# §18.6 Symmetry, Special Values, and Limits to Monomials

## §18.6(i) Symmetry and Special Values

For Jacobi, ultraspherical, Chebyshev, Legendre, and Hermite polynomials, see Table 18.6.1.

### Laguerre

 18.6.1 $L^{(\alpha)}_{n}\left(0\right)=\frac{{\left(\alpha+1\right)_{n}}}{n!}.$ ⓘ 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), $!$: factorial (as in $n!$) and $n$: nonnegative integer A&S Ref: 22.4.7 (see column for $f_{n}(0)$) Referenced by: §18.17(i), §18.6(i), §18.6(ii) Permalink: http://dlmf.nist.gov/18.6.E1 Encodings: TeX, pMML, png See also: Annotations for §18.6(i), §18.6(i), §18.6 and Ch.18

## §18.6(ii) Limits to Monomials

 18.6.2 $\displaystyle\lim_{\alpha\to\infty}\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}% {P^{(\alpha,\beta)}_{n}\left(1\right)}$ $\displaystyle=\left(\frac{1+x}{2}\right)^{n},$ ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.6(ii) Permalink: http://dlmf.nist.gov/18.6.E2 Encodings: TeX, pMML, png See also: Annotations for §18.6(ii), §18.6 and Ch.18 18.6.3 $\displaystyle\lim_{\beta\to\infty}\frac{P^{(\alpha,\beta)}_{n}\left(x\right)}{% P^{(\alpha,\beta)}_{n}\left(-1\right)}$ $\displaystyle=\left(\frac{1-x}{2}\right)^{n},$ ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.6(ii) Permalink: http://dlmf.nist.gov/18.6.E3 Encodings: TeX, pMML, png See also: Annotations for §18.6(ii), §18.6 and Ch.18 18.6.4 $\displaystyle\lim_{\lambda\to\infty}\frac{C^{(\lambda)}_{n}\left(x\right)}{C^{% (\lambda)}_{n}\left(1\right)}$ $\displaystyle=x^{n},$ ⓘ Symbols: $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.18(iv), §18.6(ii) Permalink: http://dlmf.nist.gov/18.6.E4 Encodings: TeX, pMML, png See also: Annotations for §18.6(ii), §18.6 and Ch.18 18.6.5 $\displaystyle\lim_{\alpha\to\infty}\frac{L^{(\alpha)}_{n}\left(\alpha x\right)% }{L^{(\alpha)}_{n}\left(0\right)}$ $\displaystyle=(1-x)^{n}.$ ⓘ Symbols: $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$: Laguerre (or generalized Laguerre) polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.6(ii) Permalink: http://dlmf.nist.gov/18.6.E5 Encodings: TeX, pMML, png See also: Annotations for §18.6(ii), §18.6 and Ch.18