# §18.11 Relations to Other Functions

## §18.11(i) Explicit Formulas

See §§18.5(i) and 18.5(iii) for relations to trigonometric functions, the hypergeometric function, and generalized hypergeometric functions.

### Ultraspherical

 18.11.1 $\mathsf{P}^{m}_{n}\left(x\right)={\left(\tfrac{1}{2}\right)_{m}}(-2)^{m}(1-x^{% 2})^{\frac{1}{2}m}C^{(m+\frac{1}{2})}_{n-m}\left(x\right)={\left(n+1\right)_{m% }}(-2)^{-m}(1-x^{2})^{\frac{1}{2}m}P^{(m,m)}_{n-m}\left(x\right),$ $0\leq m\leq n$.

For the Ferrers function $\mathsf{P}^{m}_{n}\left(x\right)$, see §14.3(i).

Compare also (14.3.21) and (14.3.22).

### Laguerre

 18.11.2 $L^{(\alpha)}_{n}\left(x\right)=\frac{{\left(\alpha+1\right)_{n}}}{n!}M\left(-n% ,\alpha+1,x\right)=\frac{(-1)^{n}}{n!}U\left(-n,\alpha+1,x\right)=\frac{{\left% (\alpha+1\right)_{n}}}{n!}x^{-\frac{1}{2}(\alpha+1)}e^{\frac{1}{2}x}M_{n+\frac% {1}{2}(\alpha+1),\frac{1}{2}\alpha}\left(x\right)=\frac{(-1)^{n}}{n!}x^{-\frac% {1}{2}(\alpha+1)}e^{\frac{1}{2}x}W_{n+\frac{1}{2}(\alpha+1),\frac{1}{2}\alpha}% \left(x\right).$

For the confluent hypergeometric functions $M\left(a,b,x\right)$ and $U\left(a,b,x\right)$, see §13.2(i), and for the Whittaker functions $M_{\kappa,\mu}\left(x\right)$ and $W_{\kappa,\mu}\left(x\right)$ see §13.14(i).

### Hermite

 18.11.3 $\displaystyle H_{n}\left(x\right)$ $\displaystyle=2^{n}U\left(-\tfrac{1}{2}n,\tfrac{1}{2},x^{2}\right)=2^{n}xU% \left(-\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{3}{2},x^{2}\right)=2^{\frac{1}{2}n}e^% {\frac{1}{2}x^{2}}U\left(-n-\tfrac{1}{2},2^{\frac{1}{2}}x\right).$ ⓘ Symbols: $H_{\NVar{n}}\left(\NVar{x}\right)$: Hermite polynomial, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$: parabolic cylinder function, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.5.55, 22.5.58 ((factor $x$ missing on RHS of 22.5.55)) Referenced by: §18.11(i), §18.15(v) Permalink: http://dlmf.nist.gov/18.11.E3 Encodings: TeX, pMML, png See also: Annotations for 18.11(i), 18.11(i), 18.11 and 18 18.11.4 $\displaystyle\mathit{He}_{n}\left(x\right)$ $\displaystyle=2^{\frac{1}{2}n}U\left(-\tfrac{1}{2}n,\tfrac{1}{2},\tfrac{1}{2}x% ^{2}\right)=2^{\frac{1}{2}(n-1)}xU\left(-\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{3}{% 2},\tfrac{1}{2}x^{2}\right)=e^{\tfrac{1}{4}x^{2}}U\left(-n-\tfrac{1}{2},x% \right).$

For the parabolic cylinder function $U\left(a,z\right)$, see §12.2.

## §18.11(ii) Formulas of Mehler–Heine Type

### Jacobi

 18.11.5 $\lim_{n\to\infty}\frac{1}{n^{\alpha}}P^{(\alpha,\beta)}_{n}\left(1-\frac{z^{2}% }{2n^{2}}\right)=\lim_{n\to\infty}\frac{1}{n^{\alpha}}P^{(\alpha,\beta)}_{n}% \left(\cos\frac{z}{n}\right)=\frac{2^{\alpha}}{z^{\alpha}}J_{\alpha}\left(z% \right).$ ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial, $\cos\NVar{z}$: cosine function, $z$: complex variable and $n$: nonnegative integer A&S Ref: 22.15.1 Referenced by: §18.11(ii), §18.11(ii) Permalink: http://dlmf.nist.gov/18.11.E5 Encodings: TeX, pMML, png See also: Annotations for 18.11(ii), 18.11(ii), 18.11 and 18

### Laguerre

 18.11.6 $\lim_{n\to\infty}\frac{1}{n^{\alpha}}L^{(\alpha)}_{n}\left(\frac{z}{n}\right)=% \frac{1}{z^{\frac{1}{2}\alpha}}J_{\alpha}\left(2z^{\frac{1}{2}}\right).$ ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}\right)$: Laguerre (or generalized Laguerre) polynomial, $z$: complex variable and $n$: nonnegative integer A&S Ref: 22.15.2 Referenced by: §18.11(ii) Permalink: http://dlmf.nist.gov/18.11.E6 Encodings: TeX, pMML, png See also: Annotations for 18.11(ii), 18.11(ii), 18.11 and 18

### Hermite

 18.11.7 $\displaystyle\lim_{n\to\infty}\frac{(-1)^{n}n^{\frac{1}{2}}}{2^{2n}n!}H_{2n}% \left(\frac{z}{2n^{\frac{1}{2}}}\right)$ $\displaystyle=\frac{1}{\pi^{\frac{1}{2}}}\cos z,$ 18.11.8 $\displaystyle\lim_{n\to\infty}\frac{(-1)^{n}}{2^{2n}n!}H_{2n+1}\left(\frac{z}{% 2n^{\frac{1}{2}}}\right)$ $\displaystyle=\frac{2}{\pi^{\frac{1}{2}}}\sin z.$

For the Bessel function $J_{\nu}\left(z\right)$, see §10.2(ii). The limits (18.11.5)–(18.11.8) hold uniformly for $z$ in any bounded subset of $\mathbb{C}$.