A New Proof of Winquist's Identity

✍ Scribed by Soon-Yi Kang

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 217 KB
• Volume: 78
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1996.2781

✦ Synopsis

Elementary proofs for (1.1) and (1.2) due to Ramanujan can be found in , but it was not until 1969 that the first simple proof of (1.3) of the same nature as those for (1.1) and (1.2) was given by Winquist . Winquist found and proved an identity that played an essential role in proving (1.3), as Euler's and Jacobi's theorems did in Ramanujan's proofs for (1.1) and (1.2).

Later, it was shown that Winquist's identity is a special case of the Macdonald identities for affine root systems. For Macdonald identities and their proofs, readers are refered to papers of Dyson [3], Macdonald and Stanton [8].

Carlitz and Subbarao [2] and Hirschhorn [5] found four-parameter generalizations of Winquist's identity.

In this note we give a short proof of Winquist's identity by simply multiplying two pairs of quintuple product identities and adding them. To the author's knowledge, this is the simplest and shortest proof. Thus, a simple article no. TA962781 313 0097-3165Â97 25.00