A three-term theta function identity and its applications

Liu, Zhi-Guo

doi:10.1016/j.aim.2004.07.006

Cited by 33 publications

(23 citation statements)

References 15 publications

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“…(−q; q 2 ) ∞ (q; q 2 ) ∞ − (−q 9 ; q 18 ) ∞ (q 9 ; q 18 ) ∞ = 2q (−q 2 ; q 2 ) ∞ (q; q 2 ) ∞ (−q 9 ; q 9 ) ∞ (−q 18 ; q 18 ) ∞ (−q 6 ; q 6 ) ∞ .…”

Section: Example 13 (Theta Function Identity)unclassified

Common Source of Numerous Theta Function Identities

Chu

2007

Glasgow Math. J.

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Abstract. Motivated by the recent work due to Warnaar (2005), two new and elementary proofs are presented for a very useful q-difference equation on eight shifted factorials of infinite order. As the common source of theta function identities, this qdifference equation is systematically explored to review old and establish new identities on Ramanujan's partition functions. Most of the identities obtained can be interpreted in terms of theorems on classical partitions.2000 Mathematics Subject Classification. Primary 05A30. Secondary 14K25.For two indeterminates q and z with |q| < 1, the q-shifted factorial of infinite order and the theta function are defined respectively byTheir product forms are abbreviated respectively asRecently, a very useful q-difference equation on eight shifted factorials of infinite order was established in [11, Theorem 1.1], where its applications to Ramanujan's congruences on the partition function are investigated. The purpose of this paper is twofold. First, we shall present two new and elementary proofs for that q-difference equation, inspired by Warnaar's recent paper [23]. Second, we shall explore its applications along another direction: theta function identities. We shall systematically review old and establish new theta function identities from that q-difference equation, which turns out to be a natural source for theta function identities.The contents of the paper will be as follows. The first section will enunciate the q-difference equation on shifted factorials and its main implications. The two new and elementary proofs will be presented in the second section. The third section will be dedicated to its applications to various theta function identities.

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“…(−q; q 2 ) ∞ (q; q 2 ) ∞ − (−q 9 ; q 18 ) ∞ (q 9 ; q 18 ) ∞ = 2q (−q 2 ; q 2 ) ∞ (q; q 2 ) ∞ (−q 9 ; q 9 ) ∞ (−q 18 ; q 18 ) ∞ (−q 6 ; q 6 ) ∞ .…”

Section: Example 13 (Theta Function Identity)unclassified

Common Source of Numerous Theta Function Identities

Chu

2007

Glasgow Math. J.

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“…This is an alternate form of the quintuple product identity and has appeared in [3], [16] and [20]. To obtain (1.1), replace e 2iz by a, q 2 by q, and use the infinite product expansions of θ i (z|τ ) listed in (2.2).…”

Section: Example 41 (Quintuple Product Identity)mentioning

confidence: 99%

GENERALIZED mth ORDER JACOBI THETA FUNCTIONS AND THE MACDONALD IDENTITIES

Toh

2008

Int. J. Number Theory

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We describe a m-th order generalization of Jacobi's theta functions and use these functions to construct classes of theta function identities in multiple variables. These identities are equivalent to the Macdonald identities for the seven infinite families of irreducible affine root systems. They are also equivalent to some elliptic determinant evaluations proven recently by H. Rosengren and M. Schlosser.

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“…Winquist [1969] derived an important identity, now known as the Winquist identity, which he then used to get the following identity for η 10 (τ ): Winquist then used this identity to give a simple proof of Ramanujan's partition congruence p(11n + 6) ≡ 0 (mod 11). Berndt, Chan, Liu, and Yesilyurt [2004] used two results from Ramanujan's notebooks, and Liu [2005] and then derived a proof of p(11n + 6) ≡ 0 (mod 11). Chan, Cooper, and Toh [2007] provided the following formula by using a theta function identity: 6(q; q) …”

Section: The Proof Of Theorem 16mentioning

confidence: 99%

A three-term theta function identity and its applications

Cited by 33 publications

References 15 publications

Common Source of Numerous Theta Function Identities

Common Source of Numerous Theta Function Identities

GENERALIZED mth ORDER JACOBI THETA FUNCTIONS AND THE MACDONALD IDENTITIES

Addition formulas for Jacobi theta functions, Dedekind's eta function, and Ramanujan's congruences

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