# §18.1 Notation

## §18.1(i) Special Notation

(For other notation see Notation for the Special Functions.)

$x,y$ real variables. complex variable. real variable such that $0, unless stated otherwise. nonnegative integers. nonnegative integer, except in §18.30. positive integer. Dirac delta (§1.17). arbitrary small positive constant. polynomial in $x$ of degree $n$. $0$. weight function $(\geq 0)$ on an open interval $(a,b)$. weights $(>0)$ at points $x\in X$ of a finite or countably infinite subset of $\mathbb{R}$. orthogonal polynomials.

### $x$-Differences

Forward differences:

 $\displaystyle\Delta_{x}\left(f(x)\right)$ $\displaystyle=f(x+1)-f(x),$ $\displaystyle\Delta_{x}^{n+1}\left(f(x)\right)$ $\displaystyle=\Delta_{x}\left(\Delta_{x}^{n}(f(x))\right).$

Backward differences:

 $\displaystyle\nabla_{x}\left(f(x)\right)$ $\displaystyle=f(x)-f(x-1),$ $\displaystyle\nabla_{x}^{n+1}\left(f(x)\right)$ $\displaystyle=\nabla_{x}\left(\nabla_{x}^{n}(f(x))\right).$

Central differences in imaginary direction:

 $\displaystyle\delta_{x}\left(f(x)\right)$ $\displaystyle=\left(f(x+\tfrac{1}{2}\mathrm{i})-f(x-\tfrac{1}{2}\mathrm{i})% \right)/\mathrm{i},$ $\displaystyle\delta_{x}^{n+1}\left(f(x)\right)$ $\displaystyle=\delta_{x}\left(\delta_{x}^{n}(f(x))\right).$

### $q$-Pochhammer Symbol

 $\displaystyle\left(z;q\right)_{0}$ $\displaystyle=1,$ $\displaystyle\left(z;q\right)_{n}$ $\displaystyle=(1-z)(1-zq)\cdots(1-zq^{n-1}),$
 $\left(z_{1},\dots,z_{k};q\right)_{n}=\left(z_{1};q\right)_{n}\cdots\left(z_{k}% ;q\right)_{n}.$

### Infinite $q$-Product

 $\displaystyle\left(z;q\right)_{\infty}$ $\displaystyle=\prod_{j=0}^{\infty}(1-zq^{j}),$ $\displaystyle\left(z_{1},\dots,z_{k};q\right)_{\infty}$ $\displaystyle=\left(z_{1};q\right)_{\infty}\cdots\left(z_{k};q\right)_{\infty}.$

## §18.1(ii) Main Functions

The main functions treated in this chapter are:

### Classical OP’s

• Jacobi: $P^{(\alpha,\beta)}_{n}\left(x\right)$.

• Ultraspherical (or Gegenbauer): $C^{(\lambda)}_{n}\left(x\right)$.

• Chebyshev of first, second, third, and fourth kinds: $T_{n}\left(x\right)$, $U_{n}\left(x\right)$, $V_{n}\left(x\right)$, $W_{n}\left(x\right)$.

• Shifted Chebyshev of first and second kinds: $T^{*}_{n}\left(x\right)$, $U^{*}_{n}\left(x\right)$.

• Legendre: $P_{n}\left(x\right)$.

• Shifted Legendre: $P^{*}_{n}\left(x\right)$.

• Laguerre: $L^{(\alpha)}_{n}\left(x\right)$ and $L_{n}\left(x\right)=L^{(0)}_{n}\left(x\right)$. ($L^{(\alpha)}_{n}\left(x\right)$ with $\alpha\neq 0$ is also called Generalized Laguerre.)

• Hermite: $H_{n}\left(x\right)$, $\mathit{He}_{n}\left(x\right)$.

### Hahn Class OP’s

• Hahn: $Q_{n}\left(x;\alpha,\beta,N\right)$.

• Krawtchouk: $K_{n}\left(x;p,N\right)$.

• Meixner: $M_{n}\left(x;\beta,c\right)$.

• Charlier: $C_{n}\left(x;a\right)$.

• Continuous Hahn: $p_{n}\left(x;a,b,\overline{a},\overline{b}\right)$.

• Meixner–Pollaczek: $P^{(\lambda)}_{n}\left(x;\phi\right)$.

### Wilson Class OP’s

• Wilson: $W_{n}\left(x;a,b,c,d\right)$.

• Racah: $R_{n}\left(x;\alpha,\beta,\gamma,\delta\right)$.

• Continuous Dual Hahn: $S_{n}\left(x;a,b,c\right)$.

• Dual Hahn: $R_{n}\left(x;\gamma,\delta,N\right)$.

### $q$-Hahn Class OP’s

• $q$-Hahn: $Q_{n}\left(x;\alpha,\beta,N;q\right)$.

• Big $q$-Jacobi: $P_{n}\left(x;a,b,c;q\right)$.

• Little $q$-Jacobi: $p_{n}\left(x;a,b;q\right)$.

• $q$-Laguerre: $L^{(\alpha)}_{n}\left(x;q\right)$.

• Stieltjes–Wigert: $S_{n}\left(x;q\right)$.

• Discrete $q$-Hermite I: $h_{n}\left(x;q\right)$.

• Discrete $q$-Hermite II: $\tilde{h}_{n}\left(x;q\right)$.

• Askey–Wilson: $p_{n}\left(x;a,b,c,d\,|\,q\right)$.

• Al-Salam–Chihara: $Q_{n}\left(x;a,b\,|\,q\right)$.

• Continuous $q$-Ultraspherical: $C_{n}\left(x;\beta\,|\,q\right)$.

• Continuous $q$-Hermite: $H_{n}\left(x\,|\,q\right)$.

• Continuous $q^{-1}$-Hermite: $h_{n}\left(x\,|\,q\right)$

• $q$-Racah: $R_{n}\left(x;\alpha,\beta,\gamma,\delta\,|\,q\right)$.

### Other OP’s

• Bessel: $y_{n}\left(x;a\right)$.

• Pollaczek: $P^{(\lambda)}_{n}\left(x;a,b\right)$.

### Classical OP’s in Two Variables

• Disk: $R^{(\alpha)}_{m,n}\left(z\right)$.

• Triangle: $P^{\alpha,\beta,\gamma}_{m,n}\left(x,y\right)$.

## §18.1(iii) Other Notations

In Szegő (1975, §4.7) the ultraspherical polynomials $C^{(\lambda)}_{n}\left(x\right)$ are denoted by $P_{n}^{(\lambda)}(x)$. The ultraspherical polynomials will not be considered for $\lambda=0$. They are defined in the literature by $C^{(0)}_{0}\left(x\right)=1$ and

 18.1.1 $C^{(0)}_{n}\left(x\right)=\frac{2}{n}T_{n}\left(x\right)=\frac{2(n-1)!}{{\left% (\tfrac{1}{2}\right)_{n}}}P^{(-\frac{1}{2},-\frac{1}{2})}_{n}\left(x\right),$ $n=1,2,3,\dots$.

Nor do we consider the shifted Jacobi polynomials:

 18.1.2 $G_{n}\left(p,q,x\right)=\frac{n!}{{\left(n+p\right)_{n}}}P^{(p-q,q-1)}_{n}% \left(2x-1\right),$ ⓘ Defines: $G_{\NVar{n}}\left(\NVar{p},\NVar{q},\NVar{x}\right)$: shifted Jacobi polynomial Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $q$: real variable, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.1.E2 Encodings: TeX, pMML, png See also: Annotations for 18.1(iii), 18.1 and 18

or the dilated Chebyshev polynomials of the first and second kinds:

 18.1.3 $\displaystyle C_{n}\left(x\right)$ $\displaystyle=2T_{n}\left(\tfrac{1}{2}x\right),$ $\displaystyle S_{n}\left(x\right)$ $\displaystyle=U_{n}\left(\tfrac{1}{2}x\right).$ ⓘ Defines: $S_{\NVar{n}}\left(\NVar{x}\right)$: dilated Chebyshev polynomial and $C_{\NVar{n}}\left(\NVar{x}\right)$: dilated Chebyshev polynomial Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the first kind, $U_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the second kind, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.1.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 18.1(iii), 18.1 and 18

In Koekoek et al. (2010) $\delta_{x}$ denotes the operator $\mathrm{i}\!\delta_{x}$.

In Mason and Handscomb (2003), the definitions of the Chebyshev polynomials of the third and fourth kinds $V_{n}\left(x\right)$ and $W_{n}\left(x\right)$ are the converse of the definitions in this chapter.