}{subsection.6.1}}
\newlabel{P.16}{{6.1}{20}{Recursion relation for the $b_{n,k,l}$\relax }{proposition.6.1}{}}
\newlabel{16.1}{{6.1}{20}{Recursion relation for the $b_{n,k,l}$\relax }{equation.6.1}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.2}Partial sums of the $b_{n,k,l}$}{21}{subsection.6.2}}
\newlabel{P.8}{{6.2}{21}{Partial sums of the $b_{n,k,l}$\relax }{proposition.6.2}{}}
\newlabel{8.1}{{6.2}{22}{Partial sums of the $b_{n,k,l}$\relax }{equation.6.2}{}}
\newlabel{8.2}{{6.3}{22}{Partial sums of the $b_{n,k,l}$\relax }{equation.6.2}{}}
\newlabel{8.3}{{6.4}{22}{Partial sums of the $b_{n,k,l}$\relax }{equation.6.2}{}}
\newlabel{8.4}{{6.5}{22}{Partial sums of the $b_{n,k,l}$\relax }{equation.6.2}{}}
\newlabel{8.5}{{6.6}{24}{Partial sums of the $b_{n,k,l}$\relax }{equation.6.6}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {6.3}The numbers $T_{n,k}$}{24}{subsection.6.3}}
\newlabel{P.7}{{6.3}{24}{The numbers $T_{n,k}$\relax }{proposition.6.3}{}}
\newlabel{7.1}{{6.7}{24}{The numbers $T_{n,k}$\relax }{equation.6.7}{}}
\newlabel{7.2}{{6.8}{24}{The numbers $T_{n,k}$\relax }{equation.6.8}{}}
\@writefile{lot}{\contentsline {table}{\numberline {3}{\ignorespaces The number triangle $T_{n,k}$}}{25}{table.3}}
\newlabel{T}{{3}{25}{The numbers $T_{n,k}$\relax }{table.3}{}}
\@writefile{toc}{\contentsline {section}{\numberline {7}The Maclaurin series of $\mathop {\rm  sech}\nolimits ^{c}t$}{25}{section.7}}
\newlabel{9.1}{{7.1}{25}{The Maclaurin series of $\func {sech}^{c}t$\relax }{equation.7.1}{}}
\newlabel{9.2}{{7.2}{25}{The Maclaurin series of $\func {sech}^{c}t$\relax }{equation.7.2}{}}
\newlabel{9.3}{{7.3}{25}{The Maclaurin series of $\func {sech}^{c}t$\relax }{equation.7.2}{}}
\newlabel{9.4}{{7.4}{25}{The Maclaurin series of $\func {sech}^{c}t$\relax }{equation.7.4}{}}
\newlabel{9.5}{{7.5}{26}{The Maclaurin series of $\func {sech}^{c}t$\relax }{equation.7.5}{}}
\newlabel{9.6b}{{7.6}{26}{The Maclaurin series of $\func {sech}^{c}t$\relax }{equation.7.6}{}}
\newlabel{9.6}{{7.7}{26}{The Maclaurin series of $\func {sech}^{c}t$\relax }{equation.7.7}{}}
\newlabel{P.9}{{7.1}{26}{The Maclaurin series of $\func {sech}^{c}t$\relax }{proposition.7.1}{}}
\newlabel{9.10}{{7.8}{26}{The Maclaurin series of $\func {sech}^{c}t$\relax }{equation.7.8}{}}
\citation{S}
\citation{Lu}
\citation{S}
\newlabel{9.7}{{7.9}{29}{The Maclaurin series of $\func {sech}^{c}t$\relax }{equation.7.9}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {7.1}Multinomial Euler numbers $E_{n}^{m}$}{29}{subsection.7.1}}
\newlabel{10.1}{{7.10}{29}{Multinomial Euler numbers $E_{n}^{m}$\relax }{equation.7.10}{}}
\newlabel{10.2}{{7.11}{29}{Multinomial Euler numbers $E_{n}^{m}$\relax }{equation.7.11}{}}
\newlabel{10.3}{{7.12}{29}{Multinomial Euler numbers $E_{n}^{m}$\relax }{equation.7.12}{}}
\newlabel{10.5}{{7.13}{29}{Multinomial Euler numbers $E_{n}^{m}$\relax }{equation.7.13}{}}
\newlabel{10.4}{{7.14}{29}{Multinomial Euler numbers $E_{n}^{m}$\relax }{equation.7.14}{}}
\citation{S}
\citation{St}
\@writefile{lot}{\contentsline {table}{\numberline {4}{\ignorespaces The number square $E_{n}^{m}$}}{30}{table.4}}
\newlabel{E}{{4}{30}{Multinomial Euler numbers $E_{n}^{m}$\relax }{table.4}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {7.2}Even multinomial parity numbers $e_{n}^{m}$}{30}{subsection.7.2}}
\newlabel{11.1}{{7.15}{30}{Even multinomial parity numbers $e_{n}^{m}$\relax }{equation.7.15}{}}
\newlabel{11.2}{{7.16}{30}{Even multinomial parity numbers $e_{n}^{m}$\relax }{equation.7.16}{}}
\newlabel{11.3}{{7.17}{30}{Even multinomial parity numbers $e_{n}^{m}$\relax }{equation.7.17}{}}
\newlabel{13}{{7.18}{30}{Even multinomial parity numbers $e_{n}^{m}$\relax }{equation.7.18}{}}
\newlabel{14}{{7.19}{30}{Even multinomial parity numbers $e_{n}^{m}$\relax }{equation.7.19}{}}
\newlabel{15a}{{7.20}{30}{Even multinomial parity numbers $e_{n}^{m}$\relax }{equation.7.20}{}}
\newlabel{15b}{{7.21}{30}{Even multinomial parity numbers $e_{n}^{m}$\relax }{equation.7.21}{}}
\@writefile{lot}{\contentsline {table}{\numberline {5}{\ignorespaces The number square $e_{n}^{m}$}}{31}{table.5}}
\newlabel{P}{{5}{31}{Even multinomial parity numbers $e_{n}^{m}$\relax }{table.5}{}}
\@writefile{toc}{\contentsline {subsection}{\numberline {7.3}Relations between the $E_{n}^{m}$ and the $e_{n}^{m}$ numbers}{31}{subsection.7.3}}
\newlabel{12}{{7.22}{31}{Relations between the $E_{n}^{m}$ and the $e_{n}^{m}$ numbers\relax }{equation.7.22}{}}
\newlabel{16a}{{7.23}{31}{Relations between the $E_{n}^{m}$ and the $e_{n}^{m}$ numbers\relax }{equation.7.23}{}}
\newlabel{16b}{{7.25}{31}{Relations between the $E_{n}^{m}$ and the $e_{n}^{m}$ numbers\relax }{equation.7.25}{}}
\citation{AS}
\newlabel{17.1}{{7.27}{32}{Relations between the $E_{n}^{m}$ and the $e_{n}^{m}$ numbers\relax }{equation.7.27}{}}
\newlabel{17.2}{{7.28}{32}{Relations between the $E_{n}^{m}$ and the $e_{n}^{m}$ numbers\relax }{equation.7.28}{}}
\citation{AS}
\citation{Lu}
\bibcite{AS}{1}
\bibcite{C}{2}
\bibcite{GF}{3}
\newlabel{18}{{7.29}{33}{Relations between the $E_{n}^{m}$ and the $e_{n}^{m}$ numbers\relax }{equation.7.29}{}}
\newlabel{19}{{7.30}{33}{Relations between the $E_{n}^{m}$ and the $e_{n}^{m}$ numbers\relax }{equation.7.30}{}}
\newlabel{19a}{{7.31}{33}{Relations between the $E_{n}^{m}$ and the $e_{n}^{m}$ numbers\relax }{equation.7.31}{}}
\newlabel{20}{{7.32}{33}{Relations between the $E_{n}^{m}$ and the $e_{n}^{m}$ numbers\relax }{equation.7.32}{}}
\bibcite{GR}{4}
\bibcite{M}{5}
\bibcite{L}{6}
\bibcite{Lu}{7}
\bibcite{R}{8}
\bibcite{Ro1984}{9}
\bibcite{Ru}{10}
\bibcite{S}{11}
\bibcite{St}{12}
\bibcite{W.St2}{13}