Winquist’s identity and Ramanujan’s partition congruence

By means of the difference equation on the modified Jacobi theta function, we review the proof of Winquist’s identity due to Kang [S.Y. Kang, A new proof Winquist’s identity, J. Combin. Theory (Series A) 78 (1997) 313–318]. Four related expansion formulae are examined and clarified equivalently in pairs. The recent double series representation for ( q ; q ) ∞ 10 due to Chan [S.H. Chan, Generalized lambert series identities, Proc. London Math. Soc. 91 (3) (2005) 598–622] is exemplified to prove the Ramanujan congruence modulo 11 on the partition function.