# §18.5 Explicit Representations

## §18.5(i) Trigonometric Functions

### Chebyshev

With $x=\cos\theta$,

 18.5.1 $\displaystyle T_{n}\left(x\right)$ $\displaystyle=\cos\left(n\theta\right),$ ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the first kind, $\cos\NVar{z}$: cosine function, $n$: nonnegative integer and $x$: real variable Referenced by: §18.18(viii), §18.5(i), §18.5(ii), §18.6(i), §18.9(i), §18.9(ii) Permalink: http://dlmf.nist.gov/18.5.E1 Encodings: TeX, pMML, png See also: Annotations for 18.5(i), 18.5(i), 18.5 and 18 18.5.2 $\displaystyle U_{n}\left(x\right)$ $\displaystyle=\ifrac{(\sin(n+1)\theta)}{\sin\theta},$ ⓘ Symbols: $U_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the second kind, $\sin\NVar{z}$: sine function, $n$: nonnegative integer and $x$: real variable Referenced by: §18.18(viii), §18.9(i) Permalink: http://dlmf.nist.gov/18.5.E2 Encodings: TeX, pMML, png See also: Annotations for 18.5(i), 18.5(i), 18.5 and 18 18.5.3 $\displaystyle V_{n}\left(x\right)$ $\displaystyle=\ifrac{(\sin(n+\tfrac{1}{2})\theta)}{\sin\left(\tfrac{1}{2}% \theta\right)},$ 18.5.4 $\displaystyle W_{n}\left(x\right)$ $\displaystyle=\ifrac{(\cos(n+\tfrac{1}{2})\theta)}{\cos\left(\tfrac{1}{2}% \theta\right)}.$ ⓘ Symbols: $W_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the fourth kind, $\cos\NVar{z}$: cosine function, $n$: nonnegative integer and $x$: real variable Referenced by: §18.5(i), §18.5(ii), §18.6(i), §18.9(ii) Permalink: http://dlmf.nist.gov/18.5.E4 Encodings: TeX, pMML, png See also: Annotations for 18.5(i), 18.5(i), 18.5 and 18

## §18.5(ii) Rodrigues Formulas

 18.5.5 $p_{n}(x)=\frac{1}{\kappa_{n}w(x)}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}% \left(w(x)(F(x))^{n}\right).$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $w(x)$: weight function, $n$: nonnegative integer, $p_{n}(x)$: polynomial of degree $n$ and $x$: real variable A&S Ref: 22.1.6 (incorrectly stated as property of general OP’s) Referenced by: §18.17(v), §18.17(vi), §18.17(vii), Table 18.5.1, §18.9(iii) Permalink: http://dlmf.nist.gov/18.5.E5 Encodings: TeX, pMML, png See also: Annotations for 18.5(ii), 18.5 and 18

In this equation $w(x)$ is as in Table 18.3.1, and $F(x)$, $\kappa_{n}$ are as in Table 18.5.1.

Related formula:

 18.5.6 $L^{(\alpha)}_{n}\left(\frac{1}{x}\right)=\frac{(-1)^{n}}{n!}x^{n+\alpha+1}e^{% \ifrac{1}{x}}\frac{{\mathrm{d}}^{n}}{{\mathrm{d}x}^{n}}\left(x^{-\alpha-1}e^{-% \ifrac{1}{x}}\right).$

## §18.5(iii) Finite Power Series, the Hypergeometric Function, and Generalized Hypergeometric Functions

For the definitions of ${{}_{2}F_{1}}$, ${{}_{1}F_{1}}$, and ${{}_{2}F_{0}}$ see §16.2.

### Jacobi

 18.5.7 $P^{(\alpha,\beta)}_{n}\left(x\right)=\sum_{\ell=0}^{n}\frac{{\left(n+\alpha+% \beta+1\right)_{\ell}}{\left(\alpha+\ell+1\right)_{n-\ell}}}{\ell!\;(n-\ell)!}% \left(\frac{x-1}{2}\right)^{\ell}=\frac{{\left(\alpha+1\right)_{n}}}{n!}{{}_{2% }F_{1}}\left({-n,n+\alpha+\beta+1\atop\alpha+1};\frac{1-x}{2}\right),$
 18.5.8 $P^{(\alpha,\beta)}_{n}\left(x\right)=2^{-n}\sum_{\ell=0}^{n}\genfrac{(}{)}{0.0% pt}{}{n+\alpha}{\ell}\genfrac{(}{)}{0.0pt}{}{n+\beta}{n-\ell}(x-1)^{n-\ell}(x+% 1)^{\ell}=\frac{{\left(\alpha+1\right)_{n}}}{n!}\left(\frac{x+1}{2}\right)^{n}% {{}_{2}F_{1}}\left({-n,-n-\beta\atop\alpha+1};\frac{x-1}{x+1}\right),$

and two similar formulas by symmetry; compare the second row in Table 18.6.1.

### Ultraspherical

 18.5.9 $C^{(\lambda)}_{n}\left(x\right)=\frac{{\left(2\lambda\right)_{n}}}{n!}{{}_{2}F% _{1}}\left({-n,n+2\lambda\atop\lambda+\tfrac{1}{2}};\frac{1-x}{2}\right),$
 18.5.10 $C^{(\lambda)}_{n}\left(x\right)=\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}% \frac{(-1)^{\ell}{\left(\lambda\right)_{n-\ell}}}{\ell!\;(n-2\ell)!}(2x)^{n-2% \ell}=(2x)^{n}\frac{{\left(\lambda\right)_{n}}}{n!}{{}_{2}F_{1}}\left({-\tfrac% {1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop 1-\lambda-n};\frac{1}{x^{2}}\right),$
 18.5.11 $C^{(\lambda)}_{n}\left(\cos\theta\right)=\sum_{\ell=0}^{n}\frac{{\left(\lambda% \right)_{\ell}}{\left(\lambda\right)_{n-\ell}}}{\ell!\;(n-\ell)!}\cos\left((n-% 2\ell)\theta\right)=e^{\mathrm{i}n\theta}\frac{{\left(\lambda\right)_{n}}}{n!}% {{}_{2}F_{1}}\left({-n,\lambda\atop 1-\lambda-n};e^{-2\mathrm{i}\theta}\right).$

### Laguerre

 18.5.12 $L^{(\alpha)}_{n}\left(x\right)=\sum_{\ell=0}^{n}\frac{{\left(\alpha+\ell+1% \right)_{n-\ell}}}{(n-\ell)!\;\ell!}(-x)^{\ell}=\frac{{\left(\alpha+1\right)_{% n}}}{n!}{{}_{1}F_{1}}\left({-n\atop\alpha+1};x\right).$

### Hermite

 18.5.13 $H_{n}\left(x\right)=n!\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{(-1)^% {\ell}(2x)^{n-2\ell}}{\ell!\;(n-2\ell)!}=(2x)^{n}{{}_{2}F_{0}}\left({-\tfrac{1% }{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop-};-\frac{1}{x^{2}}\right).$

For corresponding formulas for Chebyshev, Legendre, and the Hermite $\mathit{He}_{n}$ polynomials apply (18.7.3)–(18.7.6), (18.7.9), and (18.7.11).

Note. The first of each of equations (18.5.7) and (18.5.8) can be regarded as definitions of $P^{(\alpha,\beta)}_{n}\left(x\right)$ when the conditions $\alpha>-1$ and $\beta>-1$ are not satisfied. However, in these circumstances the orthogonality property (18.2.1) disappears. For this reason, and also in the interest of simplicity, in the case of the Jacobi polynomials $P^{(\alpha,\beta)}_{n}\left(x\right)$ we assume throughout this chapter that $\alpha>-1$ and $\beta>-1$, unless stated otherwise. Similarly in the cases of the ultraspherical polynomials $C^{(\lambda)}_{n}\left(x\right)$ and the Laguerre polynomials $L^{(\alpha)}_{n}\left(x\right)$ we assume that $\lambda>-\tfrac{1}{2},\lambda\neq 0$, and $\alpha>-1$, unless stated otherwise.

## §18.5(iv) Numerical Coefficients

### Chebyshev

 18.5.14 $\displaystyle T_{0}\left(x\right)$ $\displaystyle=1,$ $\displaystyle T_{1}\left(x\right)$ $\displaystyle=x,$ $\displaystyle T_{2}\left(x\right)$ $\displaystyle=2x^{2}-1,$ $\displaystyle T_{3}\left(x\right)$ $\displaystyle=4x^{3}-3x,$ $\displaystyle T_{4}\left(x\right)$ $\displaystyle=8x^{4}-8x^{2}+1,$ $\displaystyle T_{5}\left(x\right)$ $\displaystyle=16x^{5}-20x^{3}+5x,$ $\displaystyle T_{6}\left(x\right)$ $\displaystyle=32x^{6}-48x^{4}+18x^{2}-1.$ ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the first kind and $x$: real variable Permalink: http://dlmf.nist.gov/18.5.E14 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png See also: Annotations for 18.5(iv), 18.5(iv), 18.5 and 18
 18.5.15 $\displaystyle U_{0}\left(x\right)$ $\displaystyle=1,$ $\displaystyle U_{1}\left(x\right)$ $\displaystyle=2x,$ $\displaystyle U_{2}\left(x\right)$ $\displaystyle=4x^{2}-1,$ $\displaystyle U_{3}\left(x\right)$ $\displaystyle=8x^{3}-4x,$ $\displaystyle U_{4}\left(x\right)$ $\displaystyle=16x^{4}-12x^{2}+1,$ $\displaystyle U_{5}\left(x\right)$ $\displaystyle=32x^{5}-32x^{3}+6x,$ $\displaystyle U_{6}\left(x\right)$ $\displaystyle=64x^{6}-80x^{4}+24x^{2}-1.$ ⓘ Symbols: $U_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the second kind and $x$: real variable Permalink: http://dlmf.nist.gov/18.5.E15 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png See also: Annotations for 18.5(iv), 18.5(iv), 18.5 and 18

### Legendre

 18.5.16 $\displaystyle P_{0}\left(x\right)$ $\displaystyle=1,$ $\displaystyle P_{1}\left(x\right)$ $\displaystyle=x,$ $\displaystyle P_{2}\left(x\right)$ $\displaystyle=\tfrac{3}{2}x^{2}-\tfrac{1}{2},$ $\displaystyle P_{3}\left(x\right)$ $\displaystyle=\tfrac{5}{2}x^{3}-\tfrac{3}{2}x,$ $\displaystyle P_{4}\left(x\right)$ $\displaystyle=\tfrac{35}{8}x^{4}-\tfrac{15}{4}x^{2}+\tfrac{3}{8},$ $\displaystyle P_{5}\left(x\right)$ $\displaystyle=\tfrac{63}{8}x^{5}-\tfrac{35}{4}x^{3}+\tfrac{15}{8}x,$ $\displaystyle P_{6}\left(x\right)$ $\displaystyle=\tfrac{231}{16}x^{6}-\tfrac{315}{16}x^{4}+\tfrac{105}{16}x^{2}-% \tfrac{5}{16}.$ ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$: Legendre polynomial and $x$: real variable Permalink: http://dlmf.nist.gov/18.5.E16 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png See also: Annotations for 18.5(iv), 18.5(iv), 18.5 and 18

### Laguerre

 18.5.17 $\displaystyle L_{0}\left(x\right)$ $\displaystyle=1,$ $\displaystyle L_{1}\left(x\right)$ $\displaystyle=-x+1,$ $\displaystyle L_{2}\left(x\right)$ $\displaystyle=\tfrac{1}{2}x^{2}-2x+1,$ $\displaystyle L_{3}\left(x\right)$ $\displaystyle=-\tfrac{1}{6}x^{3}+\tfrac{3}{2}x^{2}-3x+1,$ $\displaystyle L_{4}\left(x\right)$ $\displaystyle=\tfrac{1}{24}x^{4}-\tfrac{2}{3}x^{3}+3x^{2}-4x+1,$ $\displaystyle L_{5}\left(x\right)$ $\displaystyle=-\tfrac{1}{120}x^{5}+\tfrac{5}{24}x^{4}-\tfrac{5}{3}x^{3}+5x^{2}% -5x+1,$ $\displaystyle L_{6}\left(x\right)$ $\displaystyle=\tfrac{1}{720}x^{6}-\tfrac{1}{20}x^{5}+\tfrac{5}{8}x^{4}-\tfrac{% 10}{3}x^{3}+\tfrac{15}{2}x^{2}-6x+1.$ ⓘ Symbols: $L_{\NVar{n}}\left(\NVar{x}\right)=L^{(0)}_{n}\left(x\right)$: Laguerre polynomial and $x$: real variable Permalink: http://dlmf.nist.gov/18.5.E17 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png See also: Annotations for 18.5(iv), 18.5(iv), 18.5 and 18

### Hermite

 18.5.18 $\displaystyle H_{0}\left(x\right)$ $\displaystyle=1,$ $\displaystyle H_{1}\left(x\right)$ $\displaystyle=2x,$ $\displaystyle H_{2}\left(x\right)$ $\displaystyle=4x^{2}-2,$ $\displaystyle H_{3}\left(x\right)$ $\displaystyle=8x^{3}-12x,$ $\displaystyle H_{4}\left(x\right)$ $\displaystyle=16x^{4}-48x^{2}+12,$ $\displaystyle H_{5}\left(x\right)$ $\displaystyle=32x^{5}-160x^{3}+120x,$ $\displaystyle H_{6}\left(x\right)$ $\displaystyle=64x^{6}-480x^{4}+720x^{2}-120.$ ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$: Hermite polynomial and $x$: real variable Referenced by: §13.6(v) Permalink: http://dlmf.nist.gov/18.5.E18 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png See also: Annotations for 18.5(iv), 18.5(iv), 18.5 and 18
 18.5.19 $\displaystyle\mathit{He}_{0}\left(x\right)$ $\displaystyle=1,$ $\displaystyle\mathit{He}_{1}\left(x\right)$ $\displaystyle=x,$ $\displaystyle\mathit{He}_{2}\left(x\right)$ $\displaystyle=x^{2}-1,$ $\displaystyle\mathit{He}_{3}\left(x\right)$ $\displaystyle=x^{3}-3x,$ $\displaystyle\mathit{He}_{4}\left(x\right)$ $\displaystyle=x^{4}-6x^{2}+3,$ $\displaystyle\mathit{He}_{5}\left(x\right)$ $\displaystyle=x^{5}-10x^{3}+15x,$ $\displaystyle\mathit{He}_{6}\left(x\right)$ $\displaystyle=x^{6}-15x^{4}+45x^{2}-15.$ ⓘ Symbols: $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$: Hermite polynomial and $x$: real variable Permalink: http://dlmf.nist.gov/18.5.E19 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png See also: Annotations for 18.5(iv), 18.5(iv), 18.5 and 18

For the corresponding polynomials of degrees 7 through 12 see Abramowitz and Stegun (1964, Tables 22.3, 22.5, 22.9, 22.10, 22.12).