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Table 2 Identities of divisor functions

From: Convolution identities for twisted Eisenstein series and twisted divisor functions

Identities of Eisenstein series

Reference

( n + 2 ) ( n + 3 ) 2 n ( n 1 ) S n + 2 = 20 ( n 2 2 ) S 4 S n 2 ( n + 2 ) ( n + 3 ) 2 n ( n 1 ) S n + 2 = + r = 1 [ ( n 2 ) / 4 ] ( n 2 2 r ) { ( n + 3 5 r ) ( n 8 5 r ) ( n + 2 ) ( n + 3 ) 2 n ( n 1 ) S n + 2 = 5 ( r 2 ) ( r + 3 ) } S 2 r + 2 S n 2 r

[[1], Entry 14, p. 332]

S 2 n + 2 ( τ ) = D ( S 2 n ( τ ) ) S 2 n + 2 ( τ ) = + 2 s = 0 n 1 ( 2 n 2 s + 1 ) S 2 n 2 s ( τ ) S 2 s + 2 ( τ )

Theorem 3.2

S 2 k + 2 , χ 0 ( τ ) = 4 s = 0 k 1 ( 2 k 2 s + 1 ) S 2 k 2 s , χ 1 ( τ ) S 2 s + 2 , χ 1 ( τ ) S 2 k + 2 , χ 0 ( τ ) = 4 s = 0 k 1 ( 2 k 2 s + 1 ) S 2 k 2 s , χ 1 ( τ ) S 2 s + 2 , χ 1 ( τ )

Lemma 3.3

D ( S 2 k , χ 0 ( τ ) ) = 2 s = 0 k 1 ( 2 k 2 s + 1 ) D ( S 2 k , χ 0 ( τ ) ) = × ( S 2 k 2 s , χ 1 ( τ ) S 2 k 2 s , χ 0 ( τ ) ) S 2 s + 2 ( τ )

Lemma 3.3