Septic analogues of the Rogers–Ramanujan functions

Hahn, Heekyoung

doi:10.4064/aa110-4-5

Cited by 10 publications

(11 citation statements)

References 6 publications

(7 reference statements)

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“…Identities (3.1)-(3.13) of [20] are special cases of Theorem 1.4. Next we prove (3.1) of [20] as an example and omit the proofs of the others.…”

Section: )mentioning

confidence: 94%

Integer Matrix Exact Covering Systems and Product Identities for Theta Functions

Cao¹

2010

International Mathematics Research Notices

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Abstract. In this paper, we prove that there is a natural correspondence between product identities for theta functions and integer matrix exact covering systems. We show that since Z n can be taken as the disjoint union of a lattice generated by n linearly independent vectors in Z n and a finite number of its translates, certain products of theta functions can be written as linear combinations of other products of theta functions. We firstly give a general theorem to write a product of n theta functions as a linear combination of other products of theta functions. Many known identities for products of theta functions are shown to be special cases of our main theorem. Several entries in Ramanujan's notebooks as well as new identities are proved as applications, including theorems for products of three and four theta functions that have not been obtained by other methods.

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“…Identities (3.1)-(3.13) of [20] are special cases of Theorem 1.4. Next we prove (3.1) of [20] as an example and omit the proofs of the others.…”

Section: )mentioning

confidence: 94%

Integer Matrix Exact Covering Systems and Product Identities for Theta Functions

Cao¹

2010

International Mathematics Research Notices

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“…In [12] and [13], H. Hahn defined the septic analogues of the Rogers-Ramanujan functions as A(q) := ∞ n=0 q 2n 2 (q 2 ; q 2 ) n (−q; q) 2n = (q 7 ; q 7 ) ∞ (q 3 ; q 7 ) ∞ (q 4 ; q 7 ) ∞ (q 2 ; q 2 ) ∞ , (1.5) B(q) := ∞ n=0 q 2n(n+1) (q 2 ; q 2 ) n (−q; q) 2n = (q 7 ; q 7 ) ∞ (q 2 ; q 7 ) ∞ (q 5 ; q 7 ) ∞ (q 2 ; q 2 ) ∞ , (1.6) and C(q) := ∞ n=0 q 2n(n+1) (q 2 ; q 2 ) n (−q; q) 2n+1 = (q 7 ; q 7 ) ∞ (q; q 7 ) ∞ (q 6 ; q 7 ) ∞ (q 2 ; q 2 ) ∞ , (1.7) where the later equalities are due to Rogers [19,20]. She found several modular relations involving only A(q), B(q), and C(q) as well as relations that are connected with the RogersRamanujan and Göllnitz-Gordon functions.…”

Section: Introductionmentioning

confidence: 96%

Modular relations for the nonic analogues of the Rogers–Ramanujan functions with applications to partitions

Baruah

Bora

2008

Journal of Number Theory

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“…Hahn [20,21] defined the septic analogues of the Rogers-Ramanujan functions as A(q) := ∞ n=0 q 2n 2 (q 2 ; q 2 ) n (−q; q) 2n = (q 7 , q 3 , q 4 ; q 7 ) ∞ (q 2 ; q 2 ) ∞ (1.5) B(q) := ∞ n=0 q 2n(n+1) (q 2 ; q 2 ) n (−q; q) 2n = (q 7 , q 2 , q 5 ; q 7 ) ∞ (q 2 ; q 2 ) ∞ (1. 6) and C(q) := ∞ n=0 q 2n(n+1) (q 2 ; q 2 ) n (−q; q) 2n+1 = (q 7 , q, q 6 ; q 7 ) ∞ (q 2 ; q 2 ) ∞ (1.7)…”

Section: Introductionmentioning

confidence: 99%

“…Later, Slater [15] offered different proofs of these identities. Hahn [20,21] discovered and established several modular relations involving only A(q), B(q) and C(q), as well as relations that are connected with the Rogers-Ramanujan and Göllnitz-Gordon functions. Baruah and Bora [23] considered the following nonic analogues of the Rogers-Ramanujan functions:…”

Section: Introductionmentioning

confidence: 99%

Some Modular Relations Analogues to the Ramanujan’s Forty Identities with Its Applications to Partitions

Adiga

Bulkhali

2013

Axioms

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Recently, the authors have established a large class of modular relations involving the Rogers-Ramanujan type functions J(q) and K(q) of order ten. In this paper, we establish further modular relations connecting these two functions with Rogers-Ramanujan functions, Göllnitz-Gordon functions and cubic functions, which are analogues to the Ramanujan's forty identities for Rogers-Ramanujan functions. Furthermore, we give partition theoretic interpretations of some of our modular relations.

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Septic analogues of the Rogers–Ramanujan functions

Cited by 10 publications

References 6 publications

Integer Matrix Exact Covering Systems and Product Identities for Theta Functions

Integer Matrix Exact Covering Systems and Product Identities for Theta Functions

Modular relations for the nonic analogues of the Rogers–Ramanujan functions with applications to partitions

Some Modular Relations Analogues to the Ramanujan’s Forty Identities with Its Applications to Partitions

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