# §18.7 Interrelations and Limit Relations

## §18.7(i) Linear Transformations

### Ultraspherical and Jacobi

 18.7.1 $\displaystyle C^{(\lambda)}_{n}\left(x\right)$ $\displaystyle=\frac{{\left(2\lambda\right)_{n}}}{{\left(\lambda+\frac{1}{2}% \right)_{n}}}P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_{n}\left(x\right),$ 18.7.2 $\displaystyle P^{(\alpha,\alpha)}_{n}\left(x\right)$ $\displaystyle=\frac{{\left(\alpha+1\right)_{n}}}{{\left(2\alpha+1\right)_{n}}}% C^{(\alpha+\frac{1}{2})}_{n}\left(x\right).$

### Chebyshev, Ultraspherical, and Jacobi

 18.7.3 $T_{n}\left(x\right)=\ifrac{P^{(-\frac{1}{2},-\frac{1}{2})}_{n}\left(x\right)}{% P^{(-\frac{1}{2},-\frac{1}{2})}_{n}\left(1\right)},$ ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the first kind, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: 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(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 $U_{n}\left(x\right)=C^{(1)}_{n}\left(x\right)=\ifrac{(n+1)P^{(\frac{1}{2},% \frac{1}{2})}_{n}\left(x\right)}{P^{(\frac{1}{2},\frac{1}{2})}_{n}\left(1% \right)},$
 18.7.5 $V_{n}\left(x\right)=\ifrac{P^{(-\frac{1}{2},\frac{1}{2})}_{n}\left(x\right)}{P% ^{(-\frac{1}{2},\frac{1}{2})}_{n}\left(1\right)},$ ⓘ 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, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.17(viii), §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 $W_{n}\left(x\right)=\ifrac{(2n+1)P^{(\frac{1}{2},-\frac{1}{2})}_{n}\left(x% \right)}{P^{(\frac{1}{2},-\frac{1}{2})}_{n}\left(1\right)}.$ ⓘ 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, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.18(i), §18.3, §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 $\displaystyle T^{*}_{n}\left(x\right)$ $\displaystyle=T_{n}\left(2x-1\right),$ ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the first kind, $T^{*}_{\NVar{n}}\left(\NVar{x}\right)$: 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 $\displaystyle U^{*}_{n}\left(x\right)$ $\displaystyle=U_{n}\left(2x-1\right).$ ⓘ Symbols: $U_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the second kind, $U^{*}_{\NVar{n}}\left(\NVar{x}\right)$: 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

### Legendre, Ultraspherical, and Jacobi

 18.7.9 $P_{n}\left(x\right)=C^{(\frac{1}{2})}_{n}\left(x\right)=P^{(0,0)}_{n}\left(x% \right).$ ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, $P_{\NVar{n}}\left(\NVar{x}\right)$: Legendre polynomial, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$: 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 $P^{*}_{n}\left(x\right)=P_{n}\left(2x-1\right).$

### Hermite

 18.7.11 $\displaystyle\mathit{He}_{n}\left(x\right)$ $\displaystyle=2^{-\frac{1}{2}n}H_{n}\left(2^{-\frac{1}{2}}x\right),$ ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$: Hermite polynomial, $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$: 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 $\displaystyle H_{n}\left(x\right)$ $\displaystyle=2^{\frac{1}{2}n}\mathit{He}_{n}\left(2^{\frac{1}{2}}x\right).$ ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$: Hermite polynomial, $\mathit{He}_{\NVar{n}}\left(\NVar{x}\right)$: 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

 18.7.13 $\displaystyle\frac{P^{(\alpha,\alpha)}_{2n}\left(x\right)}{P^{(\alpha,\alpha)}% _{2n}\left(1\right)}$ $\displaystyle=\frac{P^{(\alpha,-\frac{1}{2})}_{n}\left(2x^{2}-1\right)}{P^{(% \alpha,-\frac{1}{2})}_{n}\left(1\right)},$ ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.7(ii), §18.7(ii) Permalink: http://dlmf.nist.gov/18.7.E13 Encodings: TeX, pMML, png See also: Annotations for §18.7(ii), §18.7 and Ch.18 18.7.14 $\displaystyle\frac{P^{(\alpha,\alpha)}_{2n+1}\left(x\right)}{P^{(\alpha,\alpha% )}_{2n+1}\left(1\right)}$ $\displaystyle=\frac{xP^{(\alpha,\frac{1}{2})}_{n}\left(2x^{2}-1\right)}{P^{(% \alpha,\frac{1}{2})}_{n}\left(1\right)}.$ ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.7(ii) Permalink: http://dlmf.nist.gov/18.7.E14 Encodings: TeX, pMML, png See also: Annotations for §18.7(ii), §18.7 and Ch.18
 18.7.15 $\displaystyle C^{(\lambda)}_{2n}\left(x\right)$ $\displaystyle=\frac{{\left(\lambda\right)_{n}}}{{\left(\tfrac{1}{2}\right)_{n}% }}P^{(\lambda-\frac{1}{2},-\frac{1}{2})}_{n}\left(2x^{2}-1\right),$ 18.7.16 $\displaystyle C^{(\lambda)}_{2n+1}\left(x\right)$ $\displaystyle=\frac{{\left(\lambda\right)_{n+1}}}{{\left(\frac{1}{2}\right)_{n% +1}}}xP^{(\lambda-\frac{1}{2},\frac{1}{2})}_{n}\left(2x^{2}-1\right).$
 18.7.17 $\displaystyle U_{2n}\left(x\right)$ $\displaystyle=W_{n}\left(2x^{2}-1\right),$ ⓘ Symbols: $W_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the fourth kind, $U_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the second kind, $V_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the third kind, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.5.30 Referenced by: §18.7(ii), Erratum (V1.0.28) for Table 18.3.1 Permalink: http://dlmf.nist.gov/18.7.E17 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, on the right-hand side we replaced $V_{n}\left(2x^{2}-1\right)$, with $W_{n}\left(2x^{2}-1\right)$. For further details see Errata. See also: Annotations for §18.7(ii), §18.7 and Ch.18 18.7.18 $\displaystyle T_{2n+1}\left(x\right)$ $\displaystyle=xV_{n}\left(2x^{2}-1\right).$ ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the first kind, $W_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the fourth kind, $V_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the third kind, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.5.29 Referenced by: §18.7(ii), Erratum (V1.0.28) for Table 18.3.1 Permalink: http://dlmf.nist.gov/18.7.E18 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, on the right-hand side we replaced $xW_{n}\left(2x^{2}-1\right)$, with $xV_{n}\left(2x^{2}-1\right)$. For further details see Errata. See also: Annotations for §18.7(ii), §18.7 and Ch.18
 18.7.19 $\displaystyle H_{2n}\left(x\right)$ $\displaystyle=(-1)^{n}2^{2n}n!L^{(-\frac{1}{2})}_{n}\left(x^{2}\right),$ 18.7.20 $\displaystyle H_{2n+1}\left(x\right)$ $\displaystyle=(-1)^{n}2^{2n+1}n!\,xL^{(\frac{1}{2})}_{n}\left(x^{2}\right).$

Equations (18.7.13)–(18.7.20) are special cases of (18.2.22)–(18.2.23).

## §18.7(iii) Limit Relations

### Jacobi $\to$ Laguerre

 18.7.21 $\lim_{\beta\to\infty}P^{(\alpha,\beta)}_{n}\left(1-\left(\ifrac{2x}{\beta}% \right)\right)=L^{(\alpha)}_{n}\left(x\right).$ ⓘ Symbols: $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$: 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 $\lim_{\alpha\to\infty}P^{(\alpha,\beta)}_{n}\left((2x/\alpha)-1\right)=(-1)^{n% }L^{(\beta)}_{n}\left(x\right).$

### Jacobi $\to$ Hermite

 18.7.23 $\lim_{\alpha\to\infty}\alpha^{-\frac{1}{2}n}P^{(\alpha,\alpha)}_{n}\left(% \alpha^{-\frac{1}{2}}x\right)=\frac{H_{n}\left(x\right)}{2^{n}n!}.$

### Ultraspherical $\to$ Hermite

 18.7.24 $\lim_{\lambda\to\infty}\lambda^{-\frac{1}{2}n}C^{(\lambda)}_{n}\left(\lambda^{% -\frac{1}{2}}x\right)=\frac{H_{n}\left(x\right)}{n!}.$

### Ultraspherical $\to$ Chebyshev

 18.7.25 $\lim_{\lambda\to 0}\frac{n+\lambda}{\lambda}C^{(\lambda)}_{n}\left(x\right)=% \begin{cases}1,&\text{n=0,}\\ 2T_{n}\left(x\right),&\text{n=1,2,\dots.}\end{cases}$ ⓘ Symbols: $T_{\NVar{n}}\left(\NVar{x}\right)$: Chebyshev polynomial of the first kind, $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$: ultraspherical (or Gegenbauer) polynomial, $n$: nonnegative integer and $x$: real variable Source: Andrews et al. (1999, (6.4.13)) Referenced by: §18.7(i), §18.7(iii), Erratum (V1.2.0) for Equation (18.7.25) Permalink: http://dlmf.nist.gov/18.7.E25 Encodings: TeX, pMML, png Addition (effective with 1.2.0): We included the case $n=0$. See also: Annotations for §18.7(iii), §18.7(iii), §18.7 and Ch.18

### Laguerre $\to$ Hermite

 18.7.26 $\lim_{\alpha\to\infty}\left(\frac{2}{\alpha}\right)^{\frac{1}{2}n}L^{(\alpha)}% _{n}\left((2\alpha)^{\frac{1}{2}}x+\alpha\right)=\frac{(-1)^{n}}{n!}H_{n}\left% (x\right).$

See Figure 18.21.1 for the Askey schematic representation of most of these limits. See §18.11(ii) for limit formulas of Mehler–Heine type.