# §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: $\mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits\!\left(x\right)$.

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

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

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

• Legendre: $\mathop{P_{n}\/}\nolimits\!\left(x\right)$.

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

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

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

### Hahn Class OP’s

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

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

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

• Charlier: $\mathop{C_{n}\/}\nolimits\!\left(x;a\right)$.

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

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

### Wilson Class OP’s

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

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

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

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

### $q$-Hahn Class OP’s

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

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

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

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

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

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

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

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

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

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

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

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

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

### Other OP’s

• Bessel: $\mathop{y_{n}\/}\nolimits\!\left(x;a\right)$.

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

### Classical OP’s in Two Variables

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

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

## §18.1(iii) Other Notations

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

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

Nor do we consider the shifted Jacobi polynomials:

 18.1.2 $\mathop{G_{n}\/}\nolimits\!\left(p,q,x\right)=\frac{n!}{{\left(n+p\right)_{n}}% }\mathop{P^{(p-q,q-1)}_{n}\/}\nolimits\!\left(2x-1\right),$ Defines: $\mathop{G_{\NVar{n}}\/}\nolimits\!\left(\NVar{p},\NVar{q},\NVar{x}\right)$: shifted Jacobi polynomial Symbols: $\mathop{P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\/}\nolimits\!\left(\NVar{x% }\right)$: 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)

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

 18.1.3 $\displaystyle\mathop{C_{n}\/}\nolimits\!\left(x\right)$ $\displaystyle=2\mathop{T_{n}\/}\nolimits\!\left(\tfrac{1}{2}x\right),$ $\displaystyle\mathop{S_{n}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{U_{n}\/}\nolimits\!\left(\tfrac{1}{2}x\right).$ Defines: $\mathop{S_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: dilated Chebyshev polynomial and $\mathop{C_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: dilated Chebyshev polynomial Symbols: $\mathop{T_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: Chebyshev polynomial of the first kind, $\mathop{U_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: 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)

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 $\mathop{V_{n}\/}\nolimits\!\left(x\right)$ and $\mathop{W_{n}\/}\nolimits\!\left(x\right)$ are the converse of the definitions in this chapter.