§22.9 Cyclic Identities

§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 $1\leq m\leq p$.

 22.9.1 $s_{m,p}^{(2)}=\operatorname{sn}\left(z+2p^{-1}(m-1)K\left(k\right),k\right),$
 22.9.2 $c_{m,p}^{(2)}=\operatorname{cn}\left(z+2p^{-1}(m-1)K\left(k\right),k\right),$
 22.9.3 $d_{m,p}^{(2)}=\operatorname{dn}\left(z+2p^{-1}(m-1)K\left(k\right),k\right),$
 22.9.4 $s_{m,p}^{(4)}=\operatorname{sn}\left(z+4p^{-1}(m-1)K\left(k\right),k\right),$
 22.9.5 $c_{m,p}^{(4)}=\operatorname{cn}\left(z+4p^{-1}(m-1)K\left(k\right),k\right),$
 22.9.6 $d_{m,p}^{(4)}=\operatorname{dn}\left(z+4p^{-1}(m-1)K\left(k\right),k\right).$

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 $1\leq m\leq p$ and $1\leq n\leq p$.

Three Points

With

 22.9.7 $\kappa=\operatorname{dn}\left(2\!K\left(k\right)/3,k\right),$ ⓘ Symbols: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$: Jacobian elliptic function, $K\left(\NVar{k}\right)$: Legendre’s complete elliptic integral of the first kind, $k$: modulus and $\kappa$ Referenced by: §22.9(iii), §22.9(iv) Permalink: http://dlmf.nist.gov/22.9.E7 Encodings: TeX, pMML, png See also: Annotations for 22.9(ii), 22.9(ii), 22.9 and 22
 22.9.8 $s_{1,2}^{(4)}s_{2,2}^{(4)}+s_{2,2}^{(4)}s_{3,2}^{(4)}+s_{3,2}^{(4)}s_{1,2}^{(4% )}=\frac{\kappa^{2}-1}{k^{2}},$ ⓘ Symbols: $k$: modulus, $s_{m,p}^{(r)}$: cyclic quantity and $\kappa$ Permalink: http://dlmf.nist.gov/22.9.E8 Encodings: TeX, pMML, png See also: Annotations for 22.9(ii), 22.9(ii), 22.9 and 22
 22.9.9 $c_{1,2}^{(4)}c_{2,2}^{(4)}+c_{2,2}^{(4)}c_{3,2}^{(4)}+c_{3,2}^{(4)}c_{1,2}^{(4% )}=-\frac{\kappa(\kappa+2)}{(1+\kappa)^{2}},$ ⓘ Symbols: $c_{m,p}^{(r)}$: cyclic quantity and $\kappa$ Permalink: http://dlmf.nist.gov/22.9.E9 Encodings: TeX, pMML, png See also: Annotations for 22.9(ii), 22.9(ii), 22.9 and 22
 22.9.10 $d_{1,2}^{(2)}d_{2,2}^{(2)}+d_{2,2}^{(2)}d_{3,2}^{(2)}+d_{3,2}^{(2)}d_{1,2}^{(2% )}=d_{1,2}^{(4)}d_{2,2}^{(4)}+d_{2,2}^{(4)}d_{3,2}^{(4)}+d_{3,2}^{(4)}d_{1,2}^% {(4)}=\kappa(\kappa+2).$ ⓘ Symbols: $d_{m,p}^{(r)}$: cyclic quantity and $\kappa$ Permalink: http://dlmf.nist.gov/22.9.E10 Encodings: TeX, pMML, png See also: Annotations for 22.9(ii), 22.9(ii), 22.9 and 22

These identities are cyclic in the sense that each of the indices $m,n$ in the first product of, for example, the form $s_{m,2}^{(4)}s_{n,2}^{(4)}$ are simultaneously permuted in the cyclic order: $m\to m+1\to m+2\to\cdots p\to 1\to 2\to\cdots m-1$; $n\to n+1\to n+2\to\cdots p\to 1\to 2\to\cdots n-1$. Many of the identities that follow also have this property.

§22.9(iii) Typical Identities of Rank 3

Two Points

 22.9.11 $\left(d_{1,2}^{(2)}\right)^{2}d_{2,2}^{(2)}\pm\left(d_{2,2}^{(2)}\right)^{2}d_% {1,2}^{(2)}=k^{\prime}\left(d_{1,2}^{(2)}\pm d_{2,2}^{(2)}\right),$ ⓘ Symbols: $k^{\prime}$: complementary modulus and $d_{m,p}^{(r)}$: cyclic quantity Permalink: http://dlmf.nist.gov/22.9.E11 Encodings: TeX, pMML, png See also: Annotations for 22.9(iii), 22.9(iii), 22.9 and 22
 22.9.12 $c_{1,2}^{(2)}s_{1,2}^{(2)}d_{2,2}^{(2)}+c_{2,2}^{(2)}s_{2,2}^{(2)}d_{1,2}^{(2)% }=0.$ ⓘ Symbols: $s_{m,p}^{(r)}$: cyclic quantity, $c_{m,p}^{(r)}$: cyclic quantity and $d_{m,p}^{(r)}$: cyclic quantity Permalink: http://dlmf.nist.gov/22.9.E12 Encodings: TeX, pMML, png See also: Annotations for 22.9(iii), 22.9(iii), 22.9 and 22

Three Points

With $\kappa$ defined as in (22.9.7),

 22.9.13 $s_{1,3}^{(4)}s_{2,3}^{(4)}s_{3,3}^{(4)}=-\frac{1}{1-\kappa^{2}}\left(s_{1,3}^{% (4)}+s_{2,3}^{(4)}+s_{3,3}^{(4)}\right),$ ⓘ Symbols: $s_{m,p}^{(r)}$: cyclic quantity and $\kappa$ Permalink: http://dlmf.nist.gov/22.9.E13 Encodings: TeX, pMML, png See also: Annotations for 22.9(iii), 22.9(iii), 22.9 and 22
 22.9.14 $c_{1,3}^{(4)}c_{2,3}^{(4)}c_{3,3}^{(4)}=\frac{\kappa^{2}}{1-\kappa^{2}}\left(c% _{1,3}^{(4)}+c_{2,3}^{(4)}+c_{3,3}^{(4)}\right),$ ⓘ Symbols: $c_{m,p}^{(r)}$: cyclic quantity and $\kappa$ Permalink: http://dlmf.nist.gov/22.9.E14 Encodings: TeX, pMML, png See also: Annotations for 22.9(iii), 22.9(iii), 22.9 and 22
 22.9.15 $d_{1,3}^{(2)}d_{2,3}^{(2)}d_{3,3}^{(2)}=\frac{\kappa^{2}+k^{2}-1}{1-\kappa^{2}% }\left(d_{1,3}^{(2)}+d_{2,3}^{(2)}+d_{3,3}^{(2)}\right),$ ⓘ Symbols: $k$: modulus, $d_{m,p}^{(r)}$: cyclic quantity and $\kappa$ Permalink: http://dlmf.nist.gov/22.9.E15 Encodings: TeX, pMML, png See also: Annotations for 22.9(iii), 22.9(iii), 22.9 and 22
 22.9.16 $s_{1,3}^{(4)}c_{2,3}^{(4)}c_{3,3}^{(4)}+s_{2,3}^{(4)}c_{3,3}^{(4)}c_{1,3}^{(4)% }+s_{3,3}^{(4)}c_{1,3}^{(4)}c_{2,3}^{(4)}=\frac{\kappa(\kappa+2)}{1-\kappa^{2}% }\left(s_{1,3}^{(4)}+s_{2,3}^{(4)}+s_{3,3}^{(4)}\right).$ ⓘ Symbols: $s_{m,p}^{(r)}$: cyclic quantity, $c_{m,p}^{(r)}$: cyclic quantity and $\kappa$ Permalink: http://dlmf.nist.gov/22.9.E16 Encodings: TeX, pMML, png See also: Annotations for 22.9(iii), 22.9(iii), 22.9 and 22

Four Points

 22.9.17 $d_{1,4}^{(2)}d_{2,4}^{(2)}d_{3,4}^{(2)}\pm d_{2,4}^{(2)}d_{3,4}^{(2)}d_{4,4}^{% (2)}+d_{3,4}^{(2)}d_{4,4}^{(2)}d_{1,4}^{(2)}\pm d_{4,4}^{(2)}d_{1,4}^{(2)}d_{2% ,4}^{(2)}=k^{\prime}{\left(\pm d_{1,4}^{(2)}+d_{2,4}^{(2)}\pm d_{3,4}^{(2)}+d_% {4,4}^{(2)}\right)},$ ⓘ Symbols: $k^{\prime}$: complementary modulus and $d_{m,p}^{(r)}$: cyclic quantity Permalink: http://dlmf.nist.gov/22.9.E17 Encodings: TeX, pMML, png See also: Annotations for 22.9(iii), 22.9(iii), 22.9 and 22
 22.9.18 $\left(d_{1,4}^{(2)}\right)^{2}d_{3,4}^{(2)}\pm\left(d_{2,4}^{(2)}\right)^{2}d_% {4,4}^{(2)}+\left(d_{3,4}^{(2)}\right)^{2}d_{1,4}^{(2)}\pm\left(d_{4,4}^{(2)}% \right)^{2}d_{2,4}^{(2)}=k^{\prime}{\left(d_{1,4}^{(2)}\pm d_{2,4}^{(2)}+d_{3,% 4}^{(2)}\pm d_{4,4}^{(2)}\right)},$ ⓘ Symbols: $k^{\prime}$: complementary modulus and $d_{m,p}^{(r)}$: cyclic quantity Permalink: http://dlmf.nist.gov/22.9.E18 Encodings: TeX, pMML, png See also: Annotations for 22.9(iii), 22.9(iii), 22.9 and 22
 22.9.19 $c_{1,4}^{(2)}s_{1,4}^{(2)}d_{3,4}^{(2)}+c_{3,4}^{(2)}s_{3,4}^{(2)}d_{1,4}^{(2)% }=c_{2,4}^{(2)}s_{2,4}^{(2)}d_{4,4}^{(2)}+c_{4,4}^{(2)}s_{4,4}^{(2)}d_{2,4}^{(% 2)}=0.$ ⓘ Symbols: $s_{m,p}^{(r)}$: cyclic quantity, $c_{m,p}^{(r)}$: cyclic quantity and $d_{m,p}^{(r)}$: cyclic quantity Permalink: http://dlmf.nist.gov/22.9.E19 Encodings: TeX, pMML, png See also: Annotations for 22.9(iii), 22.9(iii), 22.9 and 22

§22.9(iv) Typical Identities of Rank 4

Two Points

 22.9.20 $\left(d_{1,2}^{(2)}\right)^{3}d_{2,2}^{(2)}\pm\left(d_{2,2}^{(2)}\right)^{3}d_% {1,2}^{(2)}=k^{\prime}\left(\left(d_{1,2}^{(2)}\right)^{2}\pm\left(d_{2,2}^{(2% )}\right)^{2}\right),$ ⓘ Symbols: $k^{\prime}$: complementary modulus and $d_{m,p}^{(r)}$: cyclic quantity Permalink: http://dlmf.nist.gov/22.9.E20 Encodings: TeX, pMML, png See also: Annotations for 22.9(iv), 22.9(iv), 22.9 and 22
 22.9.21 $k^{2}c_{1,2}^{(2)}s_{1,2}^{(2)}c_{2,2}^{(2)}s_{2,2}^{(2)}=k^{\prime}\left(1-% \left(s_{1,2}^{(2)}\right)^{2}-\left(s_{2,2}^{(2)}\right)^{2}\right).$ ⓘ Symbols: $k$: modulus, $k^{\prime}$: complementary modulus, $s_{m,p}^{(r)}$: cyclic quantity and $c_{m,p}^{(r)}$: cyclic quantity Permalink: http://dlmf.nist.gov/22.9.E21 Encodings: TeX, pMML, png See also: Annotations for 22.9(iv), 22.9(iv), 22.9 and 22

Three Points

Again with $\kappa$ defined as in (22.9.7),

 22.9.22 $s_{1,3}^{(2)}c_{1,3}^{(2)}d_{2,3}^{(2)}d_{3,3}^{(2)}+s_{2,3}^{(2)}c_{2,3}^{(2)% }d_{3,3}^{(2)}d_{1,3}^{(2)}+s_{3,3}^{(2)}c_{3,3}^{(2)}d_{1,3}^{(2)}d_{2,3}^{(2% )}=\frac{\kappa^{2}+k^{2}-1}{1-\kappa^{2}}\left(s_{1,3}^{(2)}c_{1,3}^{(2)}+s_{% 2,3}^{(2)}c_{2,3}^{(2)}+s_{3,3}^{(2)}c_{3,3}^{(2)}\right),$
 22.9.23 $s_{1,3}^{(4)}d_{1,3}^{(4)}c_{2,3}^{(4)}c_{3,3}^{(4)}+s_{2,3}^{(4)}d_{2,3}^{(4)% }c_{3,3}^{(4)}c_{1,3}^{(4)}+s_{3,3}^{(4)}d_{3,3}^{(4)}c_{1,3}^{(4)}c_{2,3}^{(4% )}=\frac{\kappa^{2}}{1-\kappa^{2}}\left(s_{1,3}^{(4)}d_{1,3}^{(4)}+s_{2,3}^{(4% )}d_{2,3}^{(4)}+s_{2,3}^{(4)}d_{2,3}^{(4)}\right).$ ⓘ Symbols: $s_{m,p}^{(r)}$: cyclic quantity, $c_{m,p}^{(r)}$: cyclic quantity, $d_{m,p}^{(r)}$: cyclic quantity and $\kappa$ Permalink: http://dlmf.nist.gov/22.9.E23 Encodings: TeX, pMML, png See also: Annotations for 22.9(iv), 22.9(iv), 22.9 and 22

§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).