§18.7 Interrelations and Limit Relations

18.7.1		$C_n^{(\lambda )}\left(x\right)$	$={\displaystyle \frac{{\left(2\lambda \right)}_n}{{\left(\lambda +\frac{1}{2}\right)}_n}}{P}_n^{(\lambda -\frac{1}{2},\lambda -\frac{1}{2})}\left(x\right),$
ⓘ Symbols: $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.5.27 Referenced by: §18.15(ii), §18.7(i), §18.7(ii), §18.7(iii) Permalink: http://dlmf.nist.gov/18.7.E1 Encodings: TeX, pMML, png See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18
18.7.2		$P_n^{(\alpha ,\alpha )}\left(x\right)$	$={\displaystyle \frac{{\left(\alpha +1\right)}_n}{{\left(2\alpha +1\right)}_n}}{C}_n^{(\alpha +\frac{1}{2})}\left(x\right).$
ⓘ Symbols: $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.5.20 Referenced by: §18.6(i), §18.7(i) Permalink: http://dlmf.nist.gov/18.7.E2 Encodings: TeX, pMML, png See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

18.7.3		$$T_n\left(x\right)=P_n^{(-\frac{1}{2},-\frac{1}{2})}\left(x\right)/P_n^{(-\frac{1}{2},-\frac{1}{2})}\left(1\right),$$
ⓘ Symbols: $T_n\left(x\right)$: Chebyshev polynomial of the first kind, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.3.15, 22.5.31 Referenced by: §18.17(viii), §18.18(i), §18.5(iii), §18.7(i), §18.7(ii) Permalink: http://dlmf.nist.gov/18.7.E3 Encodings: TeX, pMML, png See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

18.7.4		$$U_n\left(x\right)=C_n^{(1)}\left(x\right)=(n+1)P_n^{(\frac{1}{2},\frac{1}{2})}\left(x\right)/P_n^{(\frac{1}{2},\frac{1}{2})}\left(1\right),$$
ⓘ Symbols: $U_n\left(x\right)$: Chebyshev polynomial of the second kind, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.3.16, 22.5.32 Referenced by: §18.12, §18.3, §18.7(i), §18.7(ii) Permalink: http://dlmf.nist.gov/18.7.E4 Encodings: TeX, pMML, png See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

18.7.5		$$V_n\left(x\right)=P_n^{(-\frac{1}{2},\frac{1}{2})}\left(x\right)/P_n^{(-\frac{1}{2},\frac{1}{2})}\left(1\right),$$
ⓘ Symbols: $W_n\left(x\right)$: Chebyshev polynomial of the fourth kind, $V_n\left(x\right)$: Chebyshev polynomial of the third kind, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(viii), Table 18.3.1, §18.5(i), (18.7.5), (18.7.6), §18.7(i), §18.7(ii), Erratum (V1.0.28) for Table 18.3.1 Permalink: http://dlmf.nist.gov/18.7.E5 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 (18.7.5) and (18.7.6) have been interchanged. For further details see Errata. See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

18.7.6		$$W_n\left(x\right)=(2n+1)P_n^{(\frac{1}{2},-\frac{1}{2})}\left(x\right)/P_n^{(\frac{1}{2},-\frac{1}{2})}\left(1\right).$$
ⓘ Symbols: $W_n\left(x\right)$: Chebyshev polynomial of the fourth kind, $V_n\left(x\right)$: Chebyshev polynomial of the third kind, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.18(i), §18.3, Table 18.3.1, §18.5(iii), §18.5(i), (18.7.5), (18.7.6), §18.7(i), §18.7(ii), Erratum (V1.0.28) for Table 18.3.1 Permalink: http://dlmf.nist.gov/18.7.E6 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 (18.7.5) and (18.7.6) have been interchanged. For further details see Errata. See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

18.7.7		$T_n^{*}\left(x\right)$	$=T_n\left(2x-1\right),$
ⓘ Symbols: $T_n\left(x\right)$: Chebyshev polynomial of the first kind, $T_n^{*}\left(x\right)$: shifted Chebyshev polynomial of the first kind, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.5.14 Referenced by: §18.7(i) Permalink: http://dlmf.nist.gov/18.7.E7 Encodings: TeX, pMML, png See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18
18.7.8		$U_n^{*}\left(x\right)$	$=U_n\left(2x-1\right).$
ⓘ Symbols: $U_n\left(x\right)$: Chebyshev polynomial of the second kind, $U_n^{*}\left(x\right)$: shifted Chebyshev polynomial of the second kind, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.5.15 Permalink: http://dlmf.nist.gov/18.7.E8 Encodings: TeX, pMML, png See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

See also (18.9.9)–(18.9.12).

18.7.9		$$P_n\left(x\right)=C_n^{(\frac{1}{2})}\left(x\right)=P_n^{(0,0)}\left(x\right).$$
ⓘ Symbols: $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $P_n\left(x\right)$: Legendre polynomial, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.5.35, 22.5.36 Referenced by: §10.60(iii), §18.18(ii), §18.5(iii) Permalink: http://dlmf.nist.gov/18.7.E9 Encodings: TeX, pMML, png See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

18.7.10		$$P_n^{*}\left(x\right)=P_n\left(2x-1\right).$$
ⓘ Symbols: $P_n\left(x\right)$: Legendre polynomial, $P_n^{*}\left(x\right)$: shifted Legendre polynomial, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.7.E10 Encodings: TeX, pMML, png See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

18.7.11		${\mathit{He}}_n\left(x\right)$	$=2^{-\frac{1}{2}n}{H}_n\left(2^{-\frac{1}{2}}x\right),$
ⓘ Symbols: $H_n\left(x\right)$: Hermite polynomial, ${\mathit{He}}_n\left(x\right)$: Hermite polynomial, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.5.18 Referenced by: §18.5(iii) Permalink: http://dlmf.nist.gov/18.7.E11 Encodings: TeX, pMML, png See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18
18.7.12		$H_n\left(x\right)$	$=2^{\frac{1}{2}n}{\mathit{He}}_n\left(2^{\frac{1}{2}}x\right).$
ⓘ Symbols: $H_n\left(x\right)$: Hermite polynomial, ${\mathit{He}}_n\left(x\right)$: Hermite polynomial, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.5.19 Referenced by: §18.7(i) Permalink: http://dlmf.nist.gov/18.7.E12 Encodings: TeX, pMML, png See also: Annotations for §18.7(i), §18.7(i), §18.7 and Ch.18

18.7.21		$$\underset{\beta \to \mathrm{\infty }}{lim}{P}_n^{(\alpha ,\beta )}\left(1-\left(2x/\beta \right)\right)=L_n^{(\alpha )}\left(x\right).$$
ⓘ Symbols: $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.15.5 Referenced by: §18.10(ii), §18.7(iii) Permalink: http://dlmf.nist.gov/18.7.E21 Encodings: TeX, pMML, png See also: Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

18.7.22		$$\underset{\alpha \to \mathrm{\infty }}{lim}{P}_n^{(\alpha ,\beta )}\left((2x/\alpha )-1\right)={(-1)}^n{L}_n^{(\beta )}\left(x\right).$$
ⓘ Symbols: $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.7(iii) Permalink: http://dlmf.nist.gov/18.7.E22 Encodings: TeX, pMML, png See also: Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

18.7.23		$$\underset{\alpha \to \mathrm{\infty }}{lim}{\alpha }^{-\frac{1}{2}n}{P}_n^{(\alpha ,\alpha )}\left({\alpha }^{-\frac{1}{2}}x\right)=\frac{H_n\left(x\right)}{2^{n}n!}.$$
ⓘ Symbols: $H_n\left(x\right)$: Hermite polynomial, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable Referenced by: §18.10(ii), §18.7(iii) Permalink: http://dlmf.nist.gov/18.7.E23 Encodings: TeX, pMML, png See also: Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

18.7.24		$$\underset{\lambda \to \mathrm{\infty }}{lim}{\lambda }^{-\frac{1}{2}n}{C}_n^{(\lambda )}\left({\lambda }^{-\frac{1}{2}}x\right)=\frac{H_n\left(x\right)}{n!}.$$
ⓘ Symbols: $H_n\left(x\right)$: Hermite polynomial, $!$: factorial (as in $n!$), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.15.6 Referenced by: §18.7(iii) Permalink: http://dlmf.nist.gov/18.7.E24 Encodings: TeX, pMML, png See also: Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

18.7.25		$$\underset{\lambda \to 0}{lim}\frac{1}{\lambda }{C}_n^{(\lambda )}\left(x\right)=\frac{2}{n}{T}_n\left(x\right),$$
$n\ge 1$.
ⓘ Symbols: $T_n\left(x\right)$: Chebyshev polynomial of the first kind, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.7(iii) Permalink: http://dlmf.nist.gov/18.7.E25 Encodings: TeX, pMML, png See also: Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

18.7.26		$$\underset{\alpha \to \mathrm{\infty }}{lim}{\left(\frac{2}{\alpha }\right)}^{\frac{1}{2}n}{L}_n^{(\alpha )}\left({(2\alpha )}^{\frac{1}{2}}x+\alpha \right)=\frac{{(-1)}^n}{n!}{H}_n\left(x\right).$$
ⓘ Symbols: $H_n\left(x\right)$: Hermite polynomial, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $!$: factorial (as in $n!$), $n$: nonnegative integer and $x$: real variable Referenced by: §18.7(iii) Permalink: http://dlmf.nist.gov/18.7.E26 Encodings: TeX, pMML, png See also: Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

See Figure 18.21.1 for the Askey schematic representation of most of these limits.