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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).