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22 Jacobian Elliptic FunctionsProperties

§22.9 Cyclic Identities

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§22.9(i) Notation

The following notation is a generalization of that of Khare and Sukhatme (2002).

Throughout this subsection m and p are positive integers with 1mp.

22.9.1 sm,p(2)=sn(z+2p-1(m-1)K(k),k),
22.9.2 cm,p(2)=cn(z+2p-1(m-1)K(k),k),
22.9.3 dm,p(2)=dn(z+2p-1(m-1)K(k),k),
22.9.4 sm,p(4)=sn(z+4p-1(m-1)K(k),k),
22.9.5 cm,p(4)=cn(z+4p-1(m-1)K(k),k),
22.9.6 dm,p(4)=dn(z+4p-1(m-1)K(k),k).

In the remainder of this section the rank of an identity is the largest number of elliptic function factors in any term of the identity. The value of p determines the number of points in the identity. The argument z is suppressed in the above notation, as all cyclic identities are independent of z.

§22.9(ii) Typical Identities of Rank 2

In this subsection 1mp and 1np.

Three Points

With

22.9.7 κ=dn(2K(k)/3,k),
22.9.8 s1,2(4)s2,2(4)+s2,2(4)s3,2(4)+s3,2(4)s1,2(4)=κ2-1k2,
22.9.9 c1,2(4)c2,2(4)+c2,2(4)c3,2(4)+c3,2(4)c1,2(4)=-κ(κ+2)(1+κ)2,
22.9.10 d1,2(2)d2,2(2)+d2,2(2)d3,2(2)+d3,2(2)d1,2(2)=d1,2(4)d2,2(4)+d2,2(4)d3,2(4)+d3,2(4)d1,2(4)=κ(κ+2).

These identities are cyclic in the sense that each of the indices m,n in the first product of, for example, the form sm,2(4)sn,2(4) are simultaneously permuted in the cyclic order: mm+1m+2p12m-1; nn+1n+2p12n-1. Many of the identities that follow also have this property.

§22.9(iii) Typical Identities of Rank 3

Two Points

22.9.11 (d1,2(2))2d2,2(2)±(d2,2(2))2d1,2(2)=k(d1,2(2)±d2,2(2)),
22.9.12 c1,2(2)s1,2(2)d2,2(2)+c2,2(2)s2,2(2)d1,2(2)=0.

Three Points

With κ defined as in (22.9.7),

22.9.13 s1,3(4)s2,3(4)s3,3(4)=-11-κ2(s1,3(4)+s2,3(4)+s3,3(4)),
22.9.14 c1,3(4)c2,3(4)c3,3(4)=κ21-κ2(c1,3(4)+c2,3(4)+c3,3(4)),
22.9.15 d1,3(2)d2,3(2)d3,3(2)=κ2+k2-11-κ2(d1,3(2)+d2,3(2)+d3,3(2)),
22.9.16 s1,3(4)c2,3(4)c3,3(4)+s2,3(4)c3,3(4)c1,3(4)+s3,3(4)c1,3(4)c2,3(4)=κ(κ+2)1-κ2(s1,3(4)+s2,3(4)+s3,3(4)).

Four Points

22.9.17 d1,4(2)d2,4(2)d3,4(2)±d2,4(2)d3,4(2)d4,4(2)+d3,4(2)d4,4(2)d1,4(2)±d4,4(2)d1,4(2)d2,4(2)=k(±d1,4(2)+d2,4(2)±d3,4(2)+d4,4(2)),
22.9.18 (d1,4(2))2d3,4(2)±(d2,4(2))2d4,4(2)+(d3,4(2))2d1,4(2)±(d4,4(2))2d2,4(2)=k(d1,4(2)±d2,4(2)+d3,4(2)±d4,4(2)),
22.9.19 c1,4(2)s1,4(2)d3,4(2)+c3,4(2)s3,4(2)d1,4(2)=c2,4(2)s2,4(2)d4,4(2)+c4,4(2)s4,4(2)d2,4(2)=0.

§22.9(iv) Typical Identities of Rank 4

Two Points

22.9.20 (d1,2(2))3d2,2(2)±(d2,2(2))3d1,2(2)=k((d1,2(2))2±(d2,2(2))2),
22.9.21 k2c1,2(2)s1,2(2)c2,2(2)s2,2(2)=k(1-(s1,2(2))2-(s2,2(2))2).

Three Points

Again with κ defined as in (22.9.7),

22.9.22 s1,3(2)c1,3(2)d2,3(2)d3,3(2)+s2,3(2)c2,3(2)d3,3(2)d1,3(2)+s3,3(2)c3,3(2)d1,3(2)d2,3(2)=κ2+k2-11-κ2(s1,3(2)c1,3(2)+s2,3(2)c2,3(2)+s3,3(2)c3,3(2)),
22.9.23 s1,3(4)d1,3(4)c2,3(4)c3,3(4)+s2,3(4)d2,3(4)c3,3(4)c1,3(4)+s3,3(4)d3,3(4)c1,3(4)c2,3(4)=κ21-κ2(s1,3(4)d1,3(4)+s2,3(4)d2,3(4)+s2,3(4)d2,3(4)).

§22.9(v) Identities of Higher Rank

For extensions of the identities given in §§22.9(ii)22.9(iv), and also to related elliptic functions, see Khare and Sukhatme (2002), Khare et al. (2003).