§18.15 Asymptotic Approximations

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

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

where

and

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, in which one has to take $\cos {\theta }_{n,m,\mathrm{\ell }}=1$. 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 $[-1,1]$.

The case $M=1$ of (18.15.1) goes back to Darboux. It reads:

as $n\to \mathrm{\infty }$, uniformly with respect to $\theta \in [\delta ,\pi -\delta ]$.

Next, let

Then as $n\to \mathrm{\infty }$,

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

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

where

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 $0\le n/\beta \le c$, Dunster (1999) gives asymptotic expansions of $P_n^{(\alpha ,\beta )}\left(z\right)$ 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 $P_n^{(\alpha ,\beta )}\left(z\right)$ for large $n$, fixed $\alpha$, and $0\le \beta /n\le c$. 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 $P_n^{(\alpha ,\beta )}\left(z\right)$ as $n\to \mathrm{\infty }$ 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).

For fixed $\lambda \in (0,1)$ and fixed $M=0,1,2,\mathrm{\dots }$,

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

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

Asymptotic expansions for $C_n^{(\lambda )}\left(\cos \theta \right)$ 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)).

For fixed $M=0,1,2,\mathrm{\dots }$,

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

Also, when , the right-hand side of (18.15.12) with $M=\mathrm{\infty }$ converges; paradoxically, however, the sum is $2P_n\left(\cos \theta \right)$ and not $P_n\left(\cos \theta \right)$ 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 $\lambda =\frac{1}{2}$; see Szegő (1975, Theorem 8.21.5).

For asymptotic expansions of $P_n\left(\cos \theta \right)$ and $P_n\left(\cosh \xi \right)$ that are uniformly valid when $0\le \theta \le \pi -\delta$ and see §14.15(iii) with $\mu =0$ and $\nu =n$. These expansions are in terms of Bessel functions and modified Bessel functions, respectively.

Define

and

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

as $n\to \mathrm{\infty }$, uniformly on compact $x$-intervals on $ℝ$. 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 $n\to \mathrm{\infty }$ in terms of elementary functions that apply uniformly when , $-1+\delta \le t\le 1-\delta$, or , where $t=x/\sqrt{2n+1}$ 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 $n\to \mathrm{\infty }$ in terms of Airy functions that apply uniformly when or , see §§12.10(vii) and 12.10(viii). With $\mu =\sqrt{2n+1}$ the expansions in Chapter 12 are for the parabolic cylinder function $U(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})$, which is related to the Hermite polynomials via

compare (18.11.3).

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

See also Geronimo et al. (2004).

The asymptotic behavior of the classical OP’s as $x\to \pm \mathrm{\infty }$ 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).