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15 Hypergeometric FunctionProperties

§15.8 Transformations of Variable

  1. §15.8(i) Linear Transformations
  2. §15.8(ii) Linear Transformations: Limiting Cases
  3. §15.8(iii) Quadratic Transformations
  4. §15.8(iv) Quadratic Transformations (Continued)
  5. §15.8(v) Cubic Transformations

§15.8(i) Linear Transformations

All functions in this subsection and §15.8(ii) assume their principal values.

15.8.1 𝐅(a,bc;z)=(1z)a𝐅(a,cbc;zz1)=(1z)b𝐅(ca,bc;zz1)=(1z)cab𝐅(ca,cbc;z),
15.8.2 sin(π(ba))π𝐅(a,bc;z) =(z)aΓ(b)Γ(ca)𝐅(a,ac+1ab+1;1z)(z)bΓ(a)Γ(cb)𝐅(b,bc+1ba+1;1z),
15.8.3 sin(π(ba))π𝐅(a,bc;z) =(1z)aΓ(b)Γ(ca)𝐅(a,cbab+1;11z)(1z)bΓ(a)Γ(cb)𝐅(b,caba+1;11z),
15.8.4 sin(π(cab))π𝐅(a,bc;z) =1Γ(ca)Γ(cb)𝐅(a,ba+bc+1;1z)(1z)cabΓ(a)Γ(b)𝐅(ca,cbcab+1;1z),
|phz|<π, |ph(1z)|<π.
15.8.5 sin(π(cab))π𝐅(a,bc;z) =zaΓ(ca)Γ(cb)𝐅(a,ac+1a+bc+1;11z)(1z)cabzacΓ(a)Γ(b)𝐅(ca,1acab+1;11z),
|phz|<π, |ph(1z)|<π.

For an alternative version of the transformations (15.8.2) and (15.8.3), see (15.10.25), and for an alternative version of the transformations (15.8.4) and (15.8.5), see (15.10.21).

§15.8(ii) Linear Transformations: Limiting Cases

With m=0,1,2,, polynomial cases of (15.8.2)–(15.8.5) are given by

15.8.6 F(m,bc;z) =(b)m(c)m(z)mF(m,1cm1bm;1z)=(b)m(c)m(1z)mF(m,cb1bm;11z),
15.8.7 F(m,bc;z) =(cb)m(c)mF(m,bbcm+1;1z)=(cb)m(c)mzmF(m,1cmbcm+1;11z),

with the understanding that if b=, =0,1,2,, then m.

When ba is an integer limits are taken in (15.8.2) and (15.8.3) as follows.

If ba is a nonnegative integer, then

15.8.8 𝐅(a,a+mc;z)=(z)aΓ(a+m)k=0m1(a)k(mk1)!k!Γ(cak)zk+(z)aΓ(a)k=0(a+m)kk!(k+m)!Γ(cakm)(1)kzkm×(ln(z)+ψ(k+1)+ψ(k+m+1)ψ(a+k+m)ψ(cakm)),
15.8.9 𝐅(a,a+mc;z)=(1z)aΓ(a+m)Γ(ca)k=0m1(a)k(cam)k(mk1)!k!(z1)k+(1)m(1z)amΓ(a)Γ(cam)k=0(a+m)k(ca)kk!(k+m)!(1z)k×(ln(1z)+ψ(k+1)+ψ(k+m+1)ψ(a+k+m)ψ(ca+k)),

In (15.8.8) when cakm is a nonpositive integer ψ(cakm)/Γ(cakm) is interpreted as (1)m+k+ac+1(m+k+ac)!. Also, if a is a nonpositive integer, then (15.8.6) applies.

Alternatively, if ba is a negative integer, then we interchange a and b in 𝐅(a,b;c;z).

In a similar way, when cab is an integer limits are taken in (15.8.4) and (15.8.5) as follows.

If cab is a nonnegative integer, then

15.8.10 𝐅(a,ba+b+m;z)=1Γ(a+m)Γ(b+m)k=0m1(a)k(b)k(mk1)!k!(z1)k(z1)mΓ(a)Γ(b)k=0(a+m)k(b+m)kk!(k+m)!(1z)k×(ln(1z)ψ(k+1)ψ(k+m+1)+ψ(a+k+m)+ψ(b+k+m)),
15.8.11 𝐅(a,ba+b+m;z)=zaΓ(a+m)k=0m1(a)k(mk1)!k!Γ(b+mk)(11z)kzaΓ(a)k=0(a+m)kk!(k+m)!Γ(bk)(1)k(11z)k+m×(ln(1zz)ψ(k+1)ψ(k+m+1)+ψ(a+k+m)+ψ(bk)),

In (15.8.11) when bk is a nonpositive integer, ψ(bk)/Γ(bk) is interpreted as (1)kb+1(kb)!. Also, if a or b or both are nonpositive integers, then (15.8.7) applies.

Lastly, if cab is a negative integer, then we first apply the transformation

15.8.12 𝐅(a,b;a+bm;z)=(1z)m𝐅(a~,b~;a~+b~+m;z),

§15.8(iii) Quadratic Transformations

A quadratic transformation relates two hypergeometric functions, with the variable in one a quadratic function of the variable in the other, possibly combined with a fractional linear transformation.

A necessary and sufficient condition that there exists a quadratic transformation is that at least one of the equations shown in Table 15.8.1 is satisfied.

Table 15.8.1: Quadratic transformations of the hypergeometric function.
Group 1 Group 2 Group 3 Group 4
c=ab+1 a=b+12
c=2a c=ba+1 b=a+12 c=12
c=2b c=12(a+b+1) c=a+b+12 c=32
a+b=1 c=a+b12

The hypergeometric functions that correspond to Groups 1 and 2 have z as variable. The hypergeometric functions that correspond to Groups 3 and 4 have a nonlinear function of z as variable. The transformation formulas between two hypergeometric functions in Group 2, or two hypergeometric functions in Group 3, are the linear transformations (15.8.1).

In the equations that follow in this subsection all functions take their principal values.

Group 1 Group 3

Group 2 Group 3

15.8.15 F(a,bab+1;z) =(1+z)aF(12a,12a+12ab+1;4z(1+z)2),
15.8.16 F(a,bab+1;z) =(1z)aF(12a,12ab+12ab+1;4z(1z)2),
15.8.17 F(a,b12(a+b+1);z) =(12z)aF(12a,12a+1212(a+b+1);4z(z1)(12z)2),
15.8.18 F(a,b12(a+b+1);z) =F(12a,12b12(a+b+1);4z(1z)),
15.8.19 F(a,1ac;z) =(12z)1ac(1z)c1F(12(a+c),12(a+c1)c;4z(z1)(12z)2),
15.8.20 F(a,1ac;z) =(1z)c1F(12(ca),12(a+c1)c;4z(1z)),

Group 2 Group 1

15.8.21 F(a,bab+1;z) =(1+z)2aF(a,ab+122a2b+1;4z(1+z)2),
|phz|<π, |z|<1.
15.8.22 F(a,b12(a+b+1);z) =(1z111z1+1)aF(a,12(a+b)a+b;41z1(1z1+1)2),
|ph(z)|<π, z<12.
15.8.23 F(a,1ac;z)=(1z11)1a(1z1+1)a2c+1(1z1)c1F(ca,c122c1;41z1(1z1+1)2),
|ph(z)|<π, z<12.

Group 2 Group 4

15.8.24 F(a,bab+1;z)=(1z)aΓ(ab+1)Γ(12)Γ(12a+12)Γ(12ab+1)F(12a,12ab+1212;(z+1z1)2)+(1+z)(1z)a1Γ(ab+1)Γ(12)Γ(12a)Γ(12ab+12)F(12a+12,12ab+132;(z+1z1)2),
15.8.25 F(a,b12(a+b+1);z)=Γ(12(a+b+1))Γ(12)Γ(12a+12)Γ(12b+12)F(12a,12b12;(12z)2)+(12z)Γ(12(a+b+1))Γ(12)Γ(12a)Γ(12b)F(12a+12,12b+1232;(12z)2),
|phz|<π, |ph(1z)|<π.
15.8.26 F(a,1ac;z)=(1z)c1Γ(c)Γ(12)Γ(12(ca+1))Γ(12c+12a)F(12c12a,12c+12a1212;(12z)2)+(12z)(1z)c1Γ(c)Γ(12)Γ(12c12a)Γ(12(c+a1))F(12c12a+12,12c+12a32;(12z)2),
|phz|<π, |ph(1z)|<π.

Group 4 Group 2

15.8.27 2Γ(12)Γ(a+b+12)Γ(a+12)Γ(b+12)F(a,b;12;z)=F(2a,2b;a+b+12;1212z)+F(2a,2b;a+b+12;12+12z),
|phz|<π, |ph(1z)|<π.
15.8.28 2zΓ(12)Γ(a+b12)Γ(a12)Γ(b12)F(a,b;32;z)=F(2a1,2b1;a+b12;1212z)F(2a1,2b1;a+b12;12+12z),
|phz|<π, |ph(1z)|<π.

§15.8(iv) Quadratic Transformations (Continued)

When the intersection of two groups in Table 15.8.1 is not empty there exist special quadratic transformations, with only one free parameter, between two hypergeometric functions in the same group.


b=13a+13, c=2b=ab+1 in Groups 1 and 2.

(15.8.21) becomes

15.8.29 F(a,13a+1323a+23;z)=(1+z)2aF(a,23a+1643a+13;4z(1+z)2).

This is a quadratic transformation between two cases in Group 1.

We can also use (15.8.13), followed by the inverse of (15.8.15), and obtain

15.8.30 (112z)aF(12a,12a+1213a+56;(z2z)2)=F(a,13a+1323a+23;z)=(1+z)aF(12a,12a+1223a+23;4z(1+z)2),

which is a quadratic transformation between two cases in Group 3.

For further examples see Andrews et al. (1999, pp. 130–132 and 176–177).

§15.8(v) Cubic Transformations


15.8.31 F(3a,3a+124a+23;z)=(198z)2aF(a,a+122a+56;27z2(z1)(9z8)2),

With ζ=e2πi/3(1z)/(ze4πi/3)

15.8.32 (1z3)a(z)3a(1Γ(a+23)Γ(23)F(a,a+1323;z3)+e13πizΓ(a)Γ(43)F(a+13,a+2343;z3))=332a+12e12aπiΓ(a+13)(1ζ)a2πΓ(2a+23)(ζ)2aF(a+13,3a2a+23;ζ1),
|z|>1, |ph(z)|<13π.

Ramanujan’s Cubic Transformation

15.8.33 F(13,231;1(1z1+2z)3)=(1+2z)F(13,231;z3),

provided that z lies in the intersection of the open disks |z14±143i|<123, or equivalently, |ph((1z)/(1+2z))|<π/3. This is used in a cubic analog of the arithmetic-geometric mean. See Borwein and Borwein (1991), and also Berndt et al. (1995).

For further examples and higher-order transformations see Goursat (1881), Watson (1910), Vidūnas (2005), and Tu and Yang (2013); see also Erdélyi et al. (1953a, pp. 67 and 113–114).