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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;a;z) =(1-z)-b,
F(a,b;b;z) =(1-z)-a,

where the limit interpretation (15.2.6), rather than (15.2.5), has to be taken when the third parameter is a nonpositive integer. See the final paragraph in §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.

In (15.4.17), (15.4.18) and (15.4.19) when the third entry is a nonpositive integer one has to use the limit interpretation (15.2.6), rather than (15.2.5). Compare the final paragraph in §15.2(ii).

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)=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;eiπ/3)=πeiπa/2(1627)(3a+1)/6Γ(56+a)Γ(23+a)Γ(23),

where the limit interpretation (15.2.6), rather than (15.2.5), has to be taken when a=-13,-43,-73,. Compare the final paragraph in §15.2(ii).