{\catcode`\^^M=\active % \gdef\adrbox{\catcode`\^^M=\active % \def^^M{\egroup\hbox\bgroup}}} \def\adresse{\par\nobreak% \vskip 24pt plus 6pt minus 3pt% \hbox to \hsize\bgroup\hfill% \def\fin{\egroup\egroup\hskip2em\egroup\vfill\eject}% \adrbox\vbox\bgroup\hbox\bgroup} \def\fin{\vfill\eject} \catcode'32=9 \magnification=1200 % particularite du pilote canon \voffset=1cm \hoffset=0cm %\hoffset=1cm \font\tenpc=cmcsc10 %\font\eightpc=cmcsc8 % Charge des fontes de 8 et 6 points : \font\eightrm=cmr8 \font\eighti=cmmi8 \font\eightsy=cmsy8 \font\eightbf=cmbx8 \font\eighttt=cmtt8 \font\eightit=cmti8 \font\eightsl=cmsl8 \font\sixrm=cmr6 \font\sixi=cmmi6 \font\sixsy=cmsy6 \font\sixbf=cmbx6 \skewchar\eighti='177 \skewchar\sixi='177 \skewchar\eightsy='60 \skewchar\sixsy='60 % Chargement des fontes AMS \font\tengoth=eufm10 \font\tenbboard=msbm10 \font\eightgoth=eufm8 \font\eightbboard=msbm8 \font\sevengoth=eufm7 \font\sevenbboard=msbm7 \font\sixgoth=eufm6 \font\fivegoth=eufm5 \newfam\gothfam \newfam\bboardfam \catcode`\@=11 \def\raggedbottom{\topskip 10pt plus 36pt \r@ggedbottomtrue} \def\pc#1#2|{{\bigf@ntpc #1\penalty \@MM\hskip\z@skip\smallf@ntpc #2}} \def\tenpoint{% \textfont0=\tenrm \scriptfont0=\sevenrm \scriptscriptfont0=\fiverm \def\rm{\fam\z@\tenrm}% \textfont1=\teni \scriptfont1=\seveni \scriptscriptfont1=\fivei \def\oldstyle{\fam\@ne\teni}% \textfont2=\tensy \scriptfont2=\sevensy \scriptscriptfont2=\fivesy \textfont\gothfam=\tengoth \scriptfont\gothfam=\sevengoth \scriptscriptfont\gothfam=\fivegoth \def\goth{\fam\gothfam\tengoth}% \textfont\bboardfam=\tenbboard \scriptfont\bboardfam=\sevenbboard \scriptscriptfont\bboardfam=\sevenbboard \def\bboard{\fam\bboardfam}% \textfont\itfam=\tenit \def\it{\fam\itfam\tenit}% \textfont\slfam=\tensl \def\sl{\fam\slfam\tensl}% \textfont\bffam=\tenbf \scriptfont\bffam=\sevenbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\tenbf}% \textfont\ttfam=\tentt \def\tt{\fam\ttfam\tentt}% \abovedisplayskip=12pt plus 3pt minus 9pt \abovedisplayshortskip=0pt plus 3pt \belowdisplayskip=12pt plus 3pt minus 9pt \belowdisplayshortskip=7pt plus 3pt minus 4pt \smallskipamount=3pt plus 1pt minus 1pt \medskipamount=6pt plus 2pt minus 2pt \bigskipamount=12pt plus 4pt minus 4pt \normalbaselineskip=12pt \setbox\strutbox=\hbox{\vrule height8.5pt depth3.5pt width0pt}% \let\bigf@ntpc=\tenrm \let\smallf@ntpc=\sevenrm \let\petcap=\tenpc \normalbaselines\rm} \def\eightpoint{% \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm \def\rm{\fam\z@\eightrm}% \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei \def\oldstyle{\fam\@ne\eighti}% \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy \textfont\gothfam=\eightgoth \scriptfont\gothfam=\sixgoth \scriptscriptfont\gothfam=\fivegoth \def\goth{\fam\gothfam\eightgoth}% \textfont\bboardfam=\eightbboard \scriptfont\bboardfam=\sevenbboard \scriptscriptfont\bboardfam=\sevenbboard \def\bboard{\fam\bboardfam}% \textfont\itfam=\eightit \def\it{\fam\itfam\eightit}% \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}% \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}% \textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}% \abovedisplayskip=9pt plus 2pt minus 6pt \abovedisplayshortskip=0pt plus 2pt \belowdisplayskip=9pt plus 2pt minus 6pt \belowdisplayshortskip=5pt plus 2pt minus 3pt \smallskipamount=2pt plus 1pt minus 1pt \medskipamount=4pt plus 2pt minus 1pt \bigskipamount=9pt plus 3pt minus 3pt \normalbaselineskip=9pt \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}% \let\bigf@ntpc=\eightrm \let\smallf@ntpc=\sixrm \normalbaselines\rm} \let\bb=\bboard \tenpoint \catcode`\;=\active \def;{\relax\ifhmode\ifdim\lastskip>\z@ \unskip\fi\kern\fontdimen2 \font\kern -1.2 \fontdimen3 \font\fi\string;} \catcode`\:=\active \def:{\relax\ifhmode\ifdim\lastskip>\z@\unskip\fi\penalty\@M\ \fi\string:} \catcode`\!=\active \def!{\relax\ifhmode\ifdim\lastskip>\z@ \unskip\fi\kern\fontdimen2 \font \kern -1.1 \fontdimen3 \font\fi\string!} \catcode`\?=\active \def?{\relax\ifhmode\ifdim\lastskip>\z@ \unskip\fi\kern\fontdimen2 \font \kern -1.1 \fontdimen3 \font\fi\string?} \def\^#1{\if#1i{\accent"5E\i}\else{\accent"5E #1}\fi} \def\"#1{\if#1i{\accent"7F\i}\else{\accent"7F #1}\fi} \frenchspacing \newif\ifpagetitre \newtoks\auteurcourant \auteurcourant={\hfil} \newtoks\titrecourant \titrecourant={\hfil} \def\appeln@te{} \def\vfootnote#1{\def\@parameter{#1}\insert\footins\bgroup\eightpoint \interlinepenalty\interfootnotelinepenalty \splittopskip\ht\strutbox % top baseline for broken footnotes \splitmaxdepth\dp\strutbox \floatingpenalty\@MM \leftskip\z@skip \rightskip\z@skip \ifx\appeln@te\@parameter\indent \else{\noindent #1\ }\fi \footstrut\futurelet\next\fo@t} \pretolerance=500 \tolerance=1000 \brokenpenalty=5000 \newdimen\hmargehaute \hmargehaute=0cm \newdimen\lpage \lpage=13.3cm \newdimen\hpage \hpage=20cm \newdimen\lmargeext \lmargeext=1cm \hsize=11.25cm \vsize=18cm \parskip 0pt \parindent=12pt \def\margehaute{\vbox to \hmargehaute{\vss}}% \def\margebasse{\vss} \output{\shipout\vbox to \hpage{\margehaute\nointerlineskip \corpsdepage\margebasse} \advancepageno \global\pagetitrefalse \ifnum\outputpenalty>-20000 \else\dosupereject\fi} \def\corpsdepage{\hbox to \lpage{\hss\pagetexte\hskip\lmargeext}} \def\pagetexte{\vbox{\makeheadline\pagebody\makefootline}} \headline={\ifpagetitre\titleheadline \else \ifodd\pageno\rightheadline \else\leftheadline\fi\fi} \def\leftheadline{\eightpoint\hfil\the\auteurcourant\hfil} \def\rightheadline{\eightpoint\hfil\the\titrecourant\hfil} \def\titleheadline{\hfill} \pagetitretrue \def\footnoterule{\kern-6\p@ \hrule width 2truein \kern 5.6\p@} % the \hrule is .4pt high \let\rmpc=\sevenrm \def\pd#1#2 {\pc#1#2| } \def\pointir{\discretionary{.}{}{.\kern.35em---\kern.7em}\nobreak \hskip 0em plus .3em minus .4em } \def\resume#1{\vbox{\eightpoint \pc R\'ESUM\'E|\pointir #1}} \def\abstract#1{\vbox{\eightpoint \pc ABSTRACT|\pointir #1}} \def\titre#1|{\message{#1} \par\vskip 30pt plus 24pt minus 3pt\penalty -1000 \vskip 0pt plus -24pt minus 3pt\penalty -1000 \centerline{\bf #1} \vskip 5pt \penalty 10000 } \def\section#1|{\par\vskip .3cm {\bf #1}\pointir} \def\ssection#1|{\par\vskip .2cm {\it #1}\pointir} \long\def\th#1|#2\finth{\par\medskip {\petcap #1\pointir}{\it #2}\par\smallskip} \long\def\tha#1|#2\fintha{\par\medskip {\petcap #1.}\par\nobreak{\it #2}\par\smallskip} \def\cf{{\it cf}} \def\rem#1|{\par\medskip {{\it #1}\pointir}} \def\rema#1|{\par\medskip {{\it #1.}\par\nobreak }} % \def\ieme{\raise 1ex\hbox{\pc{}i\`eme|}} \def\omini{\raise 1ex\hbox{\pc{}o|}} \def\emini{\raise 1ex\hbox{\pc{}e|}} \def\ermini{\raise 1ex\hbox{\pc{}er|}} \def\remini{\raise 1ex\hbox{\pc{}re|}} %reference pour un article : \def\article#1|#2|#3|#4|#5|#6|#7| {{\leftskip=7mm\noindent \hangindent=2mm\hangafter=1 \llap{[#1]\hskip.35em}{#2}\pointir #3, {\sl #4}, t.\nobreak\ {\bf #5}, {\oldstyle #6}, p.\nobreak\ #7.\par}} %reference pour un livre : \def\livre#1|#2|#3|#4| {{\leftskip=7mm\noindent \hangindent=2mm\hangafter=1 \llap{[#1]\hskip.35em}{#2}\pointir {\sl #3}\pointir #4.\par}} %reference complementaire : \def\divers#1|#2|#3| {{\leftskip=7mm\noindent \hangindent=2mm\hangafter=1 \llap{[#1]\hskip.35em}{#2}\pointir #3.\par}} % \mathchardef\conj="0365 \def\dem{\par{\it D\'emonstration}\pointir} \def\qed{\quad\raise -2pt\hbox{\vrule\vbox to 10pt{\hrule width 4pt \vfill\hrule}\vrule}} \def\virg{\raise 2pt\hbox{,}} % virgule apr\`es une fraction \def\cqfd{\penalty 500 \hbox{\qed}\par\smallskip} \def\decale#1|{\par\noindent\hskip 28pt\llap{#1}\kern 5pt} \def\decaledecale#1|{\par\noindent\hskip 34pt\llap{#1}\kern 5pt} % pour les titres en deux lignes et les sections sans point-tiret : \def\titrea#1|#2|{\message{#1 #2} \par\vskip.5cm plus .1cm minus .1cm\penalty -1000 \centerline{\bf #1} \centerline{\bf #2} \vskip 5pt \penalty 10000 } \def\sectiona#1|{\par\vskip .3cm {\bf #1.} \par\nobreak\vskip 3pt } \def\ssectiona#1|{\par\vskip .2cm {\it #1.} \par\nobreak\vskip 2pt } \catcode`\@=12 %end format %some macros from LaTeX \def\grille{\noalign{\nointerlineskip\Grille\nointerlineskip}} %\to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to %\def\fleche(#1,#2)\dir(#3,#4)\long#5{% %\noalign{\nointerlineskip\leftput(#1,#2){\vector(#3,#4){#5}}\nointerlineskip}} %\to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \to \def\diagram#1{\def\normalbaselines{\baselineskip=0pt\lineskip=5pt} \matrix{#1}} \def\hfl#1#2#3{\smash{\mathop{\hbox to#3{\rightarrowfill}}\limits ^{\scriptstyle#1}_{\scriptstyle#2}}} \def\gfl#1#2#3{\smash{\mathop{\hbox to#3{\leftarrowfill}}\limits ^{\scriptstyle#1}_{\scriptstyle#2}}} \def\vfl#1#2#3{\llap{$\scriptstyle #1$} \left\downarrow\vbox to#3{}\right.\rlap{$\scriptstyle #2$}} \message{`lline' & `vector' macros from LaTeX} \catcode`@=11 \def\{{\relax\ifmmode\lbrace\else$\lbrace$\fi} \def\}{\relax\ifmmode\rbrace\else$\rbrace$\fi} \def\newcount{\alloc@0\count\countdef\insc@unt} \def\newdimen{\alloc@1\dimen\dimendef\insc@unt} \def\newwrite{\alloc@7\write\chardef\sixt@@n} \newwrite\@unused \newcount\@tempcnta \newcount\@tempcntb \newdimen\@tempdima \newdimen\@tempdimb \newbox\@tempboxa \def\@spaces{\space\space\space\space} \def\@whilenoop#1{} \def\@whiledim#1\do #2{\ifdim #1\relax#2\@iwhiledim{#1\relax#2}\fi} \def\@iwhiledim#1{\ifdim #1\let\@nextwhile=\@iwhiledim \else\let\@nextwhile=\@whilenoop\fi\@nextwhile{#1}} \def\@badlinearg{\@latexerr{Bad \string\line\space or \string\vector \space argument}} \def\@latexerr#1#2{\begingroup \edef\@tempc{#2}\expandafter\errhelp\expandafter{\@tempc}% %% error help message pieces. \def\@eha{Your command was ignored. ^^JType \space I \space to replace it with another command,^^Jor \space \space to continue without it.} \def\@ehb{You've lost some text. \space \@ehc} \def\@ehc{Try typing \space \space to proceed.^^JIf that doesn't work, type \space X \space to quit.} \def\@ehd{You're in trouble here. \space\@ehc} \typeout{LaTeX error. \space See LaTeX manual for explanation.^^J \space\@spaces\@spaces\@spaces Type \space H \space for immediate help.}\errmessage{#1}\endgroup} \def\typeout#1{{\let\protect\string\immediate\write\@unused{#1}}} % line & circle fonts \font\tenln = line10 \font\tenlnw = linew10 %\font\tencirc = circle10 %\font\tencircw = circlew10 \newdimen\@wholewidth \newdimen\@halfwidth \newdimen\unitlength \unitlength =1pt %\newbox\@picbox %\newdimen\@picht \def\thinlines{\let\@linefnt\tenln \let\@circlefnt\tencirc \@wholewidth\fontdimen8\tenln \@halfwidth .5\@wholewidth} \def\thicklines{\let\@linefnt\tenlnw \let\@circlefnt\tencircw \@wholewidth\fontdimen8\tenlnw \@halfwidth .5\@wholewidth} \def\linethickness#1{\@wholewidth #1\relax \@halfwidth .5\@wholewidth} \newif\if@negarg \def\lline(#1,#2)#3{\@xarg #1\relax \@yarg #2\relax \@linelen=#3\unitlength \ifnum\@xarg =0 \@vline \else \ifnum\@yarg =0 \@hline \else \@sline\fi \fi} \def\@sline{\ifnum\@xarg< 0 \@negargtrue \@xarg -\@xarg \@yyarg -\@yarg \else \@negargfalse \@yyarg \@yarg \fi \ifnum \@yyarg >0 \@tempcnta\@yyarg \else \@tempcnta -\@yyarg \fi \ifnum\@tempcnta>6 \@badlinearg\@tempcnta0 \fi \setbox\@linechar\hbox{\@linefnt\@getlinechar(\@xarg,\@yyarg)}% \ifnum \@yarg >0 \let\@upordown\raise \@clnht\z@ \else\let\@upordown\lower \@clnht \ht\@linechar\fi \@clnwd=\wd\@linechar \if@negarg \hskip -\wd\@linechar \def\@tempa{\hskip -2\wd\@linechar}\else \let\@tempa\relax \fi \@whiledim \@clnwd <\@linelen \do {\@upordown\@clnht\copy\@linechar \@tempa \advance\@clnht \ht\@linechar \advance\@clnwd \wd\@linechar}% \advance\@clnht -\ht\@linechar \advance\@clnwd -\wd\@linechar \@tempdima\@linelen\advance\@tempdima -\@clnwd \@tempdimb\@tempdima\advance\@tempdimb -\wd\@linechar \if@negarg \hskip -\@tempdimb \else \hskip \@tempdimb \fi \multiply\@tempdima \@m \@tempcnta \@tempdima \@tempdima \wd\@linechar \divide\@tempcnta \@tempdima \@tempdima \ht\@linechar \multiply\@tempdima \@tempcnta \divide\@tempdima \@m \advance\@clnht \@tempdima \ifdim \@linelen <\wd\@linechar \hskip \wd\@linechar \else\@upordown\@clnht\copy\@linechar\fi} \def\@hline{\ifnum \@xarg <0 \hskip -\@linelen \fi \vrule height \@halfwidth depth \@halfwidth width \@linelen \ifnum \@xarg <0 \hskip -\@linelen \fi} \def\@getlinechar(#1,#2){\@tempcnta#1\relax\multiply\@tempcnta 8 \advance\@tempcnta -9 \ifnum #2>0 \advance\@tempcnta #2\relax\else \advance\@tempcnta -#2\relax\advance\@tempcnta 64 \fi \char\@tempcnta} \def\vector(#1,#2)#3{\@xarg #1\relax \@yarg #2\relax \@linelen=#3\unitlength \ifnum\@xarg =0 \@vvector \else \ifnum\@yarg =0 \@hvector \else \@svector\fi \fi} \def\@hvector{\@hline\hbox to 0pt{\@linefnt \ifnum \@xarg <0 \@getlarrow(1,0)\hss\else \hss\@getrarrow(1,0)\fi}} \def\@vvector{\ifnum \@yarg <0 \@downvector \else \@upvector \fi} \def\@svector{\@sline \@tempcnta\@yarg \ifnum\@tempcnta <0 \@tempcnta=-\@tempcnta\fi \ifnum\@tempcnta <5 \hskip -\wd\@linechar \@upordown\@clnht \hbox{\@linefnt \if@negarg \@getlarrow(\@xarg,\@yyarg) \else \@getrarrow(\@xarg,\@yyarg) \fi}% \else\@badlinearg\fi} \def\@getlarrow(#1,#2){\ifnum #2 =\z@ \@tempcnta='33\else \@tempcnta=#1\relax\multiply\@tempcnta \sixt@@n \advance\@tempcnta -9 \@tempcntb=#2\relax\multiply\@tempcntb \tw@ \ifnum \@tempcntb >0 \advance\@tempcnta \@tempcntb\relax \else\advance\@tempcnta -\@tempcntb\advance\@tempcnta 64 \fi\fi\char\@tempcnta} \def\@getrarrow(#1,#2){\@tempcntb=#2\relax \ifnum\@tempcntb < 0 \@tempcntb=-\@tempcntb\relax\fi \ifcase \@tempcntb\relax \@tempcnta='55 \or \ifnum #1<3 \@tempcnta=#1\relax\multiply\@tempcnta 24 \advance\@tempcnta -6 \else \ifnum #1=3 \@tempcnta=49 \else\@tempcnta=58 \fi\fi\or \ifnum #1<3 \@tempcnta=#1\relax\multiply\@tempcnta 24 \advance\@tempcnta -3 \else \@tempcnta=51\fi\or \@tempcnta=#1\relax\multiply\@tempcnta \sixt@@n \advance\@tempcnta -\tw@ \else \@tempcnta=#1\relax\multiply\@tempcnta \sixt@@n \advance\@tempcnta 7 \fi\ifnum #2<0 \advance\@tempcnta 64 \fi \char\@tempcnta} \def\@vline{\ifnum \@yarg <0 \@downline \else \@upline\fi} \def\@upline{\hbox to \z@{\hskip -\@halfwidth \vrule width \@wholewidth height \@linelen depth \z@\hss}} \def\@downline{\hbox to \z@{\hskip -\@halfwidth \vrule width \@wholewidth height \z@ depth \@linelen \hss}} \def\@upvector{\@upline\setbox\@tempboxa\hbox{\@linefnt\char'66}\raise \@linelen \hbox to\z@{\lower \ht\@tempboxa\box\@tempboxa\hss}} \def\@downvector{\@downline\lower \@linelen \hbox to \z@{\@linefnt\char'77\hss}} %INITIALIZATION \thinlines \newcount\@xarg \newcount\@yarg \newcount\@yyarg \newcount\@multicnt \newdimen\@xdim \newdimen\@ydim \newbox\@linechar \newdimen\@linelen \newdimen\@clnwd \newdimen\@clnht \newdimen\@dashdim \newbox\@dashbox \newcount\@dashcnt \catcode`@=12 % macros supplementaires (J.D.) \newbox\tbox \newbox\tboxa \def\leftzer#1{\setbox\tbox=\hbox to 0pt{#1\hss}% \ht\tbox=0pt \dp\tbox=0pt \box\tbox} \def\rightzer#1{\setbox\tbox=\hbox to 0pt{\hss #1}% \ht\tbox=0pt \dp\tbox=0pt \box\tbox} \def\centerzer#1{\setbox\tbox=\hbox to 0pt{\hss #1\hss}% \ht\tbox=0pt \dp\tbox=0pt \box\tbox} \def\image(#1,#2)#3{\vbox to #1{\offinterlineskip \vss #3 \vskip #2}} \def\leftput(#1,#2)#3{\setbox\tboxa=\hbox{% \kern #1\unitlength \raise #2\unitlength\hbox{\leftzer{#3}}}% \ht\tboxa=0pt \wd\tboxa=0pt \dp\tboxa=0pt\box\tboxa} \def\rightput(#1,#2)#3{\setbox\tboxa=\hbox{% \kern #1\unitlength \raise #2\unitlength\hbox{\rightzer{#3}}}% \ht\tboxa=0pt \wd\tboxa=0pt \dp\tboxa=0pt\box\tboxa} \def\centerput(#1,#2)#3{\setbox\tboxa=\hbox{% \kern #1\unitlength \raise #2\unitlength\hbox{\centerzer{#3}}}% \ht\tboxa=0pt \wd\tboxa=0pt \dp\tboxa=0pt\box\tboxa} \unitlength=1mm \def\cput(#1,#2)#3{\noalign{\nointerlineskip\centerput(#1,#2){#3} \nointerlineskip}} \def\fleche(#1,#2)\dir(#3,#4)\long#5{% {\leftput(#1,#2){\vector(#3,#4){#5}}}} \def\segment(#1,#2)\dir(#3,#4)\long#5{% \leftput(#1,#2){\lline(#3,#4){#5}}} \def\boucle{\vbox{\offinterlineskip \leftput(0,4){\vector(0,-1){3}} \segment(0,0)\dir(-1,1)\long{4} \segment(-4,4)\dir(1,0)\long{4} }} %end of LaTeX macros \def\Per{\mathop{\rm Per}\nolimits} \def\cyc{\mathop{\rm cyc}\nolimits} \def\Inc{\mathop{\rm Inc}\nolimits} \def\card{\mathop{\rm card}\nolimits} \def\cont{\mathop{\rm cont}\nolimits} \auteurcourant={D. FOATA} \titrecourant={POLYN\^OMES DE MEIXNER} \eightpoint \leftline{S\'eminaire Lotharingien de Combinatoire, B06c (1982) 11 pp.} \leftline{[Formerly: Publ. I.R.M.A. Strasbourg, $\oldstyle 1982$, 191/S--05, p. 114--128.]} \tenpoint \vskip 1.5cm \centerline{{\bf COMBINATOIRE DES IDENTIT\'ES SUR}} \vskip 5pt \centerline{{\bf LES POLYN\^OMES DE MEIXNER}} \vskip 2.8mm \centerline{\sevenrm PAR} \vskip 2.8mm \centerline{{\petcap Dominique} FOATA \footnote{$(^*)$}{Publi\'e gr\^ace \`a une subvention du Fonds PCAC pour l'aide et le soutien \`a la recherche, Centre de Recherche de Math\'ematiques Appliqu\'ees, Universit\'e de Montr\'eal, Montr\'eal, P.Q., Canada H3C 3J7.}} \vskip 1cm \resume{Une identit\'e bilin\'eaire prolongeant la formule du noyau de Poisson pour les polyn\^omes de Meixner est obtenue par des m\'ethodes combinatoires. Ces m\^emes m\'ethodes fournissent aussi une extension multilin\'eaire de cette identit\'e.} \bigskip \abstract{A bilinear identity extending the formula of the Poisson kernel for the Meixner polynomials is derived by means of combinatorial methods. The same methods also provide a multilinear extention of that identity.} \bigskip \section 1. Introduction|Les polyn\^omes de Meixner $m_n(x;\beta ,c)$ $(n\ge 0)$ peuvent \^etre d\'efinis par leur fonction g\'en\'eratrice $$\sum _{n\ge 0}m_n(x;\beta ,c) {u^n\over n!} =\Bigl(1-{u\over c}\Bigr)^x(1-u)^{-x-\beta }\leqno(1.1)$$ (\cf. Erd\'elyi et al. [6, vol.~2, pp.~225--226 et vol.~3, pp. 273--274] ou Chihara [4, pp.~175-177].) Notons $(a)_n$ les factorielles montantes $$\displaylines{ (a)_0=1\quad {\rm et}\quad (a)_n=a(a+1)\cdots (a+n-1) \quad (n\ge1)\cr \noalign{\hbox{et}} {}_2F_1(a_1,a_2;b_1;x)=\sum_{n\ge 0} {(a_1)_n\,(a_2)_n\over (b_1)_n}{x^n\over n!}\cr } $$ la s\'erie hyperg\'eom\'etrique usuelle. Les polyn\^omes de Meixner ont encore une repr\'esentation naturelle comme s\'erie ${}_2F_1$. On a, en effet, (\cf.~[6, vol.~2, p.~225]) $$\eqalignno{ m_n(x;\beta ,c)&=(\beta )_n\,{}_2F_1(-n,x;\beta ;1-c^{-1})&(1.2)\cr &=(\beta +x)_n\,{}_2F_1(-n,x;1-\beta -n-x;c^{-1}).\cr} $$ \goodbreak \noindent Askey [1, p. 14 et 2] utilise la notation $M_n(x;\beta ,c)={}_2F_1(-n,x;\beta ;1-c^{-1})$. Dans le pr\'esent article, seule la version $m_n(x;\beta ,c)$ sera utilis\'ee et les formules seront donc transcrites dans cette seule notation. En particulier, la formule dite du {\it noyau de Poisson} [1, p.~15 (3.40W)] s'\'ecrit $$ \displaylines{(1.3)\quad \sum_{n\ge 0} {u^n\over (\beta )_n\, n!}\,m_n(x;\beta ,c)m_n(y;\beta ,c) \hfill\cr \hfill{} =\Bigl(1-{u\over c}\Bigr)^{x+y} (1-u)^{-x-y-\beta } \,{}_2F_1\Bigl(-x,-y;\beta ;{u(c^{-1}-1)\over (1-u/c)^2}\Bigr).\quad\cr} $$ Le but de cet article et de sommer une s\'erie bilin\'eaire plus g\'en\'erale que celle donn\'ee en (1.3) en s'inspirant du mod\`ele fourni par Erd\'elyi~[5] pour les polyn\^omes de Laguerre. On obtient $$\displaylines{(1.4)\quad \sum_{n\ge 0} {u^n\over n!} (\beta )_n\,{}_2F_1(-n,-x;\gamma ;a)\, {}_2F_1(-n,-y;\delta ;b)\hfill\cr \qquad{} =(1-u)^{-\beta} \sum_{r\ge 0} {(\beta )_r\,(-x)_r\,(-y)_r\over (\gamma )_r\,(\delta )_r\,r!}\Bigl({abu\over (1-u)^2}\Bigr)^r\hfill\cr \qquad\quad{} \times {}_2F_1\Bigl(\beta +r;-x+r;\gamma +r;-{au\over 1-u}\Bigr)\, {}_2F_1\Bigl(\beta +r;-y+r;\delta +r;-{bu\over 1-u}\Bigr).\hfill\cr} $$ Lorsque $\beta =\gamma =\delta $ et $a=b=1-c^{-1}$, le premier membre de (1.4) se r\'eduit, moyennant (1.2), au premier membre de (1.3). Quant aux deux fonctions ${}_2F_1$ du second membre, elle se r\'eduisent, par le th\'eor\`eme binomial \`a $((1-u/c)(1-u)^{-1})^{x+r}$ et $((1-u/c)(1-u)^{-1})^{y+r}$, respectivement. On retrouve alors le second membre de (1.3). Ainsi $(1.4)\Rightarrow (1.3)$. Enfin, lorsque $y=0$ et $a=b=1-c^{-1}$, la formule (1.4) donne la fonction g\'en\'eratrice suivante $$(1.5)\ \sum_{n\ge 0} {u^n\over (\gamma )_nn!}\,(\beta )_n\, m_n(x;\gamma ,c) =(1-u)^{-\beta }\,{}_2F_1\Bigl(\beta ,-x;\gamma ;{u(c^{-1}-1)\over 1-u}\Bigr), $$ qui implique \`a son tour (1.1) lorsque $\gamma =\beta $. On peut enfin appliquer le mod\`ele multilin\'eaire d\'evelopp\'e pour les polyn\^omes d'Hermite [7, 8, 12] et de Laguerre [10, 11] pour prolonger (1.4) au cas du produit de plusieurs polyn\^omes de Meixner. La troisi\`eme fonction hyperg\'eom\'etrique d'Appell \`a~$k$ variables, qui s'\'ecrit [3, p.~73] $$ \displaylines{\quad F_3^{(k)}(a_1,\ldots, a_k; b_1,\ldots, b_k;c;u_1, \ldots, u_k)\hfill\cr \hfill {}=\sum_{n_1\ge 0, \ldots, n_k\ge 0} {(a_1)_{n_1}\ldots (a_k)_{n_k} (b_1)_{n_1}\ldots (b_k)_{n_k}\over (c)_{n_1+\cdots+n_k}} {u_1^{n_1}\over n_1!}\cdots {u_k^{n_k}\over n_k!}\hfill\cr} $$ est le principal ingr\'edient de la formule ainsi prolong\'ee. Dans celle-ci (voir (1.6) ci-dessous), les symboles $\sum\limits_{(n_{ij})}$ (resp. $\sum\limits_{(r_{ij})}$) indiquent des sommations sur toutes les suites $(n_{ij})$ (resp. $(r_{ij})$ $(1\le i