About the Project
8 Incomplete Gamma and Related FunctionsRelated Functions

§8.18 Asymptotic Expansions of Ix(a,b)

Contents
  1. §8.18(i) Large Parameters, Fixed x
  2. §8.18(ii) Large Parameters: Uniform Asymptotic Expansions

§8.18(i) Large Parameters, Fixed x

If b and x are fixed, with b>0 and 0<x<1, then as a

8.18.1 Ix(a,b)=Γ(a+b)xa(1x)b1(k=0n11Γ(a+k+1)Γ(bk)(x1x)k+O(1Γ(a+n+1))),

for each n=0,1,2,. If b=1,2,3, and nb, then the O-term can be omitted and the result is exact.

If b and a and x are fixed, with a>0 and 0<x<1, then (8.18.1), with a and b interchanged and x replaced by 1x, can be combined with (8.17.4).

§8.18(ii) Large Parameters: Uniform Asymptotic Expansions

Large a, Fixed b

Let

8.18.2 ξ=lnx.

Then as a, with b (>0) fixed,

8.18.3 Ix(a,b)=Γ(a+b)Γ(a)(k=0n1dkFk+O(an)F0),

uniformly for x(0,1]. The functions Fk are defined by

8.18.4 aFk+1=(k+baξ)Fk+kξFk1,

with

8.18.5 F0 =abQ(b,aξ),
F1 =baξaF0+ξbeaξaΓ(b),

and Q(a,z) as in §8.2(i). The coefficients dk are defined by the generating function

8.18.6 (1ett)b1=k=0dk(tξ)k.

In particular,

8.18.7 d0 =(1xξ)b1,
d1 =xξ+x1(1x)ξ(b1)d0.

Compare also §24.16(i). A recurrence relation for the dk can be found in Nemes and Olde Daalhuis (2016).

Symmetric Case

Let

8.18.8 x0=a/(a+b).

Then as a+b,

8.18.9 Ix(a,b)12erfc(ηb/2)+12π(a+b)(xx0)a(1x1x0)bk=0(1)kck(η)(a+b)k,

uniformly for x(0,1) and a/(a+b), b/(a+b)[δ,1δ], where δ again denotes an arbitrary small positive constant. For erfc see §7.2(i). Also,

8.18.10 12η2=x0ln(xx0)+(1x0)ln(1x1x0),

with η/(xx0)>0, and

8.18.11 c0(η)=1ηx0(1x0)xx0,

with limiting value

8.18.12 c0(0)=12x03x0(1x0).

For this result, and for higher coefficients ck(η) see Temme (1996b, §11.3.3.2). All of the ck(η) are analytic at η=0.

General Case

For the scaled gamma function Γ*(z) see (5.11.3).

8.18.13 See (5.11.3).

Let μ=b/a, and x0 again be as in (8.18.8). Then as a

8.18.14 Ix(a,b)Q(b,aζ)(2πb)1/2Γ*(b)(xx0)a(1x1x0)bk=0hk(ζ,μ)ak,

uniformly for b(0,) and x(0,1). Here

8.18.15 μlnζζ=lnx+μln(1x)+(1+μ)ln(1+μ)μ,

with (ζμ)/(x0x)>0, and

8.18.16 h0(ζ,μ)=μ(1ζμ(1+μ)3/2x0x),

with limiting value

8.18.17 h0(μ,μ)=13(1μ1+μ1).

For this result and higher coefficients hk(ζ,μ) see Temme (1996b, §11.3.3.3). All of the hk(ζ,μ) are analytic at ζ=μ (corresponding to x=x0).

Inverse Function

For asymptotic expansions for large values of a and/or b of the x-solution of the equation

8.18.18 Ix(a,b)=p,
0p1,

see Temme (1992b).