# Krawtchouk polynomials

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##### 1: 18.24 Hahn Class: Asymptotic Approximations
With $x=\lambda N$ and $\nu=n/N$, Li and Wong (2000) gives an asymptotic expansion for $K_{n}\left(x;p,N\right)$ as $n\to\infty$, that holds uniformly for $\lambda$ and $\nu$ in compact subintervals of $(0,1)$. … With $\mu=N/n$ and $x$ fixed, Qiu and Wong (2004) gives an asymptotic expansion for $K_{n}\left(x;p,N\right)$ as $n\to\infty$, that holds uniformly for $\mu\in[1,\infty)$. …Asymptotic approximations are also provided for the zeros of $K_{n}\left(x;p,N\right)$ in various cases depending on the values of $p$ and $\mu$. … Similar approximations are included for Jacobi, Krawtchouk, and Meixner polynomials.
##### 2: 18.21 Hahn Class: Interrelations
$K_{n}\left(x;p,N\right)=K_{x}\left(n;p,N\right),$ $n,x=0,1,\dots,N$.
18.21.3 $\lim_{t\to\infty}Q_{n}\left(x;pt,(1-p)t,N\right)=K_{n}\left(x;p,N\right).$
##### 3: 18.19 Hahn Class: Definitions
###### Hahn, Krawtchouk, Meixner, and Charlier
Tables 18.19.1 and 18.19.2 provide definitions via orthogonality and normalization (§§18.2(i), 18.2(iii)) for the Hahn polynomials $Q_{n}\left(x;\alpha,\beta,N\right)$, Krawtchouk polynomials $K_{n}\left(x;p,N\right)$, Meixner polynomials $M_{n}\left(x;\beta,c\right)$, and Charlier polynomials $C_{n}\left(x;a\right)$.
##### 4: 18.22 Hahn Class: Recurrence Relations and Differences
18.22.21 $\Delta_{x}K_{n}\left(x;p,N\right)=-\frac{n}{pN}K_{n-1}\left(x;p,N-1\right),$
18.22.22 $\nabla_{x}\left(\genfrac{(}{)}{0.0pt}{}{N}{x}p^{x}(1-p)^{N-x}K_{n}\left(x;p,N% \right)\right)=\genfrac{(}{)}{0.0pt}{}{N+1}{x}p^{x}{(1-p)^{N-x}}K_{n+1}\left(x% ;p,N+1\right).$
##### 5: 18.38 Mathematical Applications
###### Coding Theory
For applications of Krawtchouk polynomials $K_{n}\left(x;p,N\right)$ and $q$-Racah polynomials $R_{n}\left(x;\alpha,\beta,\gamma,\delta\,|\,q\right)$ to coding theory see Bannai (1990, pp. 38–43), Leonard (1982), and Chihara (1987).
##### 6: 18.20 Hahn Class: Explicit Representations
For the Krawtchouk, Meixner, and Charlier polynomials, $F(x)$ and $\kappa_{n}$ are as in Table 18.20.1.
18.20.6 $K_{n}\left(x;p,N\right)={{}_{2}F_{1}}\left({-n,-x\atop-N};p^{-1}\right),$ $n=0,1,\dots,N$.
##### 7: Bibliography Q
• W. Qiu and R. Wong (2004) Asymptotic expansion of the Krawtchouk polynomials and their zeros. Comput. Methods Funct. Theory 4 (1), pp. 189–226.
• ##### 8: 18.23 Hahn Class: Generating Functions
18.23.3 $\left(1-\frac{1-p}{p}z\right)^{x}(1+z)^{N-x}=\sum_{n=0}^{N}\genfrac{(}{)}{0.0% pt}{}{N}{n}K_{n}\left(x;p,N\right)z^{n},$ $x=0,1,\dots,N$.
##### 9: 18.1 Notation
$\left(z_{1},\dots,z_{k};q\right)_{\infty}=\left(z_{1};q\right)_{\infty}\cdots% \left(z_{k};q\right)_{\infty}.$
• Krawtchouk: $K_{n}\left(x;p,N\right)$.

• ##### 10: 18.26 Wilson Class: Continued
18.26.11 $\lim_{t\to\infty}R_{n}\left(x(x+t+1);pt,(1-p)t,N\right)=K_{n}\left(x;p,N\right).$