§12.10 Uniform Asymptotic Expansions for Large Parameter

In this section we give asymptotic expansions of PCFs for large values of the parameter $a$ that are uniform with respect to the variable $z$, when both $a$ and $z$ $(=x)$ are real. These expansions follow from Olver (1959), where detailed information is also given for complex variables.

With the transformations

(12.2.2) becomes

With the upper sign in (12.10.2), expansions can be constructed for large $\mu$ in terms of elementary functions that are uniform for $t\in (-\mathrm{\infty },\mathrm{\infty })$ (§2.8(ii)). With the lower sign there are turning points at $t=\pm 1$, which need to be excluded from the regions of validity. These cases are treated in §§12.10(ii)–12.10(vi).

The turning points can be included if expansions in terms of Airy functions are used instead of elementary functions (§2.8(iii)). These cases are treated in §§12.10(vii)–12.10(viii).

Throughout this section the symbol $\delta$ again denotes an arbitrary small positive constant.

As $a\to -\mathrm{\infty }$

uniformly for $t\in [1+\delta ,\mathrm{\infty })$, where

The coefficients are given by

where $u_s(t)$ and $v_s(t)$ are polynomials in $t$ of degree $3s$, ($s$ odd), $3s-2$ ($s$ even, $s\ge 2$). For $s=0,1,2$,

Higher polynomials $u_s(t)$ can be calculated from the recurrence relation

where

and the $v_s(t)$ then follow from

Lastly, the function $g(\mu )$ in (12.10.3) and (12.10.4) has the asymptotic expansion:

where

and the coefficients ${\gamma }_s$ are defined by

compare (5.11.8). For $s\le 4$

When $\mu \to \mathrm{\infty }$, asymptotic expansions for the functions $U(-\frac{1}{2}{\mu }^2,-\mu t\sqrt{2})$ and $V(-\frac{1}{2}{\mu }^2,-\mu t\sqrt{2})$ that are uniform for $t\in [1+\delta ,\mathrm{\infty })$ are obtainable by substitution into (12.2.15) and (12.2.16) by means of (12.10.3) and (12.10.5). Similarly for $U^{\prime }(-\frac{1}{2}{\mu }^2,-\mu t\sqrt{2})$ and $V^{\prime }(-\frac{1}{2}{\mu }^2,-\mu t\sqrt{2})$.

As $a\to -\mathrm{\infty }$

uniformly for $t\in [-1+\delta ,1-\delta ]$. The quantities $\kappa$ and $\chi$ are defined by

where

and the coefficients ${\widetilde{𝒜}}_s(t)$ and ${\widetilde{ℬ}}_s(t)$ are given by

compare (12.10.8).

As $a\to \mathrm{\infty }$

uniformly for $t\in ℝ$. Here bars do not denote complex conjugates; instead

and the function $\overline{g}(\mu )$ has the asymptotic expansion

where $h(\mu )$ and ${\gamma }_s$ are as in §12.10(ii).

With the same conditions

where

In Temme (2000) modifications are given of Olver’s expansions. An example is the following modification of (12.10.3)

where $\xi$ and $h(\mu )$ are as in (12.10.7) and (12.10.15) ,

and the coefficients ${𝖠}_s(\tau )$ are the product of ${\tau }^s$ and a polynomial in $\tau$ of degree $2s$. They satisfy the recursion

starting with ${𝖠}_0(\tau )=1$. Explicitly,

The modified expansion (12.10.31) shares the property of (12.10.3) that it applies when $\mu \to \mathrm{\infty }$ uniformly with respect to $t\in [1+\delta ,\mathrm{\infty })$. In addition, it enjoys a double asymptotic property: it holds if either or both $\mu$ and $t$ tend to infinity. Observe that if $t\to \mathrm{\infty }$, then ${𝖠}_s(\tau )=O\left(t^{-2s}\right)$, whereas ${𝒜}_s(t)=O(1)$ or $O\left(t^{-2}\right)$ according as $s$ is even or odd. The proof of the double asymptotic property then follows with the aid of error bounds; compare §10.41(iv).

For additional information see Temme (2000). See also Olver (1997b, pp. 206–208) and Jones (2006).

The following expansions hold for large positive real values of $\mu$, uniformly for $t\in [-1+\delta ,\mathrm{\infty })$. (For complex values of $\mu$ and $t$ see Olver (1959).)

The variable $\zeta$ is defined by

where $\xi ,\eta$ are given by (12.10.7), (12.10.23), respectively, and

The function $\zeta =\zeta (t)$ is real for $t>-1$ and analytic at $t=1$. Inversely, with $w=2^{-\frac{1}{3}}\zeta$,

For $g(\mu )$ see (12.10.14). The coefficients $A_s(\zeta )$ and $B_s(\zeta )$ are given by

where $\varphi (\zeta )$ is as in (12.10.40), $u_k(t)$ is as in §12.10(ii), ${\alpha }_0=1$, and

The coefficients $C_s(\zeta )$ and $D_s(\zeta )$ in (12.10.36) and (12.10.38) are given by

where

Explicitly,

where $v_k(t)$ is as in §12.10(ii).