§18.5 Explicit Representations

§18.5(i) Trigonometric Functions

Chebyshev

With $x=\cos\theta=\tfrac{1}{2}(z+z^{-1})$,

 18.5.1 $\displaystyle T_{n}\left(x\right)$ $\displaystyle=\cos\left(n\theta\right)=\tfrac{1}{2}\left(z^{n}+z^{-n}\right),$ 18.5.2 $\displaystyle U_{n}\left(x\right)$ $\displaystyle=\frac{\sin\left((n+1)\theta\right)}{\sin\theta}=\dfrac{z^{n+1}-z% ^{-n-1}}{z-z^{-1}},$ 18.5.3 $\displaystyle V_{n}\left(x\right)$ $\displaystyle=\frac{\cos\left((n+\tfrac{1}{2})\theta\right)}{\cos\left(\tfrac{% 1}{2}\theta\right)}=\dfrac{z^{n+1}+z^{-n}}{z+1},$ ⓘ Symbols: $W_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the fourth kind, $V_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the third kind, $\cos\NVar{z}$: cosine function, $z$: complex variable, $n$: nonnegative integer and $x$: real variable Referenced by: (18.5.3), (18.5.4), Erratum (V1.0.28) for Table 18.3.1, Erratum (V1.2.0) for Equations (18.5.1), (18.5.2), (18.5.3), (18.5.4) Permalink: http://dlmf.nist.gov/18.5.E3 Encodings: TeX, pMML, png Correction (effective with 1.0.28): The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_{n}\left(x\right)$, $W_{n}\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_{n}\left(x\right)$, $W_{n}\left(x\right)$, having been interchanged, the right-hand sides of (18.5.3) and (18.5.4) have been swapped. For further details see Errata. See also: Annotations for §18.5(i), §18.5(i), §18.5 and Ch.18 18.5.4 $\displaystyle W_{n}\left(x\right)$ $\displaystyle=\frac{\sin\left((n+\tfrac{1}{2})\theta\right)}{\sin\left(\tfrac{% 1}{2}\theta\right)}=\dfrac{z^{n+1}-z^{-n}}{z-1}.$ ⓘ Symbols: $W_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the fourth kind, $V_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the third kind, $\sin\NVar{z}$: sine function, $z$: complex variable, $n$: nonnegative integer and $x$: real variable Referenced by: §18.16(iii), (18.5.3), (18.5.4), §18.5(i), §18.5(ii), §18.6(i), §18.9(ii), Erratum (V1.0.28) for Table 18.3.1, Erratum (V1.2.0) for Equations (18.5.1), (18.5.2), (18.5.3), (18.5.4) Permalink: http://dlmf.nist.gov/18.5.E4 Encodings: TeX, pMML, png Correction (effective with 1.0.28): The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_{n}\left(x\right)$, $W_{n}\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_{n}\left(x\right)$, $W_{n}\left(x\right)$, having been interchanged, the right-hand sides of (18.5.3) and (18.5.4) have been swapped. For further details see Errata. See also: Annotations for §18.5(i), §18.5(i), §18.5 and Ch.18
 18.5.4_5 ${\mathrm{i}}^{n}U_{n}\left(\tfrac{1}{2\mathrm{i}}\right)=F_{n+1}.$ ⓘ Symbols: $U_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the second kind, $\mathrm{i}$: imaginary unit and $n$: nonnegative integer Proof sketch: Use (18.5.2) and (26.11.7). Referenced by: §18.5(i), §18.5(i), Erratum (V1.2.0) §18.5 Permalink: http://dlmf.nist.gov/18.5.E4_5 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.5(i), §18.5(i), §18.5 and Ch.18

In (18.5.4_5) see §26.11 for the Fibonacci numbers $F_{n}$.

§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 $x$, $p_{n}(x)$: polynomial of degree $n$, $w(x)$: weight function, $n$: nonnegative integer 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), item 3., §18.39(ii), §18.39(ii), 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 Ch.18

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

Related formula:

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

See (Erdélyi et al., 1953b, §10.9(37)) for a related formula for ultraspherical polynomials.

§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 $C^{(\lambda)}_{n}\left(\cos\theta\right)=\sum_{\ell=0}^{n}\frac{{\left(\lambda% \right)_{\ell}}{\left(\lambda\right)_{n-\ell}}}{\ell!\;(n-\ell)!}\cos\left((n-% 2\ell)\theta\right)={\mathrm{e}}^{\mathrm{i}n\theta}\frac{{\left(\lambda\right% )_{n}}}{n!}{{}_{2}F_{1}}\left({-n,\lambda\atop 1-\lambda-n};{\mathrm{e}}^{-2% \mathrm{i}\theta}\right).$

Chebyshev

 18.5.11_1 $T_{n}\left(x\right)=\tfrac{1}{2}n\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}% \frac{(-1)^{\ell}(n-\ell-1)!}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}=2^{n-1}x^{n}{{}% _{2}F_{1}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop 1-n};\frac{1}% {x^{2}}\right),$ $n\geq 1$,
 18.5.11_2 $T_{n}\left(x\right)={{}_{2}F_{1}}\left({-n,n\atop\frac{1}{2}};\frac{1-x}{2}% \right),$ ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the first kind, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$: $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.5.E11_2 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18
 18.5.11_3 $U_{n}\left(x\right)=\sum_{\ell=0}^{\left\lfloor n/2\right\rfloor}\frac{(-1)^{% \ell}(n-\ell)!}{\ell!\;(n-2\ell)!}(2x)^{n-2\ell}=\left(2x\right)^{n}{{}_{2}F_{% 1}}\left({-\tfrac{1}{2}n,-\tfrac{1}{2}n+\tfrac{1}{2}\atop-n};\frac{1}{x^{2}}% \right),$
 18.5.11_4 $U_{n}\left(x\right)=\left(n+1\right){{}_{2}F_{1}}\left({-n,n+2\atop\frac{3}{2}% };\frac{1-x}{2}\right).$ ⓘ Symbols: $U_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the second kind, ${{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$: $=F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ notation for Gauss’ hypergeometric function, $n$: nonnegative integer and $x$: real variable Referenced by: §18.5(iii), Erratum (V1.2.0) §18.5 Permalink: http://dlmf.nist.gov/18.5.E11_4 Encodings: TeX, pMML, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.5(iii), §18.5(iii), §18.5 and Ch.18

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 Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.5(iv), §18.5(iv), §18.5 and Ch.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 Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.5(iv), §18.5(iv), §18.5 and Ch.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 Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.5(iv), §18.5(iv), §18.5 and Ch.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 Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.5(iv), §18.5(iv), §18.5 and Ch.18
 18.5.17_5 $\displaystyle L^{(\alpha)}_{0}\left(x\right)$ $\displaystyle=1,$ $\displaystyle L^{(\alpha)}_{1}\left(x\right)$ $\displaystyle=-x+\alpha+1,$ $\displaystyle L^{(\alpha)}_{2}\left(x\right)$ $\displaystyle=\tfrac{1}{2}x^{2}-(\alpha+2)x+\tfrac{1}{2}(\alpha+1)(\alpha+2),$ $\displaystyle L^{(\alpha)}_{3}\left(x\right)$ $\displaystyle=-\tfrac{1}{6}x^{3}+\tfrac{1}{2}(\alpha+3)x^{2}-\tfrac{1}{2}{% \left(\alpha+2\right)_{2}}x+\tfrac{1}{6}{\left(\alpha+1\right)_{3}},$ $\displaystyle L^{(\alpha)}_{4}\left(x\right)$ $\displaystyle=\tfrac{1}{24}x^{4}-\tfrac{1}{6}(\alpha+4)x^{3}+\tfrac{1}{4}{% \left(\alpha+3\right)_{2}}x^{2}-\tfrac{1}{6}{\left(\alpha+2\right)_{3}}x+% \tfrac{1}{24}{\left(\alpha+1\right)_{4}}.$ ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial) and $x$: real variable Referenced by: §18.5(iv), Erratum (V1.2.0) §18.5 Permalink: http://dlmf.nist.gov/18.5.E17_5 Encodings: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png Addition (effective with 1.2.0): This equation was added. See also: Annotations for §18.5(iv), §18.5(iv), §18.5 and Ch.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 Ch.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 Ch.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).