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§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 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

12.10.1 a =±12μ2,
x =μt2,

(12.2.2) becomes

12.10.2 d2wdt2=μ4(t2±1)w.

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

§12.10(ii) Negative a, 2-a<x<

As a-

12.10.3 U(-12μ2,μt2)g(μ)e-μ2ξ(t2-1)14s=0𝒜s(t)μ2s,
12.10.4 U(-12μ2,μt2)-μ2g(μ)(t2-1)14e-μ2ξs=0s(t)μ2s,
12.10.5 V(-12μ2,μt2)2g(μ)Γ(12+12μ2)eμ2ξ(t2-1)14s=0(-1)s𝒜s(t)μ2s,
12.10.6 V(-12μ2,μt2)2μg(μ)Γ(12+12μ2)(t2-1)14eμ2ξs=0(-1)ss(t)μ2s,

uniformly for t[1+δ,), where

12.10.7 ξ=12tt2-1-12ln(t+t2-1).

The coefficients are given by

12.10.8 𝒜s(t)=us(t)(t2-1)32s,s(t)=vs(t)(t2-1)32s,

where us(t) and vs(t) are polynomials in t of degree 3s, (s odd), 3s-2 (s even, s2). For s=0,1,2,

12.10.9 u0(t) =1,
u1(t) =t(t2-6)24,
u2(t) =-9t4+249t2+1451152,
12.10.10 v0(t) =1,
v1(t) =t(t2+6)24,
v2(t) =15t4-327t2-1431152.

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

12.10.11 (t2-1)us(t)-3stus(t)=rs-1(t),


12.10.12 8rs(t)=(3t2+2)us(t)-12(s+1)trs-1(t)+4(t2-1)rs-1(t),

and the vs(t) then follow from

12.10.13 vs(t)=us(t)+12tus-1(t)-rs-2(t).

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

12.10.14 g(μ)h(μ)(1+12s=1γs(12μ2)s),


12.10.15 h(μ)=2-14μ2-14e-14μ2μ12μ2-12,

and the coefficients γs are defined by

12.10.16 Γ(12+z)2πe-zzzs=0γszs;

compare (5.11.8). For s4

12.10.17 γ0 =1,
γ1 =-124,
γ2 =11152,
γ3 =10034 14720,
γ4 =-4027398 13120.

§12.10(iii) Negative a, -<x<-2-a

When μ, asymptotic expansions for the functions U(-12μ2,-μt2) and V(-12μ2,-μt2) that are uniform for t[1+δ,) 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(-12μ2,-μt2) and V(-12μ2,-μt2).

§12.10(iv) Negative a, -2-a<x<2-a

As a-

12.10.18 U(-12μ2,μt2) 2g(μ)(1-t2)14(cosκs=0(-1)s𝒜~2s(t)μ4s-sinκs=0(-1)s𝒜~2s+1(t)μ4s+2),
12.10.19 U(-12μ2,μt2) μ2g(μ)(1-t2)14(sinκs=0(-1)s~2s(t)μ4s+cosκs=0(-1)s~2s+1(t)μ4s+2),
12.10.20 V(-12μ2,μt2) 2g(μ)Γ(12+12μ2)(1-t2)14(cosχs=0(-1)s𝒜~2s(t)μ4s-sinχs=0(-1)s𝒜~2s+1(t)μ4s+2),
12.10.21 V(-12μ2,μt2) μ2g(μ)(1-t2)14Γ(12+12μ2)(sinχs=0(-1)s~2s(t)μ4s+cosχs=0(-1)s~2s+1(t)μ4s+2),

uniformly for t[-1+δ,1-δ]. The quantities κ and χ are defined by

12.10.22 κ =μ2η-14π,
χ =μ2η+14π,


12.10.23 η=12arccost-12t1-t2,

and the coefficients 𝒜~s(t) and ~s(t) are given by

12.10.24 𝒜~s(t) =us(t)(1-t2)32s,
~s(t) =vs(t)(1-t2)32s;

compare (12.10.8).

§12.10(v) Positive a, -<x<

As a

12.10.25 U(12μ2,μt2)g¯(μ)e-μ2ξ¯(t2+1)14s=0u¯s(t)(t2+1)32s1μ2s,

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

12.10.26 ξ¯=12tt2+1+12ln(t+t2+1),
12.10.27 u¯s(t)=isus(-it),

and the function g¯(μ) has the asymptotic expansion

12.10.28 g¯(μ)1μ2h(μ)(1+12s=1(-1)sγs(12μ2)s),

where h(μ) and γs are as in §12.10(ii).

With the same conditions

12.10.29 U(12μ2,μt2)-μ2g¯(μ)(t2+1)14e-μ2ξ¯s=0v¯s(t)(t2+1)32s1μ2s,


12.10.30 v¯s(t)=isvs(-it).

§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)

12.10.31 U(-12μ2,μt2)h(μ)e-μ2ξ(t2-1)14s=0As(τ)μ2s,

where ξ and h(μ) are as in (12.10.7) and (12.10.15) ,

12.10.32 τ=12(tt2-1-1),

and the coefficients As(τ) are the product of τs and a polynomial in τ of degree 2s. They satisfy the recursion

12.10.33 As+1(τ)=-4τ2(τ+1)2ddτAs(τ)-140τ(20u2+20u+3)As(u)du,

starting with A0(τ)=1. Explicitly,

12.10.34 A1(τ) =-112τ(20τ2+30τ+9),
A2(τ) =1288τ2(6160τ4+18480τ3+19404τ2+8028τ+945).

The modified expansion (12.10.31) shares the property of (12.10.3) that it applies when μ uniformly with respect to t[1+δ,). In addition, it enjoys a double asymptotic property: it holds if either or both μ and t tend to infinity. Observe that if t, then As(τ)=O(t-2s), whereas 𝒜s(t)=O(1) or O(t-2) 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).

§12.10(vii) Negative a, -2-a<x<. Expansions in Terms of Airy Functions

The following expansions hold for large positive real values of μ, uniformly for t[-1+δ,). (For complex values of μ and t see Olver (1959).)

12.10.35 U(-12μ2,μt2) 2π12μ13g(μ)ϕ(ζ)(Ai(μ43ζ)s=0As(ζ)μ4s+Ai(μ43ζ)μ83s=0Bs(ζ)μ4s),
12.10.36 U(-12μ2,μt2) (2π)12μ23g(μ)ϕ(ζ)(Ai(μ43ζ)μ43s=0Cs(ζ)μ4s+Ai(μ43ζ)s=0Ds(ζ)μ4s),
12.10.37 V(-12μ2,μt2) 2π12μ13g(μ)ϕ(ζ)Γ(12+12μ2)(Bi(μ43ζ)s=0As(ζ)μ4s+Bi(μ43ζ)μ83s=0Bs(ζ)μ4s),
12.10.38 V(-12μ2,μt2) (2π)12μ23g(μ)ϕ(ζ)Γ(12+12μ2)(Bi(μ43ζ)μ43s=0Cs(ζ)μ4s+Bi(μ43ζ)s=0Ds(ζ)μ4s).

The variable ζ is defined by

12.10.39 23ζ32 =ξ,1t,(ζ0);
23(-ζ)32 =η,-1<t1,(ζ0),

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

12.10.40 ϕ(ζ)=(ζt2-1)14.

The function ζ=ζ(t) is real for t>-1 and analytic at t=1. Inversely, with w=2-13ζ,

12.10.41 t=1+w-110w2+11350w3-82363000w4+1 50653242 55000w5+,

For g(μ) see (12.10.14). The coefficients As(ζ) and Bs(ζ) are given by

12.10.42 As(ζ) =ζ-3sm=02sβm(ϕ(ζ))6(2s-m)u2s-m(t),
ζ2Bs(ζ) =-ζ-3sm=02s+1αm(ϕ(ζ))6(2s-m+1)u2s-m+1(t),

where ϕ(ζ) is as in (12.10.40), uk(t) is as in §12.10(ii), α0=1, and

12.10.43 αm =(2m+1)(2m+3)(6m-1)m!(144)m,
βm =-6m+16m-1αm.

The coefficients Cs(ζ) and Ds(ζ) in (12.10.36) and (12.10.38) are given by

12.10.44 Cs(ζ) =χ(ζ)As(ζ)+As(ζ)+ζBs(ζ),
Ds(ζ) =As(ζ)+χ(ζ)Bs-1(ζ)+Bs-1(ζ),


12.10.45 χ(ζ)=ϕ(ζ)ϕ(ζ)=1-2t(ϕ(ζ))64ζ.


12.10.46 ζCs(ζ) =-ζ-3sm=02s+1βm(ϕ(ζ))6(2s-m+1)v2s-m+1(t),
Ds(ζ) =ζ-3sm=02sαm(ϕ(ζ))6(2s-m)v2s-m(t),

where vk(t) 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 μ and t tend to infinity. This is provable by the methods used in §10.41(v).

§12.10(viii) Negative a, -<x<2-a. Expansions in Terms of Airy Functions

When μ, asymptotic expansions for U(-12μ2,-μt2) and V(-12μ2,-μt2) that are uniform for t[-1+δ,) are obtained by substitution into (12.2.15) and (12.2.16) by means of (12.10.35) and (12.10.37). Similarly for U(-12μ2,-μt2) and V(-12μ2,-μt2).