%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Save this file as 1929_e.tex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass[12pt,a4paper]{article}
\begin{document}
\newcommand{\binom}[2]{{#1 \choose #2}}
\begin{center}
{\large {\bf 1929 \\[0pt]
\vspace {15pt} {\sc deuxieme semestre} \\[0pt]
\vspace {30pt} {\huge COMPTES RENDUS} \\[0pt]
\vspace {10pt} {\sc hebdomadaires} \\[0pt]
\vspace {10pt} {\Large DES S\'{E}ANCES \\[0pt]
\vspace {10pt} DE L'ACAD\'{E}MIE DES SCIENCES} \\[0pt]
\vspace {10pt} {\sc par mm. les secr\'{e}taires perp\'{e}tuels} \\[0pt]
\vspace {20pt} TOME 189 \\[0pt]
\vspace {20pt} {\huge N ${}^{\mbox {o}} $. 17 (21 Octobre 1929).} \\[0pt]
\vspace {30pt} PARIS}}
\end{center}
\newpage
\noindent ANALYSE MATH\'{E}MATIQUE. - {\it Sur une g\'{e}n\'{e}ralisation des
polynomes} {\it d'Hermite.} Note de {\bf M.Krawtchouk}, transmise par M.\,
\'{E}mile Borel.\vspace{1cm}
\noindent M. Krawtchouk. C.R.Acad. Sci. 1929. T.189, No.17. P.620 - 622.
\vspace {2cm}
MATHEMATICAL ANALYSIS. - {\it On one generalization of Hermite polynomials.}
Note \footnote{%
Submitted on September 23, 1929.} by{\bf \ M.\, Krawtchouk},
submitted by M.\, \, \'{E}mile Borel.\vspace{1cm}.\noindent
\vspace{1cm}
Let's assume that the polynomial $\psi _{m}(x)$ of the $m^{\mbox{th}}$ degree
is defined by the following equalities
\begin{equation}
\left\{
\begin{array}{c}
\sum_{i=0}^{u-1}p_{i}\psi _{l}(x_{i})\psi _{m}(x_{i})=0(l\neq m),\quad
=1\;(l=m) \\
\left( x_{i+1}-x_{i}=1,\quad p_{i}\geq o,\quad
\sum_{i=0}^{u-1}p_{i}=1\right) .
\end{array}
\right. \label{1}
\end{equation}
Then we have
\[
\left\{
\begin{array}{c}
x\psi _{m}(x)=m_{-1}\psi _{m-1}(x)+m_{0}\psi _{m}(x)+m_{1}\psi _{m+1}(x)%
\label{2} \\
\left[ m_{j}=\sum_{i=0}^{u-1}p_{i}x_{i}\psi _{m}(x_{i})\psi _{m+j}(x_{i})%
\right]
\end{array}
\right.
\]
and the minimum of the expression
\begin{equation}
J_{k}(T_{0},T_{1},\ldots ,T_{k-1})=\sum_{i=0}^{u-1}p_{i}[T_{0}\psi
_{0}(x_{i})+\ldots +T_{k-1}\psi _{k-1}(x_{i})-y(x_{i})]^{2}\quad (k\leq u)%
\label{3}
\end{equation}
is equal to
\[
J_{k}(A_{0},A_{1},\ldots
,A_{k-1})=\sum_{i=0}^{u-1}p_{i}y^{2}(x_{i})-A_{0}^{2}-\ldots -A_{N-1}^{2}
\]
where
\begin{equation}
A_{m}=\sum_{i=0}^{u-1}p_{i}y(x_{i})\psi _{m}(x_{i}).\label{4}
\end{equation}
In the important case $p_{0}=p_{1}=\ldots =p_{u-1}$ which has been studied
by P.\, Tchebycheff the polynomials $\psi _{m}$ represent the
generalization of the Legendre polynomials. We want to consider other
important case where
\begin{equation}
p_{i}=P(i,u;p,q)=\binom{u-1}{i}p^{i}q^{u-1-i},\quad x_{i}=i\quad
(p>0,\;q>0,\;p+q=1).\label{5}
\end{equation}
It is possible to prove that the functions $\psi _{m}$ have according to
this hypothesis the following simple form:
\begin{eqnarray}
\varphi _{m}(x,u;p,q) &=&\sqrt{\binom{u-1}{m}(pq)^{m}}\;\Delta
^{m}P(x-m,u-m;p,q):P(x,u;p,q)\label{6} \\
&=&\sqrt{\binom{u-1}{m}^{-1}(pq)^{-m}}\sum_{i=o}^{m}(-1)^{i}\binom{u-x-1}{m-i%
}\binom{x}{i}p^{m-i}q^{i} \nonumber
\end{eqnarray}
concerning the limiting cases of the polynomials
\begin{equation}
{\rm const.}\frac{x!}{a^{x}}\Delta ^{m}\left[ \frac{a^{x-m}}{(x-m)!}\right]
\quad (u\rightarrow \infty),\;p(u-1)=a={\rm const.})\label{7}
\end{equation}
and the Hermite polynomials
\begin{equation}
{\rm const.}e^{t^{2}}\frac{d^{m}}{dt^{m}}\left( e^{-t^{2}}\right) \quad %
\left[ u\rightarrow \infty ,\;x=p(u-1)+t\sqrt{2pq(u-1)}\right] . \label{8}
\end{equation}
In accordance with (\ref{5}) the formulas (\ref{2}) and (\ref{4}) are
conversed respectively to
\begin{eqnarray}
&&\!\!\sqrt{(m+1)(u-m-1)pq}\,\varphi _{m+1}(x) \nonumber \\
&=&\left[ p(u-1)+(q-p)m-x\right] \varphi _{m}(x)-\sqrt{m(u-m)pq}\varphi
_{m-1}(x), \nonumber \\
A_{m} &=&\sqrt{\binom{u-1}{m}(pq)^{m}}\sum_{i=0}^{u-1}y(x_{i})\Delta
^{m}P(x_{i}-m,\;u-m;\;p,q)\label{9} \\
&=&(-1)^{m}\sqrt{\binom{u-1}{m}(pq)^{m}}\sum_{i=0}^{u-1}P(i-m,u-m;p,q)\Delta
^{m}y(i-m), \nonumber
\end{eqnarray}
{\it Appendices}. --- 1. Calculation {\it of the incomplete generalized
moments}
\[
R_{m}(x)=\sum_{i=0}^{x-1}P(i,u;p,q)\varphi _{m}(i,u;p,q)\quad (m=1,2,\ldots
,k)
\]
of function (\ref{5}) gives
\begin{equation}
R_{m}(x)=\sqrt{\binom{u-1}{m}(pq)^{m}}\Delta ^{m-1}P(x-m,u-m;p,q)\quad
(m=1,2,\ldots ,k)\label{10}
\end{equation}
As to $k^{\mbox{th}}$ incomplete factorial moment
\[
\rho _{k}(x)=\sum_{i=0}^{x-1}P(i,u;p,q)\binom{p\left( u-1\right) -x}{k}\quad
(k=0,1,2,\ldots ),
\]
it is equal to the linear combination of expressions (\ref{10}) and moment $%
R_{0}(x)$ [the result is trivial in a limiting case (\ref{8})]. Equality by
M.R.\, Frisch\footnote{%
See {\sc Ch.\, Jordan}, {\it Statistique math.}, 1927, p. 85}
follows from the formula (\ref{10}) as a special case at $(m=1)$.
- 2. It is necessary to note the following expansion
\begin{eqnarray}
P(x,u_{1};p_{1},q_{1}) &=&P(x,u;p,q)\sum_{m=0}^{u-1}\sqrt{\frac{(u-m-1)!}{%
m!(u-1)!}(pq)^{-m}}\varphi _{m}(x,u;p,q) \nonumber \\
&&\times \sum_{i=0}^{m}(-1)^{m-i}\binom{m}{i}p^{m-i}q^{i}\left[ \frac{{\it d}%
^{m}[s^{u-m}(p_{1}t+q_{1}s)^{u_{1}}]}{{\it d}t^{i}{\it d}s^{m-i}}\right]
_{s,t=1} \nonumber \\
(u &\geq &u_{1}), \nonumber
\end{eqnarray}
with the limiting cases which correspond to the polynomials (\ref{7}) and (%
\ref{8}) and are well known.
\end{document}
