There are a great many beautiful identities involving -series, some of which follow directly by taking the q-analog of standard combinatorial identities, e.g., the q-binomial theorem
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(, ; Andrews 1986, p. 10), a special case of an identity due to Euler
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(Gasper and Rahman 1990, p. 9; Leininger and Milne 1999), and q-Vandermonde sum
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where is a q-hypergeometric function.
Other -series identities, e.g., the Jacobi identities, Rogers-Ramanujan identities, and q-hypergeometric identity
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seem to arise out of the blue. Another such example is
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(Gordon and McIntosh 2000).
Hirschhorn (1999) gives the beautiful identity
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(OEIS A098445). Other modular identities involving the q-series include
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(Hardy and Wright 1979, Hirschhorn 1999), where
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(Hirschhorn 1999).
Zucker (1990) defines the useful notations
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A set of beautiful identities that can be expressed in this notation were found by M. Trott (pers. comm., Dec. 19, 2000),
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These are closely related to modular equation identities. For example, equation (◇) is an elegant form of Shen (1994) equation (3.12), obtained using the identities
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(OEIS A002448, A089803, and A089804). Similarly, equation (◇) is actually the classical expression
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for the Jacobi theta functions which follows from
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(J. Zucker, pers. comm., Nov. 11, 2003).
Another set of identities found by M. Trott (pers. comm., Jul. 8, 2009) are given by
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