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13 Confluent Hypergeometric FunctionsKummer Functions

§13.8 Asymptotic Approximations for Large Parameters

  1. §13.8(i) Large |b|, Fixed a and z
  2. §13.8(ii) Large b and z, Fixed a and b/z
  3. §13.8(iii) Large a
  4. §13.8(iv) Large a and b

§13.8(i) Large |b|, Fixed a and z

If b in in such a way that |b+n|δ>0 for all n=0,1,2,, then

13.8.1 M(a,b,z)=s=0n1(a)s(b)ss!zs+O(|b|n).

For fixed a and z in

13.8.2 M(a,b,z)Γ(b)Γ(ba)s=0(a)sqs(z,a)bsa,

as b in |phb|πδ, where q0(z,a)=1 and

13.8.3 (et1)a1exp(t+z(1et))=s=0qs(z,a)ts+a1.

When the foregoing results are combined with Kummer’s transformation (13.2.39), an approximation is obtained for the case when |b| is large, and |ba| and |z| are bounded.

§13.8(ii) Large b and z, Fixed a and b/z

Let λ=z/b>0 and ζ=2(λ1lnλ) with sign(ζ)=sign(λ1). Then

13.8.4 M(a,b,z)b12ae14ζ2b(λ(λ1ζ)a1U(a12,ζb)+(λ(λ1ζ)a1(ζλ1)a)U(a32,ζb)ζb)


13.8.5 U(a,b,z)b12ae14ζ2b(λ(λ1ζ)a1U(a12,ζb)(λ(λ1ζ)a1(ζλ1)a)U(a32,ζb)ζb)

as b, uniformly in compact λ-intervals of (0,) and compact real a-intervals. For the parabolic cylinder function U see §12.2, and for an extension to an asymptotic expansion see Temme (1978).

Special cases are

13.8.6 M(a,b,b)=π(b2)12a(1Γ(12(a+1))+(a+1)8/b3Γ(12a)+O(1b)),


13.8.7 U(a,b,b)=π(2b)12a(1Γ(12(a+1))(a+1)8/b3Γ(12a)+O(1b)).

To obtain approximations for M(a,b,z) and U(a,b,z) that hold as b, with a>12b and z>0 combine (13.14.4), (13.14.5) with §13.20(i).

Also, more complicated—but more powerful—uniform asymptotic approximations can be obtained by combining (13.14.4), (13.14.5) with §§13.20(iii) and 13.20(iv).

For other asymptotic expansions for large b and z see López and Pagola (2010).

For more asymptotic expansions for the cases b± see Temme (2015, §§10.4 and 22.5)

§13.8(iii) Large a

For the notation see §§10.2(ii), 10.25(ii), and 2.8(iv).

When a+ with b (1) fixed,

13.8.8 U(a,b,x)=2e12xΓ(a)(2βtanh(w2)(1ewβ)bβ1bK1b(2βa)+a1(a1+β1+β)1be2βaO(1)),

where w=arccosh(1+(2a)1x), and β=(w+sinhw)/2. (13.8.8) holds uniformly with respect to x[0,). For the case b>1 the transformation (13.2.40) can be used.

For an extension to an asymptotic expansion complete with error bounds see Temme (1990b), and for related results see §13.21(i).

When a with b (1) fixed,

13.8.9 M(a,b,x)=Γ(b)e12x((12ba)x)1212b×(Jb1(2x(b2a))+envJb1(2x(b2a))O(|a|12)),


13.8.10 U(a,b,x)=Γ(12ba+12)e12xx1212b×(cos(aπ)Jb1(2x(b2a))sin(aπ)Yb1(2x(b2a))+envYb1(2x(b2a))O(|a|12)),

uniformly with respect to bounded positive values of x in each case.

For asymptotic approximations to M(a,b,x) and U(a,b,x) as a that hold uniformly with respect to x(0,) and bounded positive values of (b1)/|a|, combine (13.14.4), (13.14.5) with §§13.21(ii), 13.21(iii).

When a in |pha|πδ and b and z fixed,

13.8.11 U(a,b,z)2(z/a)(1b)/2ez/2Γ(a)(Kb1(2az)s=0ps(z)as+z/aKb(2az)s=0qs(z)as),
13.8.12 𝐌(a,b,z)(z/a)(1b)/2ez/2Γ(1+ab)Γ(a)×(Ib1(2az)s=0ps(z)asz/aIb(2az)s=0qs(z)as),
13.8.13 𝐌(a,b,z)(z/a)(1b)/2ez/2Γ(1+a)Γ(a+b)×(Jb1(2az)s=0ps(z)(a)sz/aJb(2az)s=0qs(z)(a)s),
13.8.14 U(a,b,z)(z/a)(1b)/2ez/2Γ(1+a)×(Cb1(a,2az)s=0ps(z)(a)sz/aCb(a,2az)s=0qs(z)(a)s),

where Cν(a,ζ)=cos(πa)Jν(ζ)+sin(πa)Yν(ζ) and

13.8.15 pk(z) =s=0k(ks)(1b+s)kszsck+s(z),
qk(z) =s=0k(ks)(2b+s)kszsck+s+1(z)

where c0(z)=1 and

13.8.16 (k+1)ck+1(z)+s=0k(bBs+1(s+1)!+z(s+1)Bs+2(s+2)!)cks(z)=0,

For the Bernoulli numbers Bk see §24.2(i) and for proofs and similar results in which z can also be unbounded see Temme (2015, Chapters 10 and 27)

§13.8(iv) Large a and b

When a,b+ with |z| and ν=ab bounded

13.8.17 M(a,b,z)=eνzΓ(b)Γ(a)(1+(1ν)(1+6ν2z2)12a+O(1min(a2,b2))),
13.8.18 U(a,b+1,z)=zbe(1ν)zΓ(b)Γ(a)(1+νz(1ν)(2νz)2a+O(1min(a2,b2))),

where Γ(a) is the scaled gamma function defined in (5.11.3). These results follow from Temme (2022), which can also be used to obtain more terms in the expansions. For generalizations in which z is also allowed to be large see Temme and Veling (2022).