# §20.7 Identities

## §20.7(i) Sums of Squares

20.7.1
20.7.2
20.7.3
20.7.4

Also

20.7.5

20.7.6
20.7.7
20.7.8
20.7.9

For these and similar formulas see Lawden (1989, §1.4), Whittaker and Watson (1927, pp. 487–488), and Carlson (2011, §5).

Also, in further development along the lines of the notations of Neville (§20.1) and of Glaisher (§22.2), the identities (20.7.6)–(20.7.9) have been recast in a more symmetric manner with respect to suffices . The symmetry, applicable also to §§20.7(iii) and 20.7(vii), is obtained by modifying traditional theta functions in the manner recommended by Carlson (2011) and used for further purposes by Fukushima (2012).

20.7.10

## §20.7(iv) Reduction Formulas for Products

20.7.11
20.7.12

Addendum: For a companion equation see (20.7.34).

20.7.13
20.7.14

With

20.7.15

Next, with

20.7.20

## §20.7(vii) Derivatives of Ratios of Theta Functions

See Lawden (1989, pp. 19–20). This reference also gives the eleven additional identities for the permutations of the four theta functions.