§22.9 Cyclic Identities

Keywords:: Jacobian elliptic functions, cyclic identities
Referenced by:: §22.8(iii)
Permalink:: http://dlmf.nist.gov/22.9

§22.9(i) Notation
§22.9(ii) Typical Identities of Rank 2
§22.9(iii) Typical Identities of Rank 3
§22.9(iv) Typical Identities of Rank 4
§22.9(v) Identities of Higher Rank

§22.9(i) Notation

Keywords:: Jacobian elliptic functions, cyclic identities
Referenced by:: Version 1.0.3 (Aug 29, 2011)
Permalink:: http://dlmf.nist.gov/22.9.i
Clarification (effective with 1.0.3):: The paragraph after (22.9.6) was revised to state more precisely the definition of rank.
Reported 2011-08-21

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 $s_{{m,p}}^{{(2)}}=\mathop{\mathrm{sn}\/}\nolimits\left(z+2p^{{-1}}(m-1)\mathop{K\/}\nolimits\!\left(k\right),k\right),$

Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex, $k$ : modulus, $m$ : positive integer, $p$ : positive integer and $s_{{m,p}}^{{(r)}}$ : cyclic quantity
Permalink:: http://dlmf.nist.gov/22.9.E1
Encodings:: TeX, pMML, png

22.9.2 $c_{{m,p}}^{{(2)}}=\mathop{\mathrm{cn}\/}\nolimits\left(z+2p^{{-1}}(m-1)\mathop{K\/}\nolimits\!\left(k\right),k\right),$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex, $k$ : modulus, $m$ : positive integer, $p$ : positive integer and $c_{{m,p}}^{{(r)}}$ : cyclic quantity
Permalink:: http://dlmf.nist.gov/22.9.E2
Encodings:: TeX, pMML, png

22.9.3 $d_{{m,p}}^{{(2)}}=\mathop{\mathrm{dn}\/}\nolimits\left(z+2p^{{-1}}(m-1)\mathop{K\/}\nolimits\!\left(k\right),k\right),$

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex, $k$ : modulus, $m$ : positive integer, $p$ : positive integer and $d_{{m,p}}^{{(r)}}$ : cyclic quantity
Permalink:: http://dlmf.nist.gov/22.9.E3
Encodings:: TeX, pMML, png

22.9.4 $s_{{m,p}}^{{(4)}}=\mathop{\mathrm{sn}\/}\nolimits\left(z+4p^{{-1}}(m-1)\mathop{K\/}\nolimits\!\left(k\right),k\right),$

Symbols:: $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex, $k$ : modulus, $m$ : positive integer, $p$ : positive integer and $s_{{m,p}}^{{(r)}}$ : cyclic quantity
Permalink:: http://dlmf.nist.gov/22.9.E4
Encodings:: TeX, pMML, png

22.9.5 $c_{{m,p}}^{{(4)}}=\mathop{\mathrm{cn}\/}\nolimits\left(z+4p^{{-1}}(m-1)\mathop{K\/}\nolimits\!\left(k\right),k\right),$

Symbols:: $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex, $k$ : modulus, $m$ : positive integer, $p$ : positive integer and $c_{{m,p}}^{{(r)}}$ : cyclic quantity
Permalink:: http://dlmf.nist.gov/22.9.E5
Encodings:: TeX, pMML, png

22.9.6 $d_{{m,p}}^{{(4)}}=\mathop{\mathrm{dn}\/}\nolimits\left(z+4p^{{-1}}(m-1)\mathop{K\/}\nolimits\!\left(k\right),k\right).$

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex, $k$ : modulus, $m$ : positive integer, $p$ : positive integer and $d_{{m,p}}^{{(r)}}$ : cyclic quantity
Referenced by:: §22.9(i)
Permalink:: http://dlmf.nist.gov/22.9.E6
Encodings:: TeX, pMML, png

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

Notes:: See Khare and Sukhatme (2002).
Referenced by:: §22.9(v)
Permalink:: http://dlmf.nist.gov/22.9.ii

In this subsection $1\leq m\leq p$ and $1\leq n\leq p$ .

¶ Three Points

Keywords:: Jacobian elliptic functions, cyclic identities

With

22.9.7 $\kappa=\mathop{\mathrm{dn}\/}\nolimits\left(2\!\mathop{K\/}\nolimits\!\left(k\right)/3,k\right),$

Symbols:: $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$ : Jacobian elliptic function, $\mathop{K\/}\nolimits\!\left(k\right)$ : 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

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

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

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

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

Notes:: See Khare and Sukhatme (2002).
Permalink:: http://dlmf.nist.gov/22.9.iii

¶ 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

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

¶ 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

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

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

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

¶ 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

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

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

§22.9(iv) Typical Identities of Rank 4

Notes:: See Khare et al. (2003), Khare and Sukhatme (2002).
Referenced by:: §22.9(v)
Permalink:: http://dlmf.nist.gov/22.9.iv

¶ 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

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

¶ Three Points

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

22.9.22 $s_{{1,3}}^{{(2)}}c_{{1,3}}^{{(2)}}d_{{2,3}}^{{(2)}}d_{{3,3}}^{{(2)}}+s_{{2,3}}^{{(2)}}c_{{2,3}}^{{(2)}}d_{{3,3}}^{{(2)}}d_{{1,3}}^{{(2)}}+s_{{3,3}}^{{(2)}}c_{{3,3}}^{{(2)}}d_{{1,3}}^{{(2)}}d_{{2,3}}^{{(2)}}=\frac{\kappa^{2}+k^{2}-1}{1-\kappa^{2}}\left(s_{{1,3}}^{{(2)}}c_{{1,3}}^{{(2)}}+s_{{2,3}}^{{(2)}}c_{{2,3}}^{{(2)}}+s_{{3,3}}^{{(2)}}c_{{3,3}}^{{(2)}}\right),$

Symbols:: $k$ : modulus, $s_{{m,p}}^{{(r)}}$ : cyclic quantity, $c_{{m,p}}^{{(r)}}$ : cyclic quantity, $d_{{m,p}}^{{(r)}}$ : cyclic quantity and $\kappa$
Permalink:: http://dlmf.nist.gov/22.9.E22
Encodings:: TeX, pMML, png

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

§22.9(v) Identities of Higher Rank

Keywords:: Jacobian elliptic functions, cyclic identities
Permalink:: http://dlmf.nist.gov/22.9.v

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).

§22.9 Cyclic Identities

Contents

§22.9(i) Notation

§22.9(ii) Typical Identities of Rank 2

¶ Three Points

§22.9(iii) Typical Identities of Rank 3

¶ Two Points

¶ Three Points

¶ Four Points

§22.9(iv) Typical Identities of Rank 4

¶ Two Points

¶ Three Points

§22.9(v) Identities of Higher Rank