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15 Hypergeometric FunctionProperties

§15.4 Special Cases

Contents

§15.4(i) Elementary Functions

The following results hold for principal branches when |z|<1, and by analytic continuation elsewhere. Exceptions are (15.4.8) and (15.4.10), that hold for |z|<π/4, and (15.4.12), (15.4.14), and (15.4.16), that hold for |z|<π/2.

15.4.1 F(1,1;2;z) =-z-1ln(1-z),
15.4.2 F(12,1;32;z2) =12zln(1+z1-z),
15.4.3 F(12,1;32;-z2) =z-1arctanz,
15.4.4 F(12,12;32;z2) =z-1arcsinz,
15.4.5 F(12,12;32;-z2) =z-1ln(z+1+z2).
15.4.6 F(a,b;b;z) =(1-z)-a;

compare §15.2(ii).

15.4.7 F(a,12+a;12;z2)=12((1+z)-2a+(1-z)-2a),
15.4.8 F(a,12+a;12;-tan2z)=(cosz)2acos(2az).
15.4.9 F(a,12+a;32;z2)=1(2-4a)z((1+z)1-2a-(1-z)1-2a),
15.4.10 F(a,12+a;32;-tan2z)=(cosz)2asin((1-2a)z)(1-2a)sinz.
15.4.11 F(-a,a;12;-z2)=12((1+z2+z)2a+(1+z2-z)2a),
15.4.12 F(-a,a;12;sin2z)=cos(2az).
15.4.13 F(a,1-a;12;-z2)=121+z2((1+z2+z)2a-1+(1+z2-z)2a-1),
15.4.14 F(a,1-a;12;sin2z)=cos((2a-1)z)cosz.
15.4.15 F(a,1-a;32;-z2)=1(2-4a)z((1+z2+z)1-2a-(1+z2-z)1-2a),
15.4.16 F(a,1-a;32;sin2z)=sin((2a-1)z)(2a-1)sinz.
15.4.17 F(a,12+a;1+2a;z)=(12+121-z)-2a,
15.4.18 F(a,12+a;2a;z)=11-z(12+121-z)1-2a.
15.4.19 F(a+1,b;a;z)=(1-(1-(b/a))z)(1-z)-1-b.

For an extensive list of elementary representations see Prudnikov et al. (1990, pp. 468–488).

§15.4(ii) Argument Unity

If (c-a-b)>0, then

15.4.20 F(a,b;c;1)=Γ(c)Γ(c-a-b)Γ(c-a)Γ(c-b).

If c=a+b, then

15.4.21 limz1-F(a,b;a+b;z)-ln(1-z)=Γ(a+b)Γ(a)Γ(b).

If (c-a-b)=0 and ca+b, then

15.4.22 limz1-(1-z)a+b-c(F(a,b;c;z)-Γ(c)Γ(c-a-b)Γ(c-a)Γ(c-b))=Γ(c)Γ(a+b-c)Γ(a)Γ(b).

If (c-a-b)<0, then

15.4.23 limz1-F(a,b;c;z)(1-z)c-a-b=Γ(c)Γ(a+b-c)Γ(a)Γ(b).

Chu–Vandermonde Identity

Dougall’s Bilateral Sum

This is a generalization of (15.4.20). If a,b are not integers and (c+d-a-b)>1, then

15.4.25 n=-Γ(a+n)Γ(b+n)Γ(c+n)Γ(d+n)=π2sin(πa)sin(πb)Γ(c+d-a-b-1)Γ(c-a)Γ(d-a)Γ(c-b)Γ(d-b).

§15.4(iii) Other Arguments

15.4.26 F(a,b;a-b+1;-1)=Γ(a-b+1)Γ(12a+1)Γ(a+1)Γ(12a-b+1).
15.4.27 F(1,a;a+1;-1)=12a(ψ(12a+12)-ψ(12a)).
15.4.28 F(a,b;12a+12b+12;12)=πΓ(12a+12b+12)Γ(12a+12)Γ(12b+12).
15.4.29 F(a,b;12a+12b+1;12)=2πa-bΓ(12a+12b+1)(1Γ(12a)Γ(12b+12)-1Γ(12a+12)Γ(12b)).
15.4.30 F(a,1-a;b;12)=21-bπΓ(b)Γ(12a+12b)Γ(12b-12a+12).
15.4.31 F(a,12+a;32-2a;-13)=(89)-2aΓ(43)Γ(32-2a)Γ(32)Γ(43-2a).
15.4.32 F(a,12+a;56+23a;19)=π(34)aΓ(56+23a)Γ(12+13a)Γ(56+13a).
15.4.33 F(3a,13+a;23+2a;π/3)=ππa/2(1627)(3a+1)/6Γ(56+a)Γ(23+a)Γ(23).