About the Project
15 Hypergeometric FunctionProperties

§15.10 Hypergeometric Differential Equation

Contents
  1. §15.10(i) Fundamental Solutions
  2. §15.10(ii) Kummer’s 24 Solutions and Connection Formulas

§15.10(i) Fundamental Solutions

15.10.1 z(1z)d2wdz2+(c(a+b+1)z)dwdzabw=0.

This is the hypergeometric differential equation. It has regular singularities at z=0,1,, with corresponding exponent pairs {0,1c}, {0,cab}, {a,b}, respectively. When none of the exponent pairs differ by an integer, that is, when none of c, cab, ab is an integer, we have the following pairs f1(z), f2(z) of fundamental solutions. They are also numerically satisfactory (§2.7(iv)) in the neighborhood of the corresponding singularity.

Singularity z=0

Singularity z=1

15.10.4 f1(z) =F(a,ba+b+1c;1z),
f2(z) =(1z)cabF(ca,cbcab+1;1z),
15.10.5 𝒲{f1(z),f2(z)}=(a+bc)zc(1z)cab1.

Singularity z=

15.10.6 f1(z) =zaF(a,ac+1ab+1;1z),
f2(z) =zbF(b,bc+1ba+1;1z),
15.10.7 𝒲{f1(z),f2(z)}=(ab)zc(z1)cab1.

(a) If c equals n=1,2,3,, and a=1,2,,n1, then fundamental solutions in the neighborhood of z=0 are given by (15.10.2) with the interpretation (15.2.5) for f2(z).

(b) If c equals n=1,2,3,, and a1,2,,n1, then fundamental solutions in the neighborhood of z=0 are given by F(a,b;n;z) and

15.10.8 F(a,bn;z)lnzk=1n1(n1)!(k1)!(nk1)!(1a)k(1b)k(z)k+k=0(a)k(b)k(n)kk!zk(ψ(a+k)+ψ(b+k)ψ(1+k)ψ(n+k)),
a,bn1,n2,,0,1,2,,

or

15.10.9 F(m,bn;z)lnzk=1n1(n1)!(k1)!(nk1)!(m+1)k(1b)k(z)k+k=0m(m)k(b)k(n)kk!zk(ψ(1+mk)+ψ(b+k)ψ(1+k)ψ(n+k))+(1)mm!k=m+1(k1m)!(b)k(n)kk!zk,
a=m, m=0,1,2,; bn1,n2,,0,1,2,,

or

15.10.10 F(m,n;z)lnzk=1n1(n1)!(k1)!(nk1)!(m+1)k(+1)k(z)k+k=0(m)k()k(n)kk!zk(ψ(1+mk)+ψ(1+k)ψ(1+k)ψ(n+k))+(1)!k=+1m(k1)!(m)k(n)kk!zk,
a=m, m=0,1,2,; b=, =0,1,2,,m.

Moreover, in (15.10.9) and (15.10.10) the symbols a and b are interchangeable.

(c) If the parameter c in the differential equation equals 2n=0,1,2,, then fundamental solutions in the neighborhood of z=0 are given by zn1 times those in (a) and (b), with a and b replaced throughout by a+n1 and b+n1, respectively.

(d) If a+b+1c equals n=1,2,3,, or 2n=0,1,2,, then fundamental solutions in the neighborhood of z=1 are given by those in (a), (b), and (c) with z replaced by 1z.

(e) Finally, if ab+1 equals n=1,2,3,, or 2n=0,1,2,, then fundamental solutions in the neighborhood of z= are given by za times those in (a), (b), and (c) with b and z replaced by ac+1 and 1/z, respectively.

§15.10(ii) Kummer’s 24 Solutions and Connection Formulas

The three pairs of fundamental solutions given by (15.10.2), (15.10.4), and (15.10.6) can be transformed into 18 other solutions by means of (15.8.1), leading to a total of 24 solutions known as Kummer’s solutions.

15.10.11 w1(z) =F(a,bc;z)=(1z)cabF(ca,cbc;z)=(1z)aF(a,cbc;zz1)=(1z)bF(ca,bc;zz1).
15.10.12 w2(z) =z1cF(ac+1,bc+12c;z)
=z1c(1z)cabF(1a,1b2c;z)
=z1c(1z)ca1F(ac+1,1b2c;zz1)
=z1c(1z)cb1F(1a,bc+12c;zz1).
15.10.13 w3(z) =F(a,ba+bc+1;1z)=z1cF(ac+1,bc+1a+bc+1;1z)=zaF(a,ac+1a+bc+1;11z)=zbF(b,bc+1a+bc+1;11z).
15.10.14 w4(z) =(1z)cabF(ca,cbcab+1;1z)
=z1c(1z)cabF(1a,1bcab+1;1z)
=zac(1z)cabF(1a,cacab+1;11z)
=zbc(1z)cabF(1b,cbcab+1;11z).
15.10.15 w5(z) =eaπizaF(a,ac+1ab+1;1z)
=e(cb)πizbc(1z)cabF(1b,cbab+1;1z)
=(1z)aF(a,cbab+1;11z)
=e(c1)πiz1c(1z)ca1F(1b,ac+1ab+1;11z).
15.10.16 w6(z) =ebπizbF(b,bc+1ba+1;1z)
=e(ca)πizac(1z)cabF(1a,caba+1;1z)
=(1z)bF(b,caba+1;11z)
=e(c1)πiz1c(1z)cb1F(1a,bc+1ba+1;11z).

The (63)=20 connection formulas for the principal branches of Kummer’s solutions are:

15.10.17 w3(z) =Γ(1c)Γ(a+bc+1)Γ(ac+1)Γ(bc+1)w1(z)+Γ(c1)Γ(a+bc+1)Γ(a)Γ(b)w2(z),
15.10.18 w4(z) =Γ(1c)Γ(cab+1)Γ(1a)Γ(1b)w1(z)+Γ(c1)Γ(cab+1)Γ(ca)Γ(cb)w2(z),
15.10.19 w5(z) =Γ(1c)Γ(ab+1)Γ(ac+1)Γ(1b)w1(z)+e(c1)πiΓ(c1)Γ(ab+1)Γ(a)Γ(cb)w2(z),
15.10.20 w6(z) =Γ(1c)Γ(ba+1)Γ(bc+1)Γ(1a)w1(z)+e(c1)πiΓ(c1)Γ(ba+1)Γ(b)Γ(ca)w2(z).
15.10.21 w1(z) =Γ(c)Γ(cab)Γ(ca)Γ(cb)w3(z)+Γ(c)Γ(a+bc)Γ(a)Γ(b)w4(z),
15.10.22 w2(z) =Γ(2c)Γ(cab)Γ(1a)Γ(1b)w3(z)+Γ(2c)Γ(a+bc)Γ(ac+1)Γ(bc+1)w4(z),
15.10.23 w5(z) =eaπiΓ(ab+1)Γ(cab)Γ(1b)Γ(cb)w3(z)+e(cb)πiΓ(ab+1)Γ(a+bc)Γ(a)Γ(ac+1)w4(z),
15.10.24 w6(z) =ebπiΓ(ba+1)Γ(cab)Γ(1a)Γ(ca)w3(z)+e(ca)πiΓ(ba+1)Γ(a+bc)Γ(b)Γ(bc+1)w4(z).
15.10.25 w1(z) =Γ(c)Γ(ba)Γ(b)Γ(ca)w5(z)+Γ(c)Γ(ab)Γ(a)Γ(cb)w6(z),
15.10.26 w2(z) =e(1c)πiΓ(2c)Γ(ba)Γ(1a)Γ(bc+1)w5(z)+e(1c)πiΓ(2c)Γ(ab)Γ(1b)Γ(ac+1)w6(z),
15.10.27 w3(z) =eaπiΓ(a+bc+1)Γ(ba)Γ(b)Γ(bc+1)w5(z)+ebπiΓ(a+bc+1)Γ(ab)Γ(a)Γ(ac+1)w6(z),
15.10.28 w4(z) =e(bc)πiΓ(cab+1)Γ(ba)Γ(1a)Γ(ca)w5(z)+e(ac)πiΓ(cab+1)Γ(ab)Γ(1b)Γ(cb)w6(z).
15.10.29 w1(z) =ebπiΓ(c)Γ(ac+1)Γ(a+bc+1)Γ(cb)w3(z)+e(bc)πiΓ(c)Γ(ac+1)Γ(b)Γ(ab+1)w5(z),
15.10.30 w1(z) =eaπiΓ(c)Γ(bc+1)Γ(a+bc+1)Γ(ca)w3(z)+e(ac)πiΓ(c)Γ(bc+1)Γ(a)Γ(ba+1)w6(z),
15.10.31 w2(z) =e(bc+1)πiΓ(2c)Γ(a)Γ(a+bc+1)Γ(1b)w3(z)+e(bc)πiΓ(2c)Γ(a)Γ(ab+1)Γ(bc+1)w5(z),
15.10.32 w2(z) =e(ac+1)πiΓ(2c)Γ(b)Γ(a+bc+1)Γ(1a)w3(z)+e(ac)πiΓ(2c)Γ(b)Γ(ba+1)Γ(ac+1)w6(z).
15.10.33 w1(z) =e(ca)πiΓ(c)Γ(1b)Γ(a)Γ(cab+1)w4(z)+eaπiΓ(c)Γ(1b)Γ(ab+1)Γ(ca)w5(z),
15.10.34 w1(z) =e(cb)πiΓ(c)Γ(1a)Γ(b)Γ(cab+1)w4(z)+ebπiΓ(c)Γ(1a)Γ(ba+1)Γ(cb)w6(z),
15.10.35 w2(z) =e(1a)πiΓ(2c)Γ(cb)Γ(ac+1)Γ(cab+1)w4(z)+expeaπiΓ(2c)Γ(cb)Γ(ab+1)Γ(1a)w5(z),
15.10.36 w2(z) =e(1b)πiΓ(2c)Γ(ca)Γ(bc+1)Γ(cab+1)w4(z)+ebπiΓ(2c)Γ(ca)Γ(ba+1)Γ(1b)w6(z).