New theta function identities for a continued fraction of Ramanujan and their applications
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New theta function identities for a continued fraction of Ramanujan and their applications Chayanika Boruah 1 · Nipen Saikia1 Received: 18 April 2018 / Accepted: 14 August 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract We prove three new theta function identities for the continued fraction H (q) defined by H (q) := q 1/8 −
q2 q3 q4 q 7/8 , |q| < 1. 2 3 1 − q + 1 + q − 1 − q + 1 + q 4 −···
The theta-function identities are then used to prove integral representations for the continued fraction H (q). We also prove general theorems and reciprocity formulas for the explicit evaluation of the continued fraction H (q). The results are analogous to those of the famous Rogers-Ramanujan continued fraction. Keywords Ramanujan’s continued fraction · Theta-function identities · Reciprocity formulas · Explicit evaluation Mathematics Subject Classification 33D90 · 11F20
1 Introduction Let the continued fraction H (q) be defined by H (q) := q 1/8 −
q2 q3 q4 q 7/8 , 2 3 1 − q + 1 + q − 1 − q + 1 + q 4 −···
|q| < 1.
(1.1)
The continued fraction H (q) is first studied by Vasuki and Shivashankara [14]. They proved that H (q) =
B
f (−q) , f (−q 4 )
(1.2)
q 1/8
Nipen Saikia [email protected] Chayanika Boruah [email protected]
1
Department of Mathematics, Rajiv Gandhi University, Rono Hills, Doimukh, Arunachal Pradesh 791112, India
123
C. Boruah, N. Saikia
where f (−q) := f (−q, −q 2 ) =
∞
(1 − q n+1 ) =: q −1/24 η(z), q = e2πi z , Im(z) > 0,
n=0
(1.3) where (η(z) denotes the Dedekind eta-function), is a special case of Ramanujan’s general theta-function f (a, b) defined by ∞
f (a, b) =
a n(n+1)/2 bn(n−1)/2 , |ab| < 1.
(1.4)
n=−∞
Two other useful special cases of f (a, b) are the theta-functions φ and ψ given by
φ(q) := f (q, q) =
∞
qn
2
(1.5)
n=−∞
and
ψ(q) := f (q, q 3 ) =
∞
q n(n+1)/2 .
(1.6)
n=0 √
Vasuki and Shivashankara [14] found some explicit values of H (e−π n ) for any positive integer n by using theta-function identities and transformation formulas. Baruah and Saikia √ [1] also found many explicit values of H (e−π n ) and proved some relations connecting H (q) and H (q n ). In this paper, we study further properties of the continued fraction H (q) and prove some results analogous to those of the famous Rogers-Ramanujan continued fraction R(q) defined by
R(q) :=
q 1/5 q q 2 q 3 , |q| < 1. 1 + 1 + 1 + 1 +···
(1.7)
In Sect. 3, we prove three new theta-function identities for the continued fraction H (q). As application, in Sect. 4 we give two integral representations of the continued fraction H (q) and in Sect. 5, we prove new general theorems and reciprocity formulas for the explicit evaluations of H (q).
2 Preliminaries Lemma 2.1 We have f 3 (−q 2 ) , f (−q) f (−q 4 ) f 2 (−q 2 ) ψ(q) = , f (−q) f (q) =
123
(2.1) (2.2)
New theta function identities for a continued fraction of Ramanujan. . .
φ(q) =
f 5 (−q 2 ) f 2 (−q) f 2 (−q 4 )
,
(2.3)
f (−q) f (−q 4 ) , f (−q 2 ) f 2 (−q) φ(−q) = . f (−q 2 )
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