# §22.9 Cyclic Identities

## §22.9(i) Notation

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

Throughout this subsection and are positive integers with .

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 determines the number of points in the identity. The argument is suppressed in the above notation, as all cyclic identities are independent of .

## §22.9(ii) Typical Identities of Rank 2

In this subsection and .

### ¶ Three Points

These identities are cyclic in the sense that each of the indices in the first product of, for example, the form are simultaneously permuted in the cyclic order: ; . Many of the identities that follow also have this property.

## §22.9(iii) Typical Identities of Rank 3

### ¶ Three Points

With defined as in (22.9.7),

22.9.13
22.9.14
22.9.15
22.9.16

## §22.9(iv) Typical Identities of Rank 4

### ¶ Three Points

Again with defined as in (22.9.7),

## §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).