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13 Confluent Hypergeometric FunctionsKummer Functions

§13.2 Definitions and Basic Properties


§13.2(i) Differential Equation

Kummer’s Equation

13.2.1 zd2wdz2+(b-z)dwdz-aw=0.

This equation has a regular singularity at the origin with indices 0 and 1-b, and an irregular singularity at infinity of rank one. It can be regarded as the limiting form of the hypergeometric differential equation (§15.10(i)) that is obtained on replacing z by z/b, letting b, and subsequently replacing the symbol c by b. In effect, the regular singularities of the hypergeometric differential equation at b and coalesce into an irregular singularity at .

Standard Solutions

The first two standard solutions are:

13.2.2 M(a,b,z)=s=0(a)s(b)ss!zs=1+abz+a(a+1)b(b+1)2!z2+,


13.2.3 M(a,b,z)=s=0(a)sΓ(b+s)s!zs,

except that M(a,b,z) does not exist when b is a nonpositive integer. In other cases

13.2.4 M(a,b,z)=Γ(b)M(a,b,z).

The series (13.2.2) and (13.2.3) converge for all z. M(a,b,z) is entire in z and a, and is a meromorphic function of b. M(a,b,z) is entire in z, a, and b.

Although M(a,b,z) does not exist when b=-n, n=0,1,2,, many formulas containing M(a,b,z) continue to apply in their limiting form. In particular,

13.2.5 limb-nM(a,b,z)Γ(b)=M(a,-n,z)=(a)n+1(n+1)!zn+1M(a+n+1,n+2,z).

When a=-n, n=0,1,2,, M(a,b,z) is a polynomial in z of degree not exceeding n; this is also true of M(a,b,z) provided that b is not a nonpositive integer.

Another standard solution of (13.2.1) is U(a,b,z), which is determined uniquely by the property

13.2.6 U(a,b,z)z-a,
z, |phz|32π-δ,

where δ is an arbitrary small positive constant. In general, U(a,b,z) has a branch point at z=0. The principal branch corresponds to the principal value of z-a in (13.2.6), and has a cut in the z-plane along the interval (-,0]; compare §4.2(i).

When a=-m, m=0,1,2,, U(a,b,z) is a polynomial in z of degree m:

13.2.7 U(-m,b,z)=(-1)m(b)mM(-m,b,z)=(-1)ms=0m(ms)(b+s)m-s(-z)s.

Similarly, when a-b+1=-n, n=0,1,2,,

13.2.8 U(a,a+n+1,z)=(-1)n(1-a-n)nza+nM(-n,1-a-n,z)=z-as=0n(ns)(a)sz-s.

When b=n+1, n=0,1,2,, and a0,-1,-2,,

13.2.9 U(a,n+1,z)=(-1)n+1n!Γ(a-n)k=0(a)k(n+1)kk!zk(lnz+ψ(a+k)-ψ(1+k)-ψ(n+k+1))+1Γ(a)k=1n(k-1)!(1-a+k)n-k(n-k)!z-k.

When b=n+1, n=0,1,2,, and a=-m, m=0,1,2,,

13.2.10 U(-m,n+1,z)=(-1)m(n+1)mM(-m,n+1,z)=(-1)ms=0m(ms)(n+s+1)m-s(-z)s.

When b=-n, n=0,1,2,, the following equation can be combined with (13.2.9) and (13.2.10):

13.2.11 U(a,-n,z)=zn+1U(a+n+1,n+2,z).

§13.2(ii) Analytic Continuation

When m,

13.2.12 U(a,b,ze2πim)=2πie-πibmsin(πbm)Γ(1+a-b)sin(πb)M(a,b,z)+e-2πibmU(a,b,z).

Except when z=0 each branch of U(a,b,z) is entire in a and b. Unless specified otherwise, however, U(a,b,z) is assumed to have its principal value.

§13.2(iii) Limiting Forms as z0

Next, in cases when a=-n or -n+b-1, where n is a nonnegative integer,

13.2.14 U(-n,b,z)=(-1)n(b)n+O(z),
13.2.15 U(-n+b-1,b,z)=(-1)n(2-b)nz1-b+O(z2-b).

In all other cases

13.2.16 U(a,b,z) =Γ(b-1)Γ(a)z1-b+O(z2-b),
b2, b2,
13.2.17 U(a,2,z) =1Γ(a)z-1+O(lnz),
13.2.18 U(a,b,z) =Γ(b-1)Γ(a)z1-b+Γ(1-b)Γ(a-b+1)+O(z2-b),
1b<2, b1,
13.2.19 U(a,1,z) =-1Γ(a)(lnz+ψ(a)+2γ)+O(zlnz),
13.2.20 U(a,b,z) =Γ(1-b)Γ(a-b+1)+O(z1-b),
13.2.21 U(a,0,z) =1Γ(a+1)+O(zlnz),
13.2.22 U(a,b,z) =Γ(1-b)Γ(a-b+1)+O(z),
b0, b0.

§13.2(iv) Limiting Forms as z

Except when a=0,-1, (polynomial cases),

13.2.23 M(a,b,z)ezza-b/Γ(a),

where δ is an arbitrary small positive constant.

For U(a,b,z) see (13.2.6).

§13.2(v) Numerically Satisfactory Solutions

Fundamental pairs of solutions of (13.2.1) that are numerically satisfactory (§2.7(iv)) in the neighborhood of infinity are

13.2.24 U(a,b,z),
13.2.25 U(a,b,z),

A fundamental pair of solutions that is numerically satisfactory near the origin is

13.2.26 M(a,b,z),z1-bM(a-b+1,2-b,z),

When b=n+1=1,2,3,, a fundamental pair that is numerically satisfactory near the origin is M(a,n+1,z) and

13.2.27 k=1nn!(k-1)!(n-k)!(1-a)kz-k-k=0(a)k(n+1)kk!zk(lnz+ψ(a+k)-ψ(1+k)-ψ(n+k+1)),

if a-n0,-1,-2,, or M(a,n+1,z) and

13.2.28 k=1nn!(k-1)!(n-k)!(1-a)kz-k-k=0-a(a)k(n+1)kk!zk(lnz+ψ(1-a-k)-ψ(1+k)-ψ(n+k+1))+(-1)1-a(-a)!k=1-a(k-1+a)!(n+1)kk!zk,

if a=0,-1,-2,, or M(a,n+1,z) and

13.2.29 k=an(k-1)!(n-k)!(k-a)!z-k,

if a=1,2,,n.

When b=-n=0,-1,-2,, a fundamental pair that is numerically satisfactory near the origin is zn+1M(a+n+1,n+2,z) and

13.2.30 k=1n+1(n+1)!(k-1)!(n-k+1)!(-a-n)kzn-k+1-k=0(a+n+1)k(n+2)kk!zn+k+1(lnz+ψ(a+n+k+1)-ψ(1+k)-ψ(n+k+2)),

if a0,-1,-2,, or zn+1M(a+n+1,n+2,z) and

13.2.31 k=1n+1(n+1)!(k-1)!(n-k+1)!(-a-n)kzn-k+1-k=0-a-n-1(a+n+1)k(n+2)kk!zn+k+1(lnz+ψ(-a-n-k)-ψ(1+k)-ψ(n+k+2))+(-1)n-a(-a-n-1)!k=-a-n(k+a+n)!(n+2)kk!zn+k+1,

if a=-n-1,-n-2,-n-3,, or zn+1M(a+n+1,n+2,z) and

13.2.32 k=a+n+1n+1(k-1)!(n-k+1)!(k-a-n-1)!zn-k+1,

if a=0,-1,,-n.

§13.2(vi) Wronskians

13.2.33 𝒲{M(a,b,z),z1-bM(a-b+1,2-b,z)} =sin(πb)z-bez/π,
13.2.34 𝒲{M(a,b,z),U(a,b,z)} =-z-bez/Γ(a),
13.2.35 𝒲{M(a,b,z),ezU(b-a,b,e±πiz)} =ebπiz-bez/Γ(b-a),
13.2.36 𝒲{z1-bM(a-b+1,2-b,z),U(a,b,z)} =-z-bez/Γ(a-b+1),
13.2.37 𝒲{z1-bM(a-b+1,2-b,z),ezU(b-a,b,e±πiz)} =-z-bez/Γ(1-a),
13.2.38 𝒲{U(a,b,z),ezU(b-a,b,e±πiz)} =e±(a-b)πiz-bez.

§13.2(vii) Connection Formulas

Kummer’s Transformations

13.2.39 M(a,b,z) =ezM(b-a,b,-z),
13.2.40 U(a,b,z) =z1-bU(a-b+1,2-b,z).
13.2.41 1Γ(b)M(a,b,z)=eaπiΓ(b-a)U(a,b,z)+e±(b-a)πiΓ(a)ezU(b-a,b,e±πiz).

Also, when b is not an integer

13.2.42 U(a,b,z)=Γ(1-b)Γ(a-b+1)M(a,b,z)+Γ(b-1)Γ(a)z1-bM(a-b+1,2-b,z).