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15 Hypergeometric FunctionProperties

§15.12 Asymptotic Approximations

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§15.12(i) Large Variable

For the asymptotic behavior of F(a,b;c;z) as z with a, b, c fixed, combine (15.2.2) with (15.8.2) or (15.8.8).

§15.12(ii) Large c

Let δ denote an arbitrary small positive constant. Also let a,b,z be real or complex and fixed, and at least one of the following conditions be satisfied:

Then for fixed m{0,1,2,},

15.12.2 F(a,b;c;z)=s=0m-1(a)s(b)s(c)ss!zs+O(c-m),
|c|.

Similar results for other sectors are given in Wagner (1988). For the more general case in which a2=o(c) and b2=o(c) see Wagner (1990).

For large b and c with c>b+1 see López and Pagola (2011).

§15.12(iii) Other Large Parameters

Again, throughout this subsection δ denotes an arbitrary small positive constant, and a,b,c,z are real or complex and fixed.

As λ,

15.12.3 F(a,bc+λ;z)Γ(c+λ)Γ(c-b+λ)s=0qs(z)(b)sλ-s-b,

where q0(z)=1 and qs(z), s=1,2,, are defined by the generating function

15.12.4 (t-1t)b-1t(1-c)(1-z+z-t)-a=s=0qs(z)ts.

If |ph(1-z)|<π, then (15.12.3) applies when |phλ|12π-δ. If z12, then (15.12.3) applies when |phλ|π-δ.

If |ph(z-1)|<π, then as λ with |phλ|π-δ,

15.12.5 F(a+λ,b-λc;12-12z)=2(a+b-1)/2(z+1)(c-a-b-1)/2(z-1)c/2ζsinhζ(λ+12a-12b)1-c×(Ic-1((λ+12a-12b)ζ)(1+O(λ-2))+Ic-2((λ+12a-12b)ζ)2λ+a-b((c-12)(c-32)(1ζ-cothζ)+12(2c-a-b-1)(a+b-1)tanh(12ζ)+O(λ-2)),)

where

15.12.6 ζ=arccoshz.

For Iν(z) see §10.25(ii). For this result and an extension to an asymptotic expansion with error bounds see Jones (2001).

See also Dunster (1999) where the asymptotics of Jacobi polynomials is described; compare (15.9.1).

If |phz|<π, then as λ with |phλ|π-δ,

15.12.7 F(a,b-λc+λ;-z)=2b-c+(1/2)(z+12z)λ(λa/2U(a-12,-αλ)((1+z)c-a-bz1-c(αz-1)1-a+O(λ-1))+λ(a-1)/2αU(a-32,-αλ)((1+z)c-a-bz1-c(αz-1)1-a-2c-b-(1/2)(αz-1)a+O(λ-1)),)

where

15.12.8 α=(-2ln(1-(z-1z+1)2))1/2,

with the branch chosen to be continuous and α>0 when ((z-1)/(z+1))>0. For U(a,z) see §12.2, and for an extension to an asymptotic expansion see Olde Daalhuis (2003a).

If |phz|<π, then as λ with |phλ|12π-δ,

15.12.9 (z+1)3λ/2(2λ)c-1F(a+λ,b+2λc;-z)=λ-1/3(π(a-c+λ+(1/3))Ai(-2π/3λ2/3β2)+π(c-a-λ-(1/3))Ai(2π/3λ2/3β2))×(a0(ζ)+O(λ-1))+λ-2/3(π(a-c+λ+(2/3))Ai(-2π/3λ2/3β2)+π(c-a-λ-(2/3))Ai(2π/3λ2/3β2))×(a1(ζ)+O(λ-1)),

where

15.12.10 ζ =arccosh(14z-1),
15.12.11 β =(-32ζ+94ln(2+ζ2+-ζ))1/3,

with the branch chosen to be continuous and β>0 when ζ>0. Also,

15.12.12 a0(ζ) =12G0(β)+12G0(-β),
a1(ζ) =(12G0(β)-12G0(-β))/β,

where

15.12.13 G0(±β)=(2+±ζ)c-b-(1/2)(1+±ζ)a-c+(1/2)(z-1-±ζ)-a+(1/2)βζ--ζ.

For Ai(z) see §9.2, and for further information and an extension to an asymptotic expansion see Olde Daalhuis (2003b). (Two errors in this reference are corrected in (15.12.9).)

By combination of the foregoing results of this subsection with the linear transformations of §15.8(i) and the connection formulas of §15.10(ii), similar asymptotic approximations for F(a+e1λ,b+e2λ;c+e3λ;z) can be obtained with ej=±1 or 0, j=1,2,3. For more details see Olde Daalhuis (2010). For other extensions, see Wagner (1986) and Temme (2003).