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15 Hypergeometric FunctionProperties

§15.10 Hypergeometric Differential Equation

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§15.10(i) Fundamental Solutions

15.10.1 z(1-z)d2wdz2+(c-(a+b+1)z)dwdz-abw=0.

This is the hypergeometric differential equation. It has regular singularities at z=0,1,, with corresponding exponent pairs {0,1-c}, {0,c-a-b}, {a,b}, respectively. When none of the exponent pairs differ by an integer, that is, when none of c, c-a-b, a-b 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

Singularity z=

(a) If c equals n=1,2,3,, and a=1,2,,n-1, 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,,n-1, 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)lnz-k=1n-1(n-1)!(k-1)!(n-k-1)!(1-a)k(1-b)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,bn-1,n-2,,0,-1,-2,,

or

15.10.9 F(-m,bn;z)lnz-k=1n-1(n-1)!(k-1)!(n-k-1)!(m+1)k(1-b)k(-z)-k+k=0m(-m)k(b)k(n)kk!zk(ψ(1+m-k)+ψ(b+k)-ψ(1+k)-ψ(n+k))+(-1)mm!k=m+1(k-1-m)!(b)k(n)kk!zk,
a=-m, m=0,1,2,; bn-1,n-2,,0,-1,-2,,

or

15.10.10 F(-m,-n;z)lnz-k=1n-1(n-1)!(k-1)!(n-k-1)!(m+1)k(+1)k(-z)-k+k=0(-m)k(-)k(n)kk!zk(ψ(1+m-k)+ψ(1+-k)-ψ(1+k)-ψ(n+k))+(-1)!k=+1m(k-1-)!(-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 2-n=0,-1,-2,, then fundamental solutions in the neighborhood of z=0 are given by zn-1 times those in (a) and (b), with a and b replaced throughout by a+n-1 and b+n-1, respectively.

(d) If a+b+1-c equals n=1,2,3,, or 2-n=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 1-z.

(e) Finally, if a-b+1 equals n=1,2,3,, or 2-n=0,-1,-2,, then fundamental solutions in the neighborhood of z= are given by z-a times those in (a), (b), and (c) with b and z replaced by a-c+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)
=(1-z)c-a-bF(c-a,c-bc;z)
=(1-z)-aF(a,c-bc;zz-1)
=(1-z)-bF(c-a,bc;zz-1).
15.10.12 w2(z) =z1-cF(a-c+1,b-c+12-c;z)
=z1-c(1-z)c-a-bF(1-a,1-b2-c;z)
=z1-c(1-z)c-a-1F(a-c+1,1-b2-c;zz-1)
=z1-c(1-z)c-b-1F(1-a,b-c+12-c;zz-1).
15.10.13 w3(z) =F(a,ba+b-c+1;1-z)
=z1-cF(a-c+1,b-c+1a+b-c+1;1-z)
=z-aF(a,a-c+1a+b-c+1;1-1z)
=z-bF(b,b-c+1a+b-c+1;1-1z).
15.10.14 w4(z) =(1-z)c-a-bF(c-a,c-bc-a-b+1;1-z)
=z1-c(1-z)c-a-bF(1-a,1-bc-a-b+1;1-z)
=za-c(1-z)c-a-bF(1-a,c-ac-a-b+1;1-1z)
=zb-c(1-z)c-a-bF(1-b,c-bc-a-b+1;1-1z).
15.10.15 w5(z) =eaπiz-aF(a,a-c+1a-b+1;1z)
=e(c-b)πizb-c(1-z)c-a-bF(1-b,c-ba-b+1;1z)
=(1-z)-aF(a,c-ba-b+1;11-z)
=e(c-1)πiz1-c(1-z)c-a-1F(1-b,a-c+1a-b+1;11-z).
15.10.16 w6(z) =ebπiz-bF(b,b-c+1b-a+1;1z)
=e(c-a)πiza-c(1-z)c-a-bF(1-a,c-ab-a+1;1z)
=(1-z)-bF(b,c-ab-a+1;11-z)
=e(c-1)πiz1-c(1-z)c-b-1F(1-a,b-c+1b-a+1;11-z).

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

15.10.17 w3(z) =Γ(1-c)Γ(a+b-c+1)Γ(a-c+1)Γ(b-c+1)w1(z)+Γ(c-1)Γ(a+b-c+1)Γ(a)Γ(b)w2(z),
15.10.18 w4(z) =Γ(1-c)Γ(c-a-b+1)Γ(1-a)Γ(1-b)w1(z)+Γ(c-1)Γ(c-a-b+1)Γ(c-a)Γ(c-b)w2(z),
15.10.19 w5(z) =Γ(1-c)Γ(a-b+1)Γ(a-c+1)Γ(1-b)w1(z)+e(c-1)πiΓ(c-1)Γ(a-b+1)Γ(a)Γ(c-b)w2(z),
15.10.20 w6(z) =Γ(1-c)Γ(b-a+1)Γ(b-c+1)Γ(1-a)w1(z)+e(c-1)πiΓ(c-1)Γ(b-a+1)Γ(b)Γ(c-a)w2(z).
15.10.21 w1(z) =Γ(c)Γ(c-a-b)Γ(c-a)Γ(c-b)w3(z)+Γ(c)Γ(a+b-c)Γ(a)Γ(b)w4(z),
15.10.22 w2(z) =Γ(2-c)Γ(c-a-b)Γ(1-a)Γ(1-b)w3(z)+Γ(2-c)Γ(a+b-c)Γ(a-c+1)Γ(b-c+1)w4(z),
15.10.23 w5(z) =eaπiΓ(a-b+1)Γ(c-a-b)Γ(1-b)Γ(c-b)w3(z)+e(c-b)πiΓ(a-b+1)Γ(a+b-c)Γ(a)Γ(a-c+1)w4(z),
15.10.24 w6(z) =ebπiΓ(b-a+1)Γ(c-a-b)Γ(1-a)Γ(c-a)w3(z)+e(c-a)πiΓ(b-a+1)Γ(a+b-c)Γ(b)Γ(b-c+1)w4(z).
15.10.25 w1(z) =Γ(c)Γ(b-a)Γ(b)Γ(c-a)w5(z)+Γ(c)Γ(a-b)Γ(a)Γ(c-b)w6(z),
15.10.26 w2(z) =e(1-c)πiΓ(2-c)Γ(b-a)Γ(1-a)Γ(b-c+1)w5(z)+e(1-c)πiΓ(2-c)Γ(a-b)Γ(1-b)Γ(a-c+1)w6(z),
15.10.27 w3(z) =e-aπiΓ(a+b-c+1)Γ(b-a)Γ(b)Γ(b-c+1)w5(z)+e-bπiΓ(a+b-c+1)Γ(a-b)Γ(a)Γ(a-c+1)w6(z),
15.10.28 w4(z) =e(b-c)πiΓ(c-a-b+1)Γ(b-a)Γ(1-a)Γ(c-a)w5(z)+e(a-c)πiΓ(c-a-b+1)Γ(a-b)Γ(1-b)Γ(c-b)w6(z).
15.10.29 w1(z) =ebπiΓ(c)Γ(a-c+1)Γ(a+b-c+1)Γ(c-b)w3(z)+e(b-c)πiΓ(c)Γ(a-c+1)Γ(b)Γ(a-b+1)w5(z),
15.10.30 w1(z) =eaπiΓ(c)Γ(b-c+1)Γ(a+b-c+1)Γ(c-a)w3(z)+e(a-c)πiΓ(c)Γ(b-c+1)Γ(a)Γ(b-a+1)w6(z),
15.10.31 w2(z) =e(b-c+1)πiΓ(2-c)Γ(a)Γ(a+b-c+1)Γ(1-b)w3(z)+e(b-c)πiΓ(2-c)Γ(a)Γ(a-b+1)Γ(b-c+1)w5(z),
15.10.32 w2(z) =e(a-c+1)πiΓ(2-c)Γ(b)Γ(a+b-c+1)Γ(1-a)w3(z)+e(a-c)πiΓ(2-c)Γ(b)Γ(b-a+1)Γ(a-c+1)w6(z).
15.10.33 w1(z) =e(c-a)πiΓ(c)Γ(1-b)Γ(a)Γ(c-a-b+1)w4(z)+e-aπiΓ(c)Γ(1-b)Γ(a-b+1)Γ(c-a)w5(z),
15.10.34 w1(z) =e(c-b)πiΓ(c)Γ(1-a)Γ(b)Γ(c-a-b+1)w4(z)+e-bπiΓ(c)Γ(1-a)Γ(b-a+1)Γ(c-b)w6(z),
15.10.35 w2(z) =e(1-a)πiΓ(2-c)Γ(c-b)Γ(a-c+1)Γ(c-a-b+1)w4(z)+e-aπiΓ(2-c)Γ(c-b)Γ(a-b+1)Γ(1-a)w5(z),
15.10.36 w2(z) =e(1-b)πiΓ(2-c)Γ(c-a)Γ(b-c+1)Γ(c-a-b+1)w4(z)+e-bπiΓ(2-c)Γ(c-a)Γ(b-a+1)Γ(1-b)w6(z).