# §18.7 Interrelations and Limit Relations

## §18.7(i) Linear Transformations

### Ultraspherical and Jacobi

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

### Chebyshev, Ultraspherical, and Jacobi

 18.7.3 $\mathop{T_{n}\/}\nolimits\!\left(x\right)=\ifrac{\mathop{P^{(-\frac{1}{2},-% \frac{1}{2})}_{n}\/}\nolimits\!\left(x\right)}{\mathop{P^{(-\frac{1}{2},-\frac% {1}{2})}_{n}\/}\nolimits\!\left(1\right)},$ Symbols: $\mathop{T_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: Chebyshev polynomial of the first kind, $\mathop{P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\/}\nolimits\!\left(\NVar{x% }\right)$: 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(ii) Permalink: http://dlmf.nist.gov/18.7.E3 Encodings: TeX, pMML, png See also: Annotations for 18.7(i)
 18.7.4 $\mathop{U_{n}\/}\nolimits\!\left(x\right)=\mathop{C^{(1)}_{n}\/}\nolimits\!% \left(x\right)=\ifrac{(n+1)\mathop{P^{(\frac{1}{2},\frac{1}{2})}_{n}\/}% \nolimits\!\left(x\right)}{\mathop{P^{(\frac{1}{2},\frac{1}{2})}_{n}\/}% \nolimits\!\left(1\right)},$
 18.7.5 $\mathop{V_{n}\/}\nolimits\!\left(x\right)=\ifrac{(2n+1)\mathop{P^{(\frac{1}{2}% ,-\frac{1}{2})}_{n}\/}\nolimits\!\left(x\right)}{\mathop{P^{(\frac{1}{2},-% \frac{1}{2})}_{n}\/}\nolimits\!\left(1\right)},$
 18.7.6 $\mathop{W_{n}\/}\nolimits\!\left(x\right)=\ifrac{\mathop{P^{(-\frac{1}{2},% \frac{1}{2})}_{n}\/}\nolimits\!\left(x\right)}{\mathop{P^{(-\frac{1}{2},\frac{% 1}{2})}_{n}\/}\nolimits\!\left(1\right)}.$
 18.7.7 $\displaystyle\mathop{T^{*}_{n}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{T_{n}\/}\nolimits\!\left(2x-1\right),$ Symbols: $\mathop{T_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: Chebyshev polynomial of the first kind, $\mathop{T^{*}_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: 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) 18.7.8 $\displaystyle\mathop{U^{*}_{n}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{U_{n}\/}\nolimits\!\left(2x-1\right).$

### Legendre, Ultraspherical, and Jacobi

 18.7.9 $\mathop{P_{n}\/}\nolimits\!\left(x\right)=\mathop{C^{(\frac{1}{2})}_{n}\/}% \nolimits\!\left(x\right)=\mathop{P^{(0,0)}_{n}\/}\nolimits\!\left(x\right).$
 18.7.10 $\mathop{P^{*}_{n}\/}\nolimits\!\left(x\right)=\mathop{P_{n}\/}\nolimits\!\left% (2x-1\right).$

### Hermite

 18.7.11 $\displaystyle\mathop{\mathit{He}_{n}\/}\nolimits\!\left(x\right)$ $\displaystyle=2^{-\frac{1}{2}n}\mathop{H_{n}\/}\nolimits\!\left(2^{-\frac{1}{2% }}x\right),$ Symbols: $\mathop{H_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: Hermite polynomial, $\mathop{\mathit{He}_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: 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) 18.7.12 $\displaystyle\mathop{H_{n}\/}\nolimits\!\left(x\right)$ $\displaystyle=2^{\frac{1}{2}n}\mathop{\mathit{He}_{n}\/}\nolimits\!\left(2^{% \frac{1}{2}}x\right).$ Symbols: $\mathop{H_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: Hermite polynomial, $\mathop{\mathit{He}_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: 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)

 18.7.13 $\displaystyle\frac{\mathop{P^{(\alpha,\alpha)}_{2n}\/}\nolimits\!\left(x\right% )}{\mathop{P^{(\alpha,\alpha)}_{2n}\/}\nolimits\!\left(1\right)}$ $\displaystyle=\frac{\mathop{P^{(\alpha,-\frac{1}{2})}_{n}\/}\nolimits\!\left(2% x^{2}-1\right)}{\mathop{P^{(\alpha,-\frac{1}{2})}_{n}\/}\nolimits\!\left(1% \right)},$ Symbols: $\mathop{P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\/}\nolimits\!\left(\NVar{x% }\right)$: Jacobi polynomial, $n$: nonnegative integer and $x$: real variable Referenced by: §18.7(ii) Permalink: http://dlmf.nist.gov/18.7.E13 Encodings: TeX, pMML, png See also: Annotations for 18.7(ii) 18.7.14 $\displaystyle\frac{\mathop{P^{(\alpha,\alpha)}_{2n+1}\/}\nolimits\!\left(x% \right)}{\mathop{P^{(\alpha,\alpha)}_{2n+1}\/}\nolimits\!\left(1\right)}$ $\displaystyle=\frac{x\mathop{P^{(\alpha,\frac{1}{2})}_{n}\/}\nolimits\!\left(2% x^{2}-1\right)}{\mathop{P^{(\alpha,\frac{1}{2})}_{n}\/}\nolimits\!\left(1% \right)}.$ Symbols: $\mathop{P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\/}\nolimits\!\left(\NVar{x% }\right)$: 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)
 18.7.15 $\displaystyle\mathop{C^{(\lambda)}_{2n}\/}\nolimits\!\left(x\right)$ $\displaystyle=\frac{{\left(\lambda\right)_{n}}}{{\left(\tfrac{1}{2}\right)_{n}% }}\mathop{P^{(\lambda-\frac{1}{2},-\frac{1}{2})}_{n}\/}\nolimits\!\left(2x^{2}% -1\right),$ 18.7.16 $\displaystyle\mathop{C^{(\lambda)}_{2n+1}\/}\nolimits\!\left(x\right)$ $\displaystyle=\frac{{\left(\lambda\right)_{n+1}}}{{\left(\frac{1}{2}\right)_{n% +1}}}x\mathop{P^{(\lambda-\frac{1}{2},\frac{1}{2})}_{n}\/}\nolimits\!\left(2x^% {2}-1\right).$
 18.7.17 $\displaystyle\mathop{U_{2n}\/}\nolimits\!\left(x\right)$ $\displaystyle=\mathop{V_{n}\/}\nolimits\!\left(2x^{2}-1\right),$ Symbols: $\mathop{U_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: Chebyshev polynomial of the second kind, $\mathop{V_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: Chebyshev polynomial of the third kind, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.5.30 Referenced by: §18.7(ii) Permalink: http://dlmf.nist.gov/18.7.E17 Encodings: TeX, pMML, png See also: Annotations for 18.7(ii) 18.7.18 $\displaystyle\mathop{T_{2n+1}\/}\nolimits\!\left(x\right)$ $\displaystyle=x\mathop{W_{n}\/}\nolimits\!\left(2x^{2}-1\right).$ Symbols: $\mathop{T_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: Chebyshev polynomial of the first kind, $\mathop{W_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: Chebyshev polynomial of the fourth kind, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.5.29 Referenced by: §18.7(ii) Permalink: http://dlmf.nist.gov/18.7.E18 Encodings: TeX, pMML, png See also: Annotations for 18.7(ii)
 18.7.19 $\displaystyle\mathop{H_{2n}\/}\nolimits\!\left(x\right)$ $\displaystyle=(-1)^{n}2^{2n}n!\mathop{L^{(-\frac{1}{2})}_{n}\/}\nolimits\!% \left(x^{2}\right),$ 18.7.20 $\displaystyle\mathop{H_{2n+1}\/}\nolimits\!\left(x\right)$ $\displaystyle=(-1)^{n}2^{2n+1}n!\,x\mathop{L^{(\frac{1}{2})}_{n}\/}\nolimits\!% \left(x^{2}\right).$

## §18.7(iii) Limit Relations

### Jacobi $\to$ Laguerre

 18.7.21 $\lim_{\beta\to\infty}\mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits\!\left(1-(% \ifrac{2x}{\beta})\right)=\mathop{L^{(\alpha)}_{n}\/}\nolimits\!\left(x\right).$
 18.7.22 $\lim_{\alpha\to\infty}\mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits\!\left((2x/% \alpha)-1\right)=(-1)^{n}\mathop{L^{(\beta)}_{n}\/}\nolimits\!\left(x\right).$

### Jacobi $\to$ Hermite

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

### Ultraspherical $\to$ Hermite

 18.7.24 $\lim_{\lambda\to\infty}\lambda^{-\frac{1}{2}n}\mathop{C^{(\lambda)}_{n}\/}% \nolimits\!\left(\lambda^{-\frac{1}{2}}x\right)=\frac{\mathop{H_{n}\/}% \nolimits\!\left(x\right)}{n!}.$
 18.7.25 $\lim_{\lambda\to 0}\frac{1}{\lambda}\mathop{C^{(\lambda)}_{n}\/}\nolimits\!% \left(x\right)=\frac{2}{n}\mathop{T_{n}\/}\nolimits\!\left(x\right),$ $n\geq 1$.

### Laguerre $\to$ Hermite

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

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