# §12.10 Uniform Asymptotic Expansions for Large Parameter

## §12.10(i) Introduction

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

With the transformations

(12.2.2) becomes

12.10.2

With the upper sign in (12.10.2), expansions can be constructed for large in terms of elementary functions that are uniform for 2.8(ii)). With the lower sign there are turning points at , 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 again denotes an arbitrary small positive constant.

## §12.10(ii) Negative ,

As

uniformly for , where

The coefficients are given by

12.10.8

where and are polynomials in of degree , ( odd), ( even, ). For ,

12.10.9
12.10.10

Higher polynomials can be calculated from the recurrence relation

where

12.10.12

and the then follow from

Lastly, the function in (12.10.3) and (12.10.4) has the asymptotic expansion:

12.10.14

where

12.10.15

and the coefficients are defined by

compare (5.11.8). For

12.10.17

## §12.10(iii) Negative ,

When , asymptotic expansions for the functions and that are uniform for are obtainable by substitution into (12.2.15) and (12.2.16) by means of (12.10.3) and (12.10.5). Similarly for and .

## §12.10(iv) Negative ,

As

uniformly for . The quantities and are defined by

12.10.22

where

12.10.23

and the coefficients and are given by

12.10.24

compare (12.10.8).

## §12.10(v) Positive ,

As

uniformly for . Here bars do not denote complex conjugates; instead

12.10.27

and the function has the asymptotic expansion

where and are as in §12.10(ii).

## §12.10(vi) Modifications of Expansions in Elementary Functions

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

where and are as in (12.10.7) and (12.10.15) ,

12.10.32

and the coefficients are the product of and a polynomial in of degree . They satisfy the recursion

starting with . Explicitly,

12.10.34

The modified expansion (12.10.31) shares the property of (12.10.3) that it applies when uniformly with respect to . In addition, it enjoys a double asymptotic property: it holds if either or both and tend to infinity. Observe that if , then , whereas or according as 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).

## §12.10(vii) Negative , . Expansions in Terms of Airy Functions

The following expansions hold for large positive real values of , uniformly for . (For complex values of and see Olver (1959).)

The variable is defined by

12.10.39

where are given by (12.10.7), (12.10.23), respectively, and

12.10.40

The function is real for and analytic at . Inversely, with ,

12.10.41

For see (12.10.14). The coefficients and are given by

where is as in (12.10.40), is as in §12.10(ii), , and

12.10.43

The coefficients and in (12.10.36) and (12.10.38) are given by

where

12.10.45

Explicitly,

where is as in §12.10(ii).

### ¶ Modified Expansions

The expansions (12.10.35)–(12.10.38) can be modified, again see Temme (2000), and the new expansions hold if either or both and tend to infinity. This is provable by the methods used in §10.41(v).

## §12.10(viii) Negative , . Expansions in Terms of Airy Functions

When , asymptotic expansions for and that are uniform for are obtained by substitution into (12.2.15) and (12.2.16) by means of (12.10.35) and (12.10.37). Similarly for and .