18 Orthogonal Polynomials Classical Orthogonal Polynomials 18.14 Inequalities 18.16 Zeros

§18.15 Asymptotic Approximations

Keywords:: classical orthogonal polynomials
Referenced by:: §18.15(i)
Permalink:: http://dlmf.nist.gov/18.15

§18.15(i) Jacobi
§18.15(ii) Ultraspherical
§18.15(iii) Legendre
§18.15(iv) Laguerre
§18.15(v) Hermite
§18.15(vi) Other Approximations

§18.15(i) Jacobi

Notes:: See Frenzen and Wong (1985).
Keywords:: Jacobi polynomials
Referenced by:: §13.27, §18.15(ii), §2.8(iv), §2.8(vi)
Permalink:: http://dlmf.nist.gov/18.15.i

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

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, : nonnegative integer, : nonnegative integer and $f_{m}(\theta)$
Referenced by:: §18.15(i)
Permalink:: http://dlmf.nist.gov/18.15.E1
Encodings:: TeX, pMML, png

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 $f_{m}(\theta)=\sum_{{\ell=0}}^{m}\frac{C_{{m,\ell}}(\alpha,\beta)}{\ell!(m-% \ell)!}\frac{\mathop{\cos\/}\nolimits\theta_{{n,m,\ell}}}{\left(\mathop{\sin\/% }\nolimits\frac{1}{2}\theta\right)^{\ell}\left(\mathop{\cos\/}\nolimits\frac{1% }{2}\theta\right)^{{m-\ell}}},$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, : : factorial, $\mathop{\sin\/}\nolimits z$ : sine function, $\ell$ : nonnegative integer, : nonnegative integer, : nonnegative integer, $f_{m}(\theta)$ , $C_{{m,\ell}}(\alpha,\beta)$ and $\theta_{{n,m,\ell}}$
Permalink:: http://dlmf.nist.gov/18.15.E2
Encodings:: TeX, pMML, png

where

18.15.3 $C_{{m,\ell}}(\alpha,\beta)=\left(\tfrac{1}{2}+\alpha\right)_{{\ell}}\left(% \tfrac{1}{2}-\alpha\right)_{{\ell}}\left(\tfrac{1}{2}+\beta\right)_{{m-\ell}}% \left(\tfrac{1}{2}-\beta\right)_{{m-\ell}},$

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\ell$ : nonnegative integer, : nonnegative integer and $C_{{m,\ell}}(\alpha,\beta)$
Permalink:: http://dlmf.nist.gov/18.15.E3
Encodings:: TeX, pMML, png

and

18.15.4 $\theta_{{n,m,\ell}}=\tfrac{1}{2}(2n+\alpha+\beta+m+1)\theta-\tfrac{1}{2}(% \alpha+\ell+\tfrac{1}{2})\pi.$

Symbols:: $\ell$ : nonnegative integer, : nonnegative integer, : nonnegative integer and $\theta_{{n,m,\ell}}$
Permalink:: http://dlmf.nist.gov/18.15.E4
Encodings:: TeX, pMML, png

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 is replaced by a complex variable that is bounded away from the orthogonality interval .

Next, let

18.15.5 $\rho=n+\tfrac{1}{2}(\alpha+\beta+1).$

Symbols:: : nonnegative integer and $\rho$
Referenced by:: ¶ ‣ §18.16(ii)
Permalink:: http://dlmf.nist.gov/18.15.E5
Encodings:: TeX, pMML, png

Then as $n\rightarrow\infty$ ,

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

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function, : : factorial, $\mathop{\sin\/}\nolimits z$ : sine function, : nonnegative integer, : nonnegative integer, $\rho$ , $\varepsilon_{M}(\rho,\theta)$ , $A_{m}(\theta)$ : coefficient and $B_{m}(\theta)$ : coefficient
Permalink:: http://dlmf.nist.gov/18.15.E6
Encodings:: TeX, pMML, png

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

18.15.7 $\varepsilon_{M}(\rho,\theta)=\begin{cases}\theta\mathop{O\/}\nolimits\!\left(% \rho^{{-2M-(3/2)}}\right),&c\rho^{{-1}}\leq\theta\leq\pi-\delta,\\ \theta^{{\alpha+(5/2)}}\mathop{O\/}\nolimits\!\left(\rho^{{-2M+\alpha}}\right)% ,&0\leq\theta\leq c\rho^{{-1}},\end{cases}$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\delta$ : arbitary small positive constant, $\rho$ and $\varepsilon_{M}(\rho,\theta)$
Permalink:: http://dlmf.nist.gov/18.15.E7
Encodings:: TeX, pMML, png

with denoting an arbitrary positive constant. Also,

18.15.8

$A_{0}(\theta)=1,$

$\theta B_{0}(\theta)=\frac{1}{4}g(\theta),$

$A_{1}(\theta)=\frac{1}{8}g^{{\prime}}(\theta)-\frac{1+2\alpha}{8}\frac{g(% \theta)}{\theta}-\frac{1}{32}(g(\theta))^{2},$

Symbols:: : nonnegative integer, $A_{m}(\theta)$ : coefficient, $B_{m}(\theta)$ : coefficient and $g(\theta)$
Permalink:: http://dlmf.nist.gov/18.15.E8
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

where

18.15.9 $g(\theta)=\left(\tfrac{1}{4}-\alpha^{2}\right)\left(\mathop{\cot\/}\nolimits\!% \left(\tfrac{1}{2}\theta\right)-\left(\tfrac{1}{2}\theta\right)^{{-1}}\right)-% \left(\tfrac{1}{4}-\beta^{2}\right)\mathop{\tan\/}\nolimits\!\left(\tfrac{1}{2% }\theta\right).$

Symbols:: $\mathop{\cot\/}\nolimits z$ : cotangent function, $\mathop{\tan\/}\nolimits z$ : tangent function and $g(\theta)$
Permalink:: http://dlmf.nist.gov/18.15.E9
Encodings:: TeX, pMML, png

For higher coefficients see Baratella and Gatteschi (1988), and for another estimate of the error term 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 -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 , fixed $\alpha$ , and $0\leq\beta/n\leq c$ . The latter expansions are in terms of Bessel functions, and are uniform in complex -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 bounded away from , see Elliott (1971). The first term of this expansion also appears in Szegö (1975, Theorem 8.21.7).

§18.15(ii) Ultraspherical

Notes:: See Szegö (1975, Theorem 8.21.11).
Keywords:: ultraspherical polynomials
Permalink:: http://dlmf.nist.gov/18.15.ii

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

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $\mathop{\cos\/}\nolimits z$ : cosine function, : : factorial, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial, : nonnegative integer, : nonnegative integer and $\theta_{{n,m}}$
Referenced by:: §18.15(ii)
Permalink:: http://dlmf.nist.gov/18.15.E10
Encodings:: TeX, pMML, png

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

18.15.11 $\theta_{{n,m}}=(n+m+\lambda)\theta-\tfrac{1}{2}(m+\lambda)\pi.$

Symbols:: : nonnegative integer, : nonnegative integer and $\theta_{{n,m}}$
Permalink:: http://dlmf.nist.gov/18.15.E11
Encodings:: TeX, pMML, png

For a bound on the error term in (18.15.10) see Szegö (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 Szegö (1933) and Szegö (1975, Eq. (8.21.14)).

§18.15(iii) Legendre

Keywords:: Legendre polynomials
Referenced by:: §14.15(iii), §2.1(v), ¶ ‣ §2.10(iv)
Permalink:: http://dlmf.nist.gov/18.15.iii

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

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$ : Legendre polynomial, $\binom{m}{n}$ : binomial coefficient, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : sine function, : nonnegative integer, : nonnegative integer and $\alpha_{{n,m}}$
Referenced by:: §18.15(iii)
Permalink:: http://dlmf.nist.gov/18.15.E12
Encodings:: TeX, pMML, png

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

18.15.13 $\alpha_{{n,m}}=(n-m+\tfrac{1}{2})\theta+(n-\tfrac{1}{2}m-\tfrac{1}{4})\pi.$

Symbols:: : nonnegative integer, : nonnegative integer and $\alpha_{{n,m}}$
Permalink:: http://dlmf.nist.gov/18.15.E13
Encodings:: TeX, pMML, png

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 Szegö (1975, §8.4(3)).

For these results and further information see Olver (1997b, pp. 311–313). For another form of the asymptotic expansion, complete with error bound, see Szegö (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

Notes:: See Szegö (1975, Theorem 8.22.2), Frenzen and Wong (1988).
Keywords:: Laguerre polynomials
Referenced by:: §2.8(iii), §2.8(iv)
Permalink:: http://dlmf.nist.gov/18.15.iv

¶ In Terms of Elementary Functions

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

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\cos\/}\nolimits z$ : cosine function, : base of exponential function, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\mathop{\sin\/}\nolimits z$ : sine function, : nonnegative integer, : nonnegative integer, $\theta_{{n}}^{{(\alpha)}}(x)$ , $a_{m}(x)$ : coefficients, $b_{m}(x)$ : coefficients and : real variable
Permalink:: http://dlmf.nist.gov/18.15.E14
Encodings:: TeX, pMML, png

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

18.15.15 $\theta_{n}^{{(\alpha)}}(x)=2(nx)^{{\frac{1}{2}}}-\left(\tfrac{1}{2}\alpha+% \tfrac{1}{4}\right)\pi.$

Symbols:: : nonnegative integer, $\theta_{{n}}^{{(\alpha)}}(x)$ and : real variable
Permalink:: http://dlmf.nist.gov/18.15.E15
Encodings:: TeX, pMML, png

The leading coefficients are given by

18.15.16

$a_{0}(x)=1,$

$a_{1}(x)=0,$

$b_{1}(x)=\frac{1}{48x^{\frac{1}{2}}}\left(4x^{2}-12\alpha^{2}-24\alpha x-24x+3% \right).$

Symbols:: : nonnegative integer, $a_{m}(x)$ : coefficients, $b_{m}(x)$ : coefficients and : real variable
Permalink:: http://dlmf.nist.gov/18.15.E16
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

¶ In Terms of Bessel Functions

Define

18.15.17 $\nu=4n+2\alpha+2,$

Symbols:: : nonnegative integer and $\nu$
Referenced by:: ¶ ‣ §18.15(iv), §18.16(iv)
Permalink:: http://dlmf.nist.gov/18.15.E17
Encodings:: TeX, pMML, png

18.15.18 $\xi=\tfrac{1}{2}\left(\sqrt{x-x^{2}}+\mathop{\mathrm{arcsin}\/}\nolimits(\sqrt% {x})\right),$ $0\leq x\leq 1$ .

Symbols:: $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function, $\xi$ and : real variable
Permalink:: http://dlmf.nist.gov/18.15.E18
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, : base of exponential function, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, : nonnegative integer, : nonnegative integer, $\nu$ , $\xi$ , $A_{m}(\xi)$ : coefficient, $B_{m}(\xi)$ : coefficient and : real variable
Permalink:: http://dlmf.nist.gov/18.15.E19
Encodings:: TeX, pMML, png

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)), $\mathop{\mathrm{env}J_{{\nu}}\/}\nolimits\!\left(z\right)$ 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 $B_{0}(\xi)=-\frac{1}{2}\left(\frac{1-4\alpha^{2}}{8}+\xi\left(\frac{1-x}{x}% \right)^{{\frac{1}{2}}}\left(\frac{4\alpha^{2}-1}{8}+\frac{1}{4}\frac{x}{1-x}+% \frac{5}{24}\left(\frac{x}{1-x}\right)^{2}\right)\right).$

Symbols:: : nonnegative integer, $\xi$ , $B_{m}(\xi)$ : coefficient and : real variable
Permalink:: http://dlmf.nist.gov/18.15.E20
Encodings:: TeX, pMML, png

¶ In Terms of Airy Functions

Keywords:: Laguerre polynomials

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

18.15.21

$\zeta=-\left(\tfrac{3}{4}\left(\mathop{\mathrm{arccos}\/}\nolimits\!\left(% \sqrt{x}\right)-\sqrt{x-x^{2}}\right)\right)^{{\frac{2}{3}}},$ $0\leq x\leq 1$ ,

$\zeta=\left(\tfrac{3}{4}\left(\sqrt{x^{2}-x}-\mathop{\mathrm{arccosh}\/}% \nolimits\!\left(\sqrt{x}\right)\right)\right)^{{\frac{2}{3}}},$ $x\geq 1$ .

Symbols:: $\mathop{\mathrm{arccosh}\/}\nolimits z$ : inverse hyperbolic cosine function, $\mathop{\mathrm{arccos}\/}\nolimits z$ : arccosine function, $\zeta$ and : real variable
Permalink:: http://dlmf.nist.gov/18.15.E21
Encodings:: TeX, TeX, pMML, pMML, png, png

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

18.15.22 $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(\nu x\right)\sim(-1)^{n}\frac{% e^{{\frac{1}{2}\nu x}}}{2^{{\alpha-\frac{1}{2}}}x^{{\frac{1}{2}\alpha+\frac{1}% {4}}}}\*\left(\frac{\zeta}{x-1}\right)^{{\frac{1}{4}}}\left(\frac{\mathop{% \mathrm{Ai}\/}\nolimits\!\left(\nu^{{\frac{2}{3}}}\zeta\right)}{\nu^{{\frac{1}% {3}}}}\sum_{{m=0}}^{{M-1}}\frac{E_{m}(\zeta)}{\nu^{{2m}}}+\frac{{\mathop{% \mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(\nu^{{\frac{2}{3}}}\zeta\right)}{% \nu^{{\frac{5}{3}}}}\sum_{{m=0}}^{{M-1}}\frac{F_{m}(\zeta)}{\nu^{{2m}}}+% \mathop{\mathrm{envAi}\/}\nolimits\!\left(\nu^{{\frac{2}{3}}}\zeta\right)% \mathop{O\/}\nolimits\!\left(\frac{1}{\nu^{{2M-\frac{2}{3}}}}\right)\right),$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, : base of exponential function, $\mathop{L^{{(\alpha)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Laguerre (or generalized Laguerre) polynomial, $\sim$ : asymptotic equality, : nonnegative integer, : nonnegative integer, $\nu$ , $\zeta$ , $E_{m}$ : coefficient, $F_{m}$ : coefficient and : real variable
Permalink:: http://dlmf.nist.gov/18.15.E22
Encodings:: TeX, pMML, png

as $n\to\infty$ uniformly for $\delta\leq x<\infty$ . Here $\mathop{\mathrm{Ai}\/}\nolimits$ denotes the Airy function (§9.2), ${\mathop{\mathrm{Ai}\/}\nolimits^{{\prime}}}$ 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 $F_{0}(\zeta)=-\frac{5}{48\zeta^{2}}+\left(\frac{x-1}{x\zeta}\right)^{{\frac{1}% {2}}}\left(\frac{1}{2}\alpha^{2}-\frac{1}{8}-\frac{1}{4}\frac{x}{x-1}+\frac{5}% {24}\left(\frac{x}{x-1}\right)^{2}\right),$ $0\leq x<\infty$ .

Symbols:: : nonnegative integer, $\zeta$ , $F_{m}$ : coefficient and : real variable
Permalink:: http://dlmf.nist.gov/18.15.E23
Encodings:: TeX, pMML, png

§18.15(v) Hermite

Notes:: See Szegö (1975, Theorem 8.22.6).
Keywords:: Hermite polynomials
Referenced by:: Other Changes
Permalink:: http://dlmf.nist.gov/18.15.v
Amendment (effective with 1.0.1):: A reference to Geronimo et al. (2004) was added at the end of this subsection.
Reported 2010-11-23

Define

18.15.24 $\mu=2n+1,$

Symbols:: : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.15.E24
Encodings:: TeX, pMML, png

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:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function and : nonnegative integer
Permalink:: http://dlmf.nist.gov/18.15.E25
Encodings:: TeX, pMML, png

and

18.15.26 $\omega_{{n,m}}(x)=\mu^{{\frac{1}{2}}}x-\tfrac{1}{2}(m+n)\pi.$

Symbols:: : nonnegative integer, : nonnegative integer, $\omega_{{n,m}}(x)$ and : real variable
Permalink:: http://dlmf.nist.gov/18.15.E26
Encodings:: TeX, pMML, png

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

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

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, $\mathop{\cos\/}\nolimits z$ : cosine function, : base of exponential function, : nonnegative integer, : nonnegative integer, $\omega_{{n,m}}(x)$ and : real variable
Permalink:: http://dlmf.nist.gov/18.15.E27
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as $n\to\infty$ , uniformly on compact -intervals on $\Real$ . The coefficients $u_{m}(x)$ are polynomials in , 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 $-\infty<t\leq-1-\delta$ , 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 $-\infty<t\leq 1-\delta$ , 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 $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)=2^{{\frac{1}{4}(\mu^{2}-1)}}e^{{% \frac{1}{2}\mu^{2}t^{2}}}\mathop{U\/}\nolimits\!\left(-\tfrac{1}{2}\mu^{2},\mu t% \sqrt{2}\right);$

Symbols:: $\mathop{H_{{n}}\/}\nolimits\!\left(x\right)$ : Hermite polynomial, : base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, : nonnegative integer and : real variable
Permalink:: http://dlmf.nist.gov/18.15.E28
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compare (18.11.3).

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

§18.15(vi) Other Approximations

Keywords:: classical orthogonal polynomials
Permalink:: http://dlmf.nist.gov/18.15.vi

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).