History of Krawtchouk polynomials $K_n(x;p,N)$ in the Askey normalisation

 - Kravchuk polynomials - Askey scheme-Definition: For rational $0<p<1$, integer $N\geq1$ and $0\leq n\leq N$, $K_n(x;p,N)$-  is the degree-$n$ Krawtchouk polynomial in the Askey normalisation, defined by the-  terminating hypergeometric sum CITE{formula-hypergeometric} of DLMF CITE{DLMFExplicit}.+Definition: For rational $0<p<1$, integer $N\geq1$ and integer $0\leq n\leq N$, the+  Krawtchouk polynomial is $K_n(x;p,N)=\sum_{j=0}^{n}\frac{(-n)_j(-x)_j}{(-N)_j\,j!}p^{-j}$,+  the terminating ${}_2F_1(-n,-x;-N;1/p)$ of DLMF CITE{DLMFExplicit}. Parameters:   p:

   formula-hypergeometric: $K_n(x;p,N)={}_2F_1\left(\begin{matrix}-n,-x\\-N\end{matrix};p^{-1}\right)=\sum_{j=0}^{n}\frac{(-n)_j(-x)_j}{(-N)_j     j!}p^{-j}$ CITE{DLMFExplicit}.-  formula-generating-function: $\sum_{n=0}^{N}\binom{N}{n}K_n(x;p,N)t^n=(1-\frac{1-p}{p}t)^x(1+t)^{N-x}$+  formula-generating-function: For $x=0,1,\ldots,N$, $\sum_{n=0}^{N}\binom{N}{n}K_n(x;p,N)t^n=(1-\frac{1-p}{p}t)^x(1+t)^{N-x}$     CITE{DLMFGenerating}.   formula-orthogonality: $\sum_{x=0}^{N}\binom{N}{x}p^x(1-p)^{N-x}K_m(x;p,N)K_n(x;p,N)=\delta_{mn}(\frac{1-p}{p})^n/\binom{N}{n}$

   formula-special-value: $K_n(0;p,N)=1$ for every $0\leq n\leq N$.   formula-duality: For integers $0\leq r,n\leq N$, $K_n(r;p,N)=K_r(n;p,N)$.+  formula-reflection: $K_n(N-x;p,N)=\left(-\frac{1-p}{p}\right)^nK_n(x;1-p,N)$. Similar tables: - table: HREF{Krawtchouk_polynomials_of_the_Hamming_scheme}[Krawtchouk polynomials     of the Hamming scheme]   relation: are the coding-theoretic rescaling $\mathcal{K}_n(x;N,q)=\binom{N}{n}(q-1)^nK_n(x;1-1/q,N)$+- table: HREF{Hahn_polynomials}[Hahn polynomials]+  relation: tend to these, $Q_n(x;pt,(1-p)t,N)\to K_n(x;p,N)$ as $t\to\infty$ CITE{DLMFHahnLimit} - table: HREF{Meixner_polynomials}[Meixner polynomials]   relation: satisfy $K_n(x;p,N)=M_n(x;-N,p/(p-1))$, a Meixner specialisation outside     the parameter range $\beta>0$, $0<c<1$ CITE{DLMFExplicit}+- table: HREF{Charlier_polynomials}[Charlier polynomials]+  relation: are the limit $K_n(x;a/N,N)\to C_n(x;a)$ as $N\to\infty$ CITE{DLMFCharlierLimit} Tags: - polynomial

     title: 'DLMF 18.23.3: Krawtchouk generating function'     url: https://dlmf.nist.gov/18.23.E3+  DLMFHahnLimit:+    title: 'DLMF 18.21.3: Hahn to Krawtchouk limit'+    url: https://dlmf.nist.gov/18.21.E3+  DLMFCharlierLimit:+    title: 'DLMF 18.21.6: Krawtchouk to Charlier limit'+    url: https://dlmf.nist.gov/18.21.E6   Wiki:     title: 'Wikipedia: Krawtchouk polynomials'

   complete: 'no'   complete-note: it holds $p\in\{\tfrac14,\tfrac13,\tfrac12,\tfrac23,\tfrac34\}$,-    every support endpoint $1\leq N\leq16$ and every degree $0\leq n\leq N$, a symmetric-    set of common binomial probabilities with short support sizes+    every support endpoint $1\leq N\leq16$ and every degree $0\leq n\leq N$; the probabilities+    $p=\tfrac12,\tfrac23,\tfrac34$ are the Hamming-scheme cases with $q=2,3,4$, and+    the chosen probabilities are closed under $p\mapsto1-p$   rigour details: The generator computes the terminating hypergeometric sum in Sage's     rational polynomial ring $\mathbb{Q}[x]$, with every division made in $\mathbb{Q}$.

     rescaling, the generating function, exact orthogonality for the binomial weight     including the norm, the value $K_n(0;p,N)=1$, self-duality on the finite support,-    and the leading coefficient. The dry run measured 760 entries, longest written-    entry 549 characters, and an entries block of 112.4 KB.+    and the leading coefficient. Display properties:   number-header: $K_n(x;p,N)$