§18.20 Hahn Class: Explicit Representations

§18.20(i) Rodrigues Formulas

For comments on the use of the forward-difference operator $\Delta_{x}$, the backward-difference operator $\nabla_{x}$, and the central-difference operator $\delta_{x}$, see §18.2(ii).

Hahn, Krawtchouk, Meixner, and Charlier

 18.20.1 $p_{n}(x)=\frac{1}{\kappa_{n}w_{x}}\nabla_{x}^{n}\left(w_{x}\prod_{\ell=0}^{n-1% }F(x+\ell)\right),$ $x\in X$.

In (18.20.1) $X$ and $w_{x}$ are as in Table 18.19.1. For the Hahn polynomials $p_{n}(x)=Q_{n}\left(x;\alpha,\beta,N\right)$ and

 18.20.2 $\displaystyle F(x)$ $\displaystyle=(x+\alpha+1)(x-N),$ $\displaystyle\kappa_{n}$ $\displaystyle={\left(-N\right)_{n}}{\left(\alpha+1\right)_{n}}.$ ⓘ Defines: $F(x)$: coefficient (locally) and $\kappa_{n}$: coefficient (locally) Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer, $N$: positive integer and $x$: real variable Referenced by: §18.20(i) Permalink: http://dlmf.nist.gov/18.20.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 18.20(i), 18.20(i), 18.20 and 18

For the Krawtchouk, Meixner, and Charlier polynomials, $F(x)$ and $\kappa_{n}$ are as in Table 18.20.1.

Continuous Hahn

 18.20.3 $w(x;a,b,\overline{a},\overline{b})p_{n}\left(x;a,b,\overline{a},\overline{b}% \right)=\frac{1}{n!}\delta_{x}^{n}\left(w(x;a+\tfrac{1}{2}n,b+\tfrac{1}{2}n,% \overline{a}+\tfrac{1}{2}n,\overline{b}+\tfrac{1}{2}n)\right).$

Meixner–Pollaczek

 18.20.4 $w^{(\lambda)}(x;\phi)P^{(\lambda)}_{n}\left(x;\phi\right)=\frac{1}{n!}\delta_{% x}^{n}\left(w^{(\lambda+\frac{1}{2}n)}(x;\phi)\right).$

§18.20(ii) Hypergeometric Function and Generalized Hypergeometric Functions

For the definition of hypergeometric and generalized hypergeometric functions see §16.2.

 18.20.5 $Q_{n}\left(x;\alpha,\beta,N\right)={{}_{3}F_{2}}\left({-n,n+\alpha+\beta+1,-x% \atop\alpha+1,-N};1\right),$ $n=0,1,\dots,N$.
 18.20.6 $\displaystyle K_{n}\left(x;p,N\right)$ $\displaystyle={{}_{2}F_{1}}\left({-n,-x\atop-N};p^{-1}\right),$ $n=0,1,\dots,N$. 18.20.7 $\displaystyle M_{n}\left(x;\beta,c\right)$ $\displaystyle={{}_{2}F_{1}}\left({-n,-x\atop\beta};1-c^{-1}\right).$ 18.20.8 $\displaystyle C_{n}\left(x;a\right)$ $\displaystyle={{}_{2}F_{0}}\left({-n,-x\atop-};-a^{-1}\right).$
 18.20.9 $p_{n}\left(x;a,b,\overline{a},\overline{b}\right)=\frac{{\mathrm{i}^{n}}{\left% (a+\overline{a}\right)_{n}}{\left(a+\overline{b}\right)_{n}}}{n!}\*{{}_{3}F_{2% }}\left({-n,n+2\Re(a+b)-1,a+\mathrm{i}x\atop a+\overline{a},a+\overline{b}};1% \right).$

(For symmetry properties of $p_{n}\left(x;a,b,\overline{a},\overline{b}\right)$ with respect to $a$, $b$, $\overline{a}$, $\overline{b}$ see Andrews et al. (1999, Corollary 3.3.4).)

 18.20.10 $P^{(\lambda)}_{n}\left(x;\phi\right)=\frac{{\left(2\lambda\right)_{n}}}{n!}e^{% \mathrm{i}n\phi}\*{{}_{2}F_{1}}\left({-n,\lambda+\mathrm{i}x\atop 2\lambda};1-% e^{-2\mathrm{i}\phi}\right).$