# §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 $\kappa=\operatorname{dn}\left(2K\left(k\right)/3,k\right),$ ⓘ Defines: $\kappa$: change of variable (locally) 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 and $k$: modulus 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 Ch.22
 22.9.8 $s_{1,3}^{(4)}s_{2,3}^{(4)}+s_{2,3}^{(4)}s_{3,3}^{(4)}+s_{3,3}^{(4)}s_{1,3}^{(4% )}=\frac{\kappa^{2}-1}{k^{2}},$ ⓘ Symbols: $k$: modulus, $s_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable Referenced by: Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10) Permalink: http://dlmf.nist.gov/22.9.E8 Encodings: TeX, pMML, png Correction (effective with 1.0.25): Originally this equation was written incorrectly for all $s_{m,p}^{(4)}$ with $p=2$. It has been corrected by writing $p=3$ throughout. Suggested 2019-11-07 by Juan Miguel Nieto See also: Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22
 22.9.9 $c_{1,3}^{(4)}c_{2,3}^{(4)}+c_{2,3}^{(4)}c_{3,3}^{(4)}+c_{3,3}^{(4)}c_{1,3}^{(4% )}=-\frac{\kappa(\kappa+2)}{(1+\kappa)^{2}},$ ⓘ Symbols: $c_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable Referenced by: Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10) Permalink: http://dlmf.nist.gov/22.9.E9 Encodings: TeX, pMML, png Correction (effective with 1.0.25): Originally this equation was written incorrectly for all $c_{m,p}^{(4)}$ with $p=2$. It has been corrected by writing $p=3$ throughout. Suggested 2019-11-07 by Juan Miguel Nieto See also: Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22
 22.9.10 $d_{1,3}^{(2)}d_{2,3}^{(2)}+d_{2,3}^{(2)}d_{3,3}^{(2)}+d_{3,3}^{(2)}d_{1,3}^{(2% )}=d_{1,3}^{(4)}d_{2,3}^{(4)}+d_{2,3}^{(4)}d_{3,3}^{(4)}+d_{3,3}^{(4)}d_{1,3}^% {(4)}=\kappa(\kappa+2).$ ⓘ Symbols: $d_{m,p}^{(r)}$: cyclic quantity and $\kappa$: change of variable Referenced by: §22.9(ii), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10) Permalink: http://dlmf.nist.gov/22.9.E10 Encodings: TeX, pMML, png Correction (effective with 1.0.25): Originally this equation was written incorrectly for all $d_{m,p}^{(2)}$ and $d_{m,p}^{(4)}$ with $p=2$. It has been corrected by writing $p=3$ throughout. Suggested 2019-11-07 by Juan Miguel Nieto See also: Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22

These identities are cyclic in the sense that each of the indices $m,n$ in the first product of, for example, the form $s_{m,p}^{(4)}s_{n,p}^{(4)}$ 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 Ch.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 Ch.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 $\kappa$: change of variable 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 Ch.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 $\kappa$: change of variable 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 Ch.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 $\kappa$: change of variable 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 Ch.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 $\kappa$: change of variable 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 Ch.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 Ch.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 Ch.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 Ch.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 Ch.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 Ch.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 $\kappa$: change of variable 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 Ch.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).