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

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

Compare also (14.3.21) and (14.3.22).

# Laguerre

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

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

# Hermite

 18.11.3 $\displaystyle\mathop{H_{n}\/}\nolimits\!\left(x\right)$ $\displaystyle=2^{n}\mathop{U\/}\nolimits\!\left(-\tfrac{1}{2}n,\tfrac{1}{2},x^% {2}\right)$ $\displaystyle=2^{n}x\mathop{U\/}\nolimits\!\left(-\tfrac{1}{2}n+\tfrac{1}{2},% \tfrac{3}{2},x^{2}\right)$ $\displaystyle=2^{\frac{1}{2}n}e^{\frac{1}{2}x^{2}}\mathop{U\/}\nolimits\!\left% (-n-\tfrac{1}{2},2^{\frac{1}{2}}x\right).$ Symbols: $\mathop{H_{n}\/}\nolimits\!\left(x\right)$: Hermite polynomial, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$: Kummer confluent hypergeometric function, $e$: base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$: 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 18.11.4 $\displaystyle\mathop{\mathit{He}_{n}\/}\nolimits\!\left(x\right)$ $\displaystyle=2^{\frac{1}{2}n}\mathop{U\/}\nolimits\!\left(-\tfrac{1}{2}n,% \tfrac{1}{2},\tfrac{1}{2}x^{2}\right)$ $\displaystyle=2^{\frac{1}{2}(n-1)}x\mathop{U\/}\nolimits\!\left(-\tfrac{1}{2}n% +\tfrac{1}{2},\tfrac{3}{2},\tfrac{1}{2}x^{2}\right)$ $\displaystyle=e^{\tfrac{1}{4}x^{2}}\mathop{U\/}\nolimits\!\left(-n-\tfrac{1}{2% },x\right).$

For the parabolic cylinder function $\mathop{U\/}\nolimits\!\left(a,z\right)$, see §12.2.

# Jacobi

 18.11.5 $\lim_{n\to\infty}\frac{1}{n^{\alpha}}\mathop{P^{(\alpha,\beta)}_{n}\/}% \nolimits\!\left(1-\frac{z^{2}}{2n^{2}}\right)=\lim_{n\to\infty}\frac{1}{n^{% \alpha}}\mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\frac{z}{n}\right)=\frac{2^{\alpha}}{z^{\alpha}}\mathop{J_{\alpha}\/}% \nolimits\!\left(z\right).$

# Laguerre

 18.11.6 $\lim_{n\to\infty}\frac{1}{n^{\alpha}}\mathop{L^{(\alpha)}_{n}\/}\nolimits\!% \left(\frac{z}{n}\right)=\frac{1}{z^{\frac{1}{2}\alpha}}\mathop{J_{\alpha}\/}% \nolimits\!\left(2z^{\frac{1}{2}}\right).$

# Hermite

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

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