# §18.15 Asymptotic Approximations

## §18.15(i) Jacobi

With the exception of the penultimate paragraph, we assume throughout this subsection that $\alpha$, $\beta$, and $M$ ($=0,1,2,\ldots$) are all fixed.

 18.15.1 $\left(\mathop{\sin\/}\nolimits\tfrac{1}{2}\theta\right)^{\alpha+\frac{1}{2}}% \left(\mathop{\cos\/}\nolimits\tfrac{1}{2}\theta\right)^{\beta+\frac{1}{2}}% \mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits% \theta\right)=\pi^{-1}2^{2n+\alpha+\beta+1}\mathop{\mathrm{B}\/}\nolimits\!% \left(n+\alpha+1,n+\beta+1\right)\*\left(\sum_{m=0}^{M-1}\frac{f_{m}(\theta)}{% 2^{m}{\left(2n+\alpha+\beta+2\right)_{m}}}+\mathop{O\/}\nolimits\!\left(n^{-M}% \right)\right),$

as $n\to\infty$, uniformly with respect to $\theta\in[\delta,\pi-\delta]$. Here, and elsewhere in §18.15, $\delta$ is an arbitrary small positive constant. Also, $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ is the beta function (§5.12) and

 18.15.2 $f_{m}(\theta)=\sum_{\ell=0}^{m}\frac{C_{m,\ell}(\alpha,\beta)}{\ell!(m-\ell)!}% \frac{\mathop{\cos\/}\nolimits\theta_{n,m,\ell}}{\left(\mathop{\sin\/}% \nolimits\frac{1}{2}\theta\right)^{\ell}\left(\mathop{\cos\/}\nolimits\frac{1}% {2}\theta\right)^{m-\ell}},$ Defines: $f_{m}(\theta)$ (locally) Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $!$: factorial (as in $n!$), $\mathop{\sin\/}\nolimits\NVar{z}$: sine function, $\ell$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $C_{m,\ell}(\alpha,\beta)$ and $\theta_{n,m,\ell}$ Permalink: http://dlmf.nist.gov/18.15.E2 Encodings: TeX, pMML, png See also: Annotations for 18.15(i)

where

 18.15.3 $C_{m,\ell}(\alpha,\beta)={\left(\tfrac{1}{2}+\alpha\right)_{\ell}}{\left(% \tfrac{1}{2}-\alpha\right)_{\ell}}{\left(\tfrac{1}{2}+\beta\right)_{m-\ell}}{% \left(\tfrac{1}{2}-\beta\right)_{m-\ell}},$ Defines: $C_{m,\ell}(\alpha,\beta)$ (locally) Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\ell$: nonnegative integer and $m$: nonnegative integer Permalink: http://dlmf.nist.gov/18.15.E3 Encodings: TeX, pMML, png See also: Annotations for 18.15(i)

and

 18.15.4 $\theta_{n,m,\ell}=\tfrac{1}{2}(2n+\alpha+\beta+m+1)\theta-\tfrac{1}{2}(\alpha+% \ell+\tfrac{1}{2})\pi.$ Defines: $\theta_{n,m,\ell}$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\ell$: nonnegative integer, $m$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/18.15.E4 Encodings: TeX, pMML, png See also: Annotations for 18.15(i)

When $\alpha,\beta\in(-\frac{1}{2},\frac{1}{2})$, the error term in (18.15.1) is less than twice the first neglected term in absolute value. See Hahn (1980), where corresponding results are given when $x$ is replaced by a complex variable $z$ that is bounded away from the orthogonality interval $[-1,1]$.

Next, let

 18.15.5 $\rho=n+\tfrac{1}{2}(\alpha+\beta+1).$ Defines: $\rho$ (locally) Symbols: $n$: nonnegative integer Referenced by: §18.16(ii) Permalink: http://dlmf.nist.gov/18.15.E5 Encodings: TeX, pMML, png See also: Annotations for 18.15(i)

Then as $n\rightarrow\infty$,

 18.15.6 $(\mathop{\sin\/}\nolimits\tfrac{1}{2}\theta)^{\alpha+\frac{1}{2}}(\mathop{\cos% \/}\nolimits\tfrac{1}{2}\theta)^{\beta+\frac{1}{2}}\mathop{P^{(\alpha,\beta)}_% {n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)=\frac{\mathop{% \Gamma\/}\nolimits\!\left(n+\alpha+1\right)}{2^{\frac{1}{2}}\rho^{\alpha}n!}\*% \left(\theta^{\frac{1}{2}}\mathop{J_{\alpha}\/}\nolimits\!\left(\rho\theta% \right)\sum_{m=0}^{M}\dfrac{A_{m}(\theta)}{\rho^{2m}}+\theta^{\frac{3}{2}}% \mathop{J_{\alpha+1}\/}\nolimits\!\left(\rho\theta\right)\sum_{m=0}^{M-1}% \dfrac{B_{m}(\theta)}{\rho^{2m+1}}+\varepsilon_{M}(\rho,\theta)\right),$

where $\mathop{J_{\nu}\/}\nolimits\!\left(z\right)$ is the Bessel function (§10.2(ii)), and

 18.15.7 $\varepsilon_{M}(\rho,\theta)=\begin{cases}\theta\mathop{O\/}\nolimits\!\left(% \rho^{-2M-(3/2)}\right),&c\rho^{-1}\leq\theta\leq\pi-\delta,\\ \theta^{\alpha+(5/2)}\mathop{O\/}\nolimits\!\left(\rho^{-2M+\alpha}\right),&0% \leq\theta\leq c\rho^{-1},\end{cases}$ Defines: $\varepsilon_{M}(\rho,\theta)$ (locally) Symbols: $\mathop{O\/}\nolimits\!\left(\NVar{x}\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\delta$: arbitary small positive constant and $\rho$ Permalink: http://dlmf.nist.gov/18.15.E7 Encodings: TeX, pMML, png See also: Annotations for 18.15(i)

with $c$ denoting an arbitrary positive constant. Also,

 18.15.8 $\displaystyle A_{0}(\theta)$ $\displaystyle=1,$ $\displaystyle\theta B_{0}(\theta)$ $\displaystyle=\frac{1}{4}g(\theta),$ $\displaystyle A_{1}(\theta)$ $\displaystyle=\frac{1}{8}g^{\prime}(\theta)-\frac{1+2\alpha}{8}\frac{g(\theta)% }{\theta}-\frac{1}{32}(g(\theta))^{2},$ Defines: $A_{m}(\theta)$: coefficient (locally) and $B_{m}(\theta)$: coefficient (locally) Symbols: $m$: nonnegative integer and $g(\theta)$ Permalink: http://dlmf.nist.gov/18.15.E8 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 18.15(i)

where

 18.15.9 $g(\theta)=\left(\tfrac{1}{4}-\alpha^{2}\right)\left(\mathop{\cot\/}\nolimits\!% \left(\tfrac{1}{2}\theta\right)-\left(\tfrac{1}{2}\theta\right)^{-1}\right)-% \left(\tfrac{1}{4}-\beta^{2}\right)\mathop{\tan\/}\nolimits\!\left(\tfrac{1}{2% }\theta\right).$ Defines: $g(\theta)$ (locally) Symbols: $\mathop{\cot\/}\nolimits\NVar{z}$: cotangent function and $\mathop{\tan\/}\nolimits\NVar{z}$: tangent function Referenced by: §18.15(i) Permalink: http://dlmf.nist.gov/18.15.E9 Encodings: TeX, pMML, png See also: Annotations for 18.15(i)

For higher coefficients see Baratella and Gatteschi (1988), and for another estimate of the error term in a related expansion see Wong and Zhao (2003). For large $\beta$, fixed $\alpha$, and $0\leq n/\beta\leq c$, Dunster (1999) gives asymptotic expansions of $\mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits\!\left(z\right)$ that are uniform in unbounded complex $z$-domains containing $z=\pm 1$. These expansions are in terms of Whittaker functions (§13.14). This reference also supplies asymptotic expansions of $\mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits\!\left(z\right)$ for large $n$, fixed $\alpha$, and $0\leq\beta/n\leq c$. The latter expansions are in terms of Bessel functions, and are uniform in complex $z$-domains not containing neighborhoods of 1. For a complementary result, see Wong and Zhao (2004). By using the symmetry property given in the second row of Table 18.6.1, the roles of $\alpha$ and $\beta$ can be interchanged.

For an asymptotic expansion of $\mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits\!\left(z\right)$ as $n\to\infty$ that holds uniformly for complex $z$ bounded away from $[-1,1]$, see Elliott (1971). The first term of this expansion also appears in Szegő (1975, Theorem 8.21.7).

## §18.15(ii) Ultraspherical

For fixed $\lambda\in(0,1)$ and fixed $M=0,1,2,\ldots,$

 18.15.10 $\mathop{C^{(\lambda)}_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta% \right)=\frac{2^{2\lambda}\mathop{\Gamma\/}\nolimits\!\left(\lambda+\frac{1}{2% }\right)}{\pi^{\frac{1}{2}}\mathop{\Gamma\/}\nolimits\!\left(\lambda+1\right)}% \frac{{\left(2\lambda\right)_{n}}}{{\left(\lambda+1\right)_{n}}}\*\left(\sum_{% m=0}^{M-1}\dfrac{{\left(\lambda\right)_{m}}{\left(1-\lambda\right)_{m}}}{m!\,{% \left(n+\lambda+1\right)_{m}}}\dfrac{\mathop{\cos\/}\nolimits\theta_{n,m}}{(2% \mathop{\sin\/}\nolimits\theta)^{m+\lambda}}+\mathop{O\/}\nolimits\!\left(% \frac{1}{n^{M}}\right)\right),$

as $n\to\infty$ uniformly with respect to $\theta\in[\delta,\pi-\delta]$, where

 18.15.11 $\theta_{n,m}=(n+m+\lambda)\theta-\tfrac{1}{2}(m+\lambda)\pi.$ Defines: $\theta_{n,m}$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $m$: nonnegative integer and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/18.15.E11 Encodings: TeX, pMML, png See also: Annotations for 18.15(ii)

For a bound on the error term in (18.15.10) see Szegő (1975, Theorem 8.21.11).

Asymptotic expansions for $\mathop{C^{(\lambda)}_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)$ can be obtained from the results given in §18.15(i) by setting $\alpha=\beta=\lambda-\frac{1}{2}$ and referring to (18.7.1). See also Szegő (1933) and Szegő (1975, Eq. (8.21.14)).

## §18.15(iii) Legendre

For fixed $M=0,1,2,\dots$,

 18.15.12 $\mathop{P_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)=\left(% \frac{2}{\mathop{\sin\/}\nolimits\theta}\right)^{\frac{1}{2}}\sum_{m=0}^{M-1}% \binom{-\tfrac{1}{2}}{m}\binom{m-\frac{1}{2}}{n}\frac{\mathop{\cos\/}\nolimits% \alpha_{n,m}}{(2\mathop{\sin\/}\nolimits\theta)^{m}}+\mathop{O\/}\nolimits\!% \left(\frac{1}{n^{M+\frac{1}{2}}}\right),$

as $n\to\infty$, uniformly with respect to $\theta\in[\delta,\pi-\delta]$, where

 18.15.13 $\alpha_{n,m}=(n-m+\tfrac{1}{2})\theta+(n-\tfrac{1}{2}m-\tfrac{1}{4})\pi.$ Defines: $\alpha_{n,m}$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $m$: nonnegative integer and $n$: nonnegative integer Referenced by: §18.15(iii) Permalink: http://dlmf.nist.gov/18.15.E13 Encodings: TeX, pMML, png See also: Annotations for 18.15(iii)

Also, when $\tfrac{1}{6}\pi<\theta<\tfrac{5}{6}\pi$, the right-hand side of (18.15.12) with $M=\infty$ converges; paradoxically, however, the sum is $2\!\mathop{P_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)$ and not $\mathop{P_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)$ as stated erroneously in Szegő (1975, §8.4(3)).

For these results and further information see Olver (1997b, pp. 311–313). Another expansion follows from (18.15.10) by taking $\lambda=\tfrac{1}{2}$; see Szegő (1975, Theorem 8.21.5).

For asymptotic expansions of $\mathop{P_{n}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right)$ and $\mathop{P_{n}\/}\nolimits\!\left(\mathop{\cosh\/}\nolimits\xi\right)$ that are uniformly valid when $0\leq\theta\leq\pi-\delta$ and $0\leq\xi<\infty$ see §14.15(iii) with $\mu=0$ and $\nu=n$. These expansions are in terms of Bessel functions and modified Bessel functions, respectively.

## §18.15(iv) Laguerre

### In Terms of Elementary Functions

For fixed $M=0,1,2,\dots$, and fixed $\alpha$,

 18.15.14 $\mathop{L^{(\alpha)}_{n}\/}\nolimits\!\left(x\right)=\frac{n^{\frac{1}{2}% \alpha-\frac{1}{4}}e^{\frac{1}{2}x}}{\pi^{\frac{1}{2}}x^{\frac{1}{2}\alpha+% \frac{1}{4}}}\left(\mathop{\cos\/}\nolimits\theta_{n}^{(\alpha)}(x)\left(\sum_% {m=0}^{M-1}\frac{a_{m}(x)}{n^{\frac{1}{2}m}}+\mathop{O\/}\nolimits\!\left(% \frac{1}{n^{\frac{1}{2}M}}\right)\right)+\mathop{\sin\/}\nolimits\theta_{n}^{(% \alpha)}(x)\left(\sum_{m=1}^{M-1}\frac{b_{m}(x)}{n^{\frac{1}{2}m}}+\mathop{O\/% }\nolimits\!\left(\frac{1}{n^{\frac{1}{2}M}}\right)\right)\right),$

as $n\to\infty$, uniformly on compact $x$-intervals in $(0,\infty)$, where

 18.15.15 $\theta_{n}^{(\alpha)}(x)=2(nx)^{\frac{1}{2}}-\left(\tfrac{1}{2}\alpha+\tfrac{1% }{4}\right)\pi.$ Defines: $\theta_{n}^{(\alpha)}(x)$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.15.E15 Encodings: TeX, pMML, png See also: Annotations for 18.15(iv)

The leading coefficients are given by

 18.15.16 $\displaystyle a_{0}(x)$ $\displaystyle=1,$ $\displaystyle a_{1}(x)$ $\displaystyle=0,$ $\displaystyle b_{1}(x)$ $\displaystyle=\frac{1}{48x^{\frac{1}{2}}}\left(4x^{2}-12\alpha^{2}-24\alpha x-% 24x+3\right).$ Defines: $a_{m}(x)$: coefficients (locally) and $b_{m}(x)$: coefficients (locally) Symbols: $m$: nonnegative integer and $x$: real variable Referenced by: §18.15(iv) Permalink: http://dlmf.nist.gov/18.15.E16 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 18.15(iv)

### In Terms of Bessel Functions

Define

 18.15.17 $\nu=4n+2\alpha+2,$ Defines: $\nu$ (locally) Symbols: $n$: nonnegative integer Referenced by: §18.15(iv), §18.16(iv) Permalink: http://dlmf.nist.gov/18.15.E17 Encodings: TeX, pMML, png See also: Annotations for 18.15(iv)
 18.15.18 $\xi=\tfrac{1}{2}\left(\sqrt{x-x^{2}}+\mathop{\mathrm{arcsin}\/}\nolimits(\sqrt% {x})\right),$ $0\leq x\leq 1$. Defines: $\xi$ (locally) Symbols: $\mathop{\mathrm{arcsin}\/}\nolimits\NVar{z}$: arcsine function and $x$: real variable Permalink: http://dlmf.nist.gov/18.15.E18 Encodings: TeX, pMML, png See also: Annotations for 18.15(iv)

Then for fixed $M=0,1,2,\dots$, and fixed $\alpha$,

 18.15.19 $\mathop{L^{(\alpha)}_{n}\/}\nolimits\!\left(\nu x\right)=\frac{e^{\frac{1}{2}% \nu x}}{2^{\alpha}x^{\frac{1}{2}\alpha+\frac{1}{4}}(1-x)^{\frac{1}{4}}}\left(% \xi^{\frac{1}{2}}\mathop{J_{\alpha}\/}\nolimits\!\left(\nu\xi\right)\sum_{m=0}% ^{M-1}\frac{A_{m}(\xi)}{\nu^{2m}}+\xi^{-\frac{1}{2}}\mathop{J_{\alpha+1}\/}% \nolimits\!\left(\nu\xi\right)\sum_{m=0}^{M-1}\frac{B_{m}(\xi)}{\nu^{2m+1}}+% \xi^{\frac{1}{2}}\mathop{\mathrm{env}\mskip-2.0mu J_{\alpha}\/}\nolimits\!% \left(\nu\xi\right)\mathop{O\/}\nolimits\!\left(\frac{1}{\nu^{2M-1}}\right)% \right),$

as $n\to\infty$ uniformly for $0\leq x\leq 1-\delta$. Here $\mathop{J_{\nu}\/}\nolimits\!\left(z\right)$ denotes the Bessel function (§10.2(ii)), $\mathop{\mathrm{env}\mskip-2.0mu J_{\nu}\/}\nolimits\!\left(z\right)$ denotes its envelope (§2.8(iv)), and $\delta$ is again an arbitrary small positive constant. The leading coefficients are given by $A_{0}(\xi)=1$ and

 18.15.20 $B_{0}(\xi)=-\frac{1}{2}\left(\frac{1-4\alpha^{2}}{8}+\xi\left(\frac{1-x}{x}% \right)^{\frac{1}{2}}\left(\frac{4\alpha^{2}-1}{8}+\frac{1}{4}\frac{x}{1-x}+% \frac{5}{24}\left(\frac{x}{1-x}\right)^{2}\right)\right).$ Defines: $B_{m}(\xi)$: coefficient (locally) Symbols: $m$: nonnegative integer, $\xi$ and $x$: real variable Permalink: http://dlmf.nist.gov/18.15.E20 Encodings: TeX, pMML, png See also: Annotations for 18.15(iv)

### In Terms of Airy Functions

Again define $\nu$ as in (18.15.17); also,

 18.15.21 $\displaystyle\zeta$ $\displaystyle=-\left(\tfrac{3}{4}\left(\mathop{\mathrm{arccos}\/}\nolimits\!% \left(\sqrt{x}\right)-\sqrt{x-x^{2}}\right)\right)^{\frac{2}{3}},$ $0\leq x\leq 1$, $\displaystyle\zeta$ $\displaystyle=\left(\tfrac{3}{4}\left(\sqrt{x^{2}-x}-\mathop{\mathrm{arccosh}% \/}\nolimits\!\left(\sqrt{x}\right)\right)\right)^{\frac{2}{3}},$ $x\geq 1$. Defines: $\zeta$ (locally) Symbols: $\mathop{\mathrm{arccosh}\/}\nolimits\NVar{z}$: inverse hyperbolic cosine function, $\mathop{\mathrm{arccos}\/}\nolimits\NVar{z}$: arccosine function and $x$: real variable Permalink: http://dlmf.nist.gov/18.15.E21 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 18.15(iv)

Then for fixed $M=0,1,2,\dots$, and fixed $\alpha$,

 18.15.22 $\mathop{L^{(\alpha)}_{n}\/}\nolimits\!\left(\nu x\right)=(-1)^{n}\frac{e^{% \frac{1}{2}\nu x}}{2^{\alpha-\frac{1}{2}}x^{\frac{1}{2}\alpha+\frac{1}{4}}}\*% \left(\frac{\zeta}{x-1}\right)^{\frac{1}{4}}\left(\frac{\mathop{\mathrm{Ai}\/}% \nolimits\!\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{1}{3}}}\sum_{m=0}^{% M-1}\frac{E_{m}(\zeta)}{\nu^{2m}}+\frac{\mathop{\mathrm{Ai}\/}\nolimits'\!% \left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{5}{3}}}\sum_{m=0}^{M-1}\frac{F% _{m}(\zeta)}{\nu^{2m}}+\mathop{\mathrm{envAi}\/}\nolimits\!\left(\nu^{\frac{2}% {3}}\zeta\right)\mathop{O\/}\nolimits\!\left(\frac{1}{\nu^{2M-\frac{2}{3}}}% \right)\right),$ Symbols: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(\NVar{z}\right)$: Airy function, $\mathop{O\/}\nolimits\!\left(\NVar{x}\right)$: order not exceeding, $\mathop{L^{(\NVar{\alpha})}_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: Laguerre (or generalized Laguerre) polynomial, $\sim$: Poincaré asymptotic expansion, $\mathop{\mathrm{envAi}\/}\nolimits\!\left(\NVar{x}\right)$: envelope of Airy function $\mathop{\mathrm{Ai}\/}\nolimits\!\left(\NVar{x}\right)$, $\mathrm{e}$: base of exponential function, $m$: nonnegative integer, $n$: nonnegative integer, $\nu$, $\zeta$, $E_{m}$: coefficient, $F_{m}$: coefficient and $x$: real variable Referenced by: Other Changes Permalink: http://dlmf.nist.gov/18.15.E22 Encodings: TeX, pMML, png Errata (effective with 1.0.11): Originally this equation was expressed in terms of the asymptotic symbol $\sim$. As a consequence of the use of the $\mathop{O\/}\nolimits$ order symbol on the right hand side, $\sim$ was replaced by $=$. Reported 2015-10-27 See also: Annotations for 18.15(iv)

as $n\to\infty$ uniformly for $\delta\leq x<\infty$. Here $\mathop{\mathrm{Ai}\/}\nolimits$ denotes the Airy function (§9.2), $\mathop{\mathrm{Ai}\/}\nolimits'$ denotes its derivative, and $\mathop{\mathrm{envAi}\/}\nolimits$ denotes its envelope (§2.8(iii)). The leading coefficients are given by $E_{0}(\zeta)=1$ and

 18.15.23 $F_{0}(\zeta)=-\frac{5}{48\zeta^{2}}+\left(\frac{x-1}{x\zeta}\right)^{\frac{1}{% 2}}\left(\frac{1}{2}\alpha^{2}-\frac{1}{8}-\frac{1}{4}\frac{x}{x-1}+\frac{5}{2% 4}\left(\frac{x}{x-1}\right)^{2}\right),$ $0\leq x<\infty$. Defines: $F_{m}$: coefficient (locally) Symbols: $m$: nonnegative integer, $\zeta$ and $x$: real variable Permalink: http://dlmf.nist.gov/18.15.E23 Encodings: TeX, pMML, png See also: Annotations for 18.15(iv)

## §18.15(v) Hermite

Define

 18.15.24 $\mu=2n+1,$ Symbols: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/18.15.E24 Encodings: TeX, pMML, png See also: Annotations for 18.15(v)
 18.15.25 $\lambda_{n}=\begin{cases}\ifrac{\mathop{\Gamma\/}\nolimits\!\left(n+1\right)}{% \mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}n+1\right)},&n\text{ even},\\ \ifrac{\mathop{\Gamma\/}\nolimits\!\left(n+2\right)}{\left(\mu^{\frac{1}{2}}% \mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}n+\frac{3}{2}\right)\right)},&n% \text{ odd},\end{cases}$ Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/18.15.E25 Encodings: TeX, pMML, png See also: Annotations for 18.15(v)

and

 18.15.26 $\omega_{n,m}(x)=\mu^{\frac{1}{2}}x-\tfrac{1}{2}(m+n)\pi.$ Defines: $\omega_{n,m}(x)$ (locally) Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable Permalink: http://dlmf.nist.gov/18.15.E26 Encodings: TeX, pMML, png See also: Annotations for 18.15(v)

Then for fixed $M=0,1,2,\dots$,

 18.15.27 $\mathop{H_{n}\/}\nolimits\!\left(x\right)=\lambda_{n}e^{\frac{1}{2}x^{2}}\left% (\sum_{m=0}^{M-1}\frac{u_{m}(x)\mathop{\cos\/}\nolimits\omega_{n,m}(x)}{\mu^{% \frac{1}{2}m}}+\mathop{O\/}\nolimits\!\left(\frac{1}{\mu^{\frac{1}{2}M}}\right% )\right),$

as $n\to\infty$, uniformly on compact $x$-intervals on $\mathbb{R}$. The coefficients $u_{m}(x)$ are polynomials in $x$, and $u_{0}(x)=1$, $u_{1}(x)=\frac{1}{6}x^{3}$.

For more powerful asymptotic expansions as $n\to\infty$ in terms of elementary functions that apply uniformly when $1+\delta\leq t<\infty$, $-1+\delta\leq t\leq 1-\delta$, or $-\infty, where $t=\ifrac{x}{\sqrt{2n+1}}$ and $\delta$ is again an arbitrary small positive constant, see §§12.10(i)12.10(iv) and 12.10(vi). And for asymptotic expansions as $n\to\infty$ in terms of Airy functions that apply uniformly when $-1+\delta\leq t<\infty$ or $-\infty, see §§12.10(vii) and 12.10(viii). With $\mu=\sqrt{2n+1}$ the expansions in Chapter 12 are for the parabolic cylinder function $\mathop{U\/}\nolimits\!\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)$, which is related to the Hermite polynomials via

 18.15.28 $\mathop{H_{n}\/}\nolimits\!\left(x\right)=2^{\frac{1}{4}(\mu^{2}-1)}e^{\frac{1% }{2}\mu^{2}t^{2}}\mathop{U\/}\nolimits\!\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{% 2}\right);$

compare (18.11.3).

For an error bound for the first term in the Airy-function expansions see Olver (1997b, p. 403).

The asymptotic behavior of the classical OP’s as $x\to\pm\infty$ with the degree and parameters fixed is evident from their explicit polynomial forms; see, for example, (18.2.7) and the last two columns of Table 18.3.1.
For asymptotic approximations of Jacobi, ultraspherical, and Laguerre polynomials in terms of Hermite polynomials, see López and Temme (1999a). These approximations apply when the parameters are large, namely $\alpha$ and $\beta$ (subject to restrictions) in the case of Jacobi polynomials, $\lambda$ in the case of ultraspherical polynomials, and $|\alpha|+|x|$ in the case of Laguerre polynomials. See also Dunster (1999), Atia et al. (2014) and Temme (2015, Chapter 32).