# §18.15 Asymptotic Approximations

## §18.15(i) Jacobi

With the exception of the penultimate paragraph, we assume throughout this subsection that , , and () are all fixed.

as , uniformly with respect to . Here, and elsewhere in §18.15, is an arbitrary small positive constant. Also, is the beta function (§5.12) and

where

and

When , 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

Then as ,

where is the Bessel function (§10.2(ii)), and

with denoting an arbitrary positive constant. Also,

18.15.8

where

18.15.9

For higher coefficients see Baratella and Gatteschi (1988), and for another estimate of the error term see Wong and Zhao (2003).

For large , fixed , and , Dunster (1999) gives asymptotic expansions of that are uniform in unbounded complex -domains containing . These expansions are in terms of Whittaker functions (§13.14). This reference also supplies asymptotic expansions of for large , fixed , and . 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 and can be interchanged.

For an asymptotic expansion of as 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

For fixed and fixed

as uniformly with respect to , where

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

Asymptotic expansions for can be obtained from the results given in §18.15(i) by setting and referring to (18.7.1). See also Szegö (1933) and Szegö (1975, Eq. (8.21.14)).

## §18.15(iii) Legendre

For fixed ,

as , uniformly with respect to , where

Also, when , the right-hand side of (18.15.12) with converges; paradoxically, however, the sum is and not 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 and that are uniformly valid when and see §14.15(iii) with and . These expansions are in terms of Bessel functions and modified Bessel functions, respectively.

## §18.15(iv) Laguerre

### ¶ In Terms of Elementary Functions

For fixed , and fixed ,

18.15.14

as , uniformly on compact -intervals in , where

18.15.15

The leading coefficients are given by

18.15.16

### ¶ In Terms of Bessel Functions

Define

18.15.17
18.15.18.

Then for fixed , and fixed ,

18.15.19

as uniformly for . Here denotes the Bessel function (§10.2(ii)), denotes its envelope (§2.8(iv)), and is again an arbitrary small positive constant. The leading coefficients are given by and

18.15.20

### ¶ In Terms of Airy Functions

Again define as in (18.15.17); also,

Then for fixed , and fixed ,

as uniformly for . Here denotes the Airy function (§9.2), denotes its derivative, and denotes its envelope (§2.8(iii)). The leading coefficients are given by and

18.15.23.

## §18.15(v) Hermite

Define

18.15.24
18.15.25

and

Then for fixed ,

as , uniformly on compact -intervals on . The coefficients are polynomials in , and , .

For more powerful asymptotic expansions as in terms of elementary functions that apply uniformly when , , or , where and is again an arbitrary small positive constant, see §§12.10(i)12.10(iv) and 12.10(vi). And for asymptotic expansions as in terms of Airy functions that apply uniformly when or , see §§12.10(vii) and 12.10(viii). With the expansions in Chapter 12 are for the parabolic cylinder function , 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).