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§20.7 Identities

Contents

§20.7(i) Sums of Squares

20.7.1 θ32(0,q)θ32(z,q)=θ42(0,q)θ42(z,q)+θ22(0,q)θ22(z,q),
20.7.2 θ32(0,q)θ42(z,q)=θ22(0,q)θ12(z,q)+θ42(0,q)θ32(z,q),
20.7.3 θ22(0,q)θ42(z,q)=θ32(0,q)θ12(z,q)+θ42(0,q)θ22(z,q),
20.7.4 θ22(0,q)θ32(z,q)=θ42(0,q)θ12(z,q)+θ32(0,q)θ22(z,q).

Also

20.7.5 θ34(0,q)=θ24(0,q)+θ44(0,q).

See also Carlson (2011, §2).

§20.7(ii) Addition Formulas

20.7.6 θ42(0,q)θ1(w+z,q)θ1(w-z,q)=θ32(w,q)θ22(z,q)-θ22(w,q)θ32(z,q),
20.7.7 θ42(0,q)θ2(w+z,q)θ2(w-z,q)=θ42(w,q)θ22(z,q)-θ12(w,q)θ32(z,q),
20.7.8 θ42(0,q)θ3(w+z,q)θ3(w-z,q)=θ42(w,q)θ32(z,q)-θ12(w,q)θ22(z,q),
20.7.9 θ42(0,q)θ4(w+z,q)θ4(w-z,q)=θ32(w,q)θ32(z,q)-θ22(w,q)θ22(z,q).

For these and similar formulas see Lawden (1989, §1.4), Whittaker and Watson (1927, pp. 487–488), and Carlson (2011, §5).

Also, in further development along the lines of the notations of Neville (§20.1) and of Glaisher (§22.2), the identities (20.7.6)–(20.7.9) have been recast in a more symmetric manner with respect to suffices 2,3,4. The symmetry, applicable also to §§20.7(iii) and 20.7(vii), is obtained by modifying traditional theta functions in the manner recommended by Carlson (2011) and used for further purposes by Fukushima (2012).

§20.7(iii) Duplication Formula

20.7.10 θ1(2z,q)=2θ1(z,q)θ2(z,q)θ3(z,q)θ4(z,q)θ2(0,q)θ3(0,q)θ4(0,q).

See also Carlson (2011, §§1 and 4).

§20.7(iv) Reduction Formulas for Products

20.7.11 θ1(z,q)θ2(z,q)θ1(2z,q2)=θ3(z,q)θ4(z,q)θ4(2z,q2)=θ4(0,q2),
20.7.12 θ1(z,q2)θ4(z,q2)θ1(z,q)=θ2(z,q2)θ3(z,q2)θ2(z,q)=12θ2(0,q).

Addendum: For a companion equation see (20.7.34).

§20.7(v) Watson’s Identities

20.7.13 θ1(z,q)θ1(w,q)=θ3(z+w,q2)θ2(z-w,q2)-θ2(z+w,q2)θ3(z-w,q2),
20.7.14 θ3(z,q)θ3(w,q)=θ3(z+w,q2)θ3(z-w,q2)+θ2(z+w,q2)θ2(z-w,q2).

§20.7(vi) Landen Transformations

With

20.7.15 AA(τ)=1/θ4(0|2τ),
20.7.16 θ1(2z|2τ) =Aθ1(z|τ)θ2(z|τ),
20.7.17 θ2(2z|2τ) =Aθ1(14π-z|τ)θ1(14π+z|τ),
20.7.18 θ3(2z|2τ) =Aθ3(14π-z|τ)θ3(14π+z|τ),
20.7.19 θ4(2z|2τ) =Aθ3(z|τ)θ4(z|τ).

Next, with

20.7.20 BB(τ)=1/(θ3(0|τ)θ4(0|τ)θ3(14π|τ)),
20.7.21 θ1(4z|4τ) =Bθ1(z|τ)θ1(14π-z|τ)θ1(14π+z|τ)θ2(z|τ),
20.7.22 θ2(4z|4τ) =Bθ2(18π-z|τ)θ2(18π+z|τ)θ2(38π-z|τ)θ2(38π+z|τ),
20.7.23 θ3(4z|4τ) =Bθ3(18π-z|τ)θ3(18π+z|τ)θ3(38π-z|τ)θ3(38π+z|τ),
20.7.24 θ4(4z|4τ) =Bθ4(z|τ)θ4(14π-z|τ)θ4(14π+z|τ)θ3(z|τ).

§20.7(vii) Derivatives of Ratios of Theta Functions

20.7.25 z(θ2(z|τ)θ4(z|τ))=-θ32(0|τ)θ1(z|τ)θ3(z|τ)θ42(z|τ).

See Lawden (1989, pp. 19–20). This reference also gives the eleven additional identities for the permutations of the four theta functions.

See also Carlson (2011, §3).

§20.7(viii) Transformations of Lattice Parameter

20.7.26 θ1(z|τ+1) =π/4θ1(z|τ),
20.7.27 θ2(z|τ+1) =π/4θ2(z|τ),
20.7.28 θ3(z|τ+1) =θ4(z|τ),
20.7.29 θ4(z|τ+1) =θ3(z|τ).

In the following equations τ=-1/τ, and all square roots assume their principal values.

20.7.30 (-τ)1/2θ1(z|τ) =-exp(τz2/π)θ1(zτ|τ),
20.7.31 (-τ)1/2θ2(z|τ) =exp(τz2/π)θ4(zτ|τ),
20.7.32 (-τ)1/2θ3(z|τ) =exp(τz2/π)θ3(zτ|τ),
20.7.33 (-τ)1/2θ4(z|τ) =exp(τz2/π)θ2(zτ|τ).

These are examples of modular transformations; see §23.15.

§20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products

20.7.34 θ1(z,q2)θ3(z,q2)θ1(z,q)=θ2(z,q2)θ4(z,q2)θ2(z,q)=-1/4θ2(0,q2)θ4(0,q2)2.

See also (20.7.11) and (20.7.12).