Ramanujan's Theta Functions by Shaun Cooper

Author:Shaun Cooper
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

We begin by showing that each of y a , y b , and y c can be expressed as eta-quotients.

Theorem 6.20.

The following infinite product expansions hold:

Proof.

By a direct calculation using the definitions (6.29), (6.6), and (0.​61), we have

Appealing to Theorem 6.19 it follows that

The expressions for y a , y b , and y c in terms of eta-quotients can now be obtained from the definitions of z a , z b , and z c as eta-quotients given in (6.23).

The results for w a y a , w b y b , and w c y c can be obtained by comparing eta function representations of the various functions: eta-quotients for z a , z b , z c , and z d are given by (6.23); eta-quotients for w a , w b , and w c are given in (6.30); and eta-quotients for y a , y b , and y c have been proved above as part of this theorem. □

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