Relations Between Ramanujan’s Cubic Continued Fraction and a Continued Fraction of Order 12 and its Evaluations
Belakavadi Radhakrishna Srivatsa Kumar^{*}, and Harekala Chandrashekara Vidya
Department of Mathematics, Manipal Institute of Technology, Manipal Academy of Higher Education, Manipal-576 104, India, e-mail: sri_vatsabr@yahoo.com and vidyaashwath@gmail.com
^{*}Corresponding author.
Received: February 10, 2017; Accepted: March 29, 2018; Published online: June 23, 2018.
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Abstract
In the present paper, we establish relationship between continued fraction U(−q) of order 12 and Ramanujan’s cubic continued fraction G(−q) and G(q^{n}) for n = 1, 2, 3, 5 and 7. Also we evaluate U(q) and U(−q) by using two parameters for Ramanujan’s theta-functions and their explicit values.
Keywords: Ramanujan cubic continued fraction, Theta functions
1. Introduction
Throughout this paper, we assume that |q| < 1 and for any complex number a,
where n is a positive integer. In Chapter 16 of his second notebook [5, p. 34], [21, p. 197], S. Ramanujan developed the theory of theta-functions and his theta-function is defined as follows:
Recently H. M. Srivastava et. al. [25] proved two q-identities which provide relationship between f(−q), ϕ(q) and ψ(q), by using well known Jacobi’s triple product identity. These q-identities are analogous to Ramanujan’s identities.
The celebrated Rogers–Ramanujan continued fraction is defined as
On page 365 of his ‘Lost’ notebook [22], Ramanujan recorded five identities showing the relationships between R(q) and five continued fractions R(−q), R(q^{2}), R(q^{3}), R(q^{4}) and R(q^{5}). He also recorded these identities at the scattered places of his Notebooks [21]. L. J. Rogers [23] established the modular equations relating R(q) and R(q^{n}) for n = 2, 3, 5 and 11. Recently, K. R. Vasuki and S. R. Swamy [28] found the modular equations relating R(q) and R(q^{11}).
Recently C. Adiga et. al. [1] have established several modular relations for the Rogers–Ramanujan type functions of order eleven which analogous to Ramanuja’s forty identities for Rogers–Ramanujan functions and also they established certain interesting partition-theoretic interpretation of some of the modular relations and H. M. Srivastava and M. P. Chaudhary [24] established a set of four new results which depicit the interrelationships between q-product identities, continued fraction identities and combinatorial partition identities.
The Ramanujan’s cubic continued fraction G(q) is defined as
The continued fraction (1.2) was first introduced by Ramanujan in his second letter to G. H. Hardy [16]. He also recorded the continued fraction (1.2) on p. 365 of his ‘Lost’ notebook [22] and claimed that there are many results for G(q) similar to the results obtained for Rogers–Ramanujan continued fraction (1.1). Motivated by Ramanujan’s claim H. H. Chan [11] proved three identities giving the relations between G(q) and three continued fractions G(−q), G(q^{2}) and G(q^{3}). Further N. D. Baruah [3], established the modular relations between G(q) and G(q^{n}) for n = 5 and 7.
The Ramanujan-Göllnitz-Gordon continued fraction [22, p. 44], [14, 15] is defined as
H. H. Chan and S. S. Hang [12] and K. R. Vasuki and B. R. Srivatsa Kumar [27] established the relationships between H(q) and H(q^{n}) with n = 3, 4, 5, 11 by using the modular equations deduced by Ramanujan. Recently, B. Cho, J. K. Koo and Y. K. Park [13] extended the results cited above for the continued fraction (1.3) to all odd prime p by computing the affine models of modular curves X(Γ) with Γ = Γ_{1}(8) ∩ Γ_{0}(16p).
was established by M. S. M. Naika et. al. [18] as a special case of fascinating continued fraction identity recorded by Ramanujan in his second notebook [21, p. 74] and they have established a modular relationship between the continued fraction U(q) and U(q^{n}) with n = 3 and 5. Recently K. R. Vasuki et. al. [26] have established a relationship between the continued fraction U(q) and U(q^{n}) with n = 7, 9, 11 and 13.
Also in his ‘Lost’ notebook [22], Ramanujan recorded the following continued fraction identity
C. Adiga et. al. [2], have established some relation between Ramanujan’s continued fraction F(a, b; q) and obtained three equivalent integral representations of F(−1, 1; q) and some modular equations for the same. Also they found continued fraction representations for the Ramanujan-Weber class invariants. Further, they deduced some algebraic numbers and transcendental numbers involving F(−1, 1; q) + 1, the Ramanujan-Göllnitz-Gordon continued fraction H(q) and the Dedekind eta function.
At the end of his brief communication [19, 20] announcing his proofs of the Rogers–Ramanujan identities, Ramanujan remarks, “I have now found an algebraic relation between G(q) and H(q), Viz,:
Each of these formulae is the simplest of a large class”. But Ramanujan has not shown how he had proved these two identities and these are two from the list of forty identities involving G(q) and H(q). Rogers [23] established ten of the identities and Watson [29] proved eight of the identities, but two of them in the group that Rogers had proved. These forty identities were first brought before the mathematical world by B. J. Birch [9] who found Watson’s hand written copy of Ramanujan list of forty identities in the Oxford University Library. D. Bressoud [10], in his Ph. D. thesis proved fifteen from the list of forty identities. After the work of Rogers, Watson and Bressoud, nine remain to be proved. A. J. F. Biagioli [8] used modular forms to prove eight of them. In 2007, B. C. Berndt et. al. [6] have proved thirty five of the forty identities and also established several new identities involving the Rogers–Ramanujan functions by using modular equations found by Ramanujan. Recently C. Adiga et. al. [1] have established several modular relations for the Rogers–Ramanujan type functions of order eleven which analogous to Ramanujan’s forty identities for Rogers–Ramanujan functions and also they established certain interesting partition-theoretic interpretation of some of the modular relations.
Next we introduce the Ramanujan–Weber class invariants. For $q={e}^{-\pi \sqrt{n}}$, where n is a positive real number, define the two class invariants G_{n} and g_{n} by
$${G}_{n}={2}^{-1/4}{q}^{-1/24}\chi (q)$$
and
$${g}_{n}={2}^{-1/4}{q}^{-1/24}\chi (-q).$$
where χ(q) is as defined earlier. Ramanujan first introduced the above class invariants in his paper ‘Modular equations and approximations to π’. Ramanujan [21] recorded the values of 107 class invariants in his first notebook. H. Weber [30] first constructed a table consisting of 50 values of class invariants earlier to Ramanujan. Weber preliminarily was motivated to calculate class so that he could construct Hilbert class fields. Ramanujan without the knowledge of Weber’s work, independently calculated class invariants for different reasons. Furthermore, S. Bhargava, K. R. Vasuki and B. R. Srivatsa Kumar [7], J. Yi [31], N. D. Baruah [4], M. S. M. naika and K. S. Bairy [17] have obtained several values for G_{n} and g_{n}. Recently J. Yi [31] defined two parametrization h_{k,n} and ${h}_{k,n}^{\prime}$ for the theta function ϕ and obtained some interesting properties of them for all real numbers k and n and they are defined as
Motivated by the above work, in the present paper, we establish the relationship between U(−q) and the other continued fractions G(−q) and G(q^{n}) for n = 1, 2, 3, 5 and 7. Since the working method is monotonous, we skip the detailed proof of U(−q) and G(q^{n}) for n = 3, 5 and 7. Also we evaluate U(q) and U(−q) by using two parameters for Ramanujan’s theta-functions and their explicit values.
Thus, the second factor does not vanish. Hence by identity theorem, one can see that, the second factor does not tend to zero, whereas the first factor tends to zero in some neighbourhood of q = 0. Hence by analytic continuation in |q| < 1, we have
Replacing q to −q in (4.1) and then using the above result, we arrive at (i).
(ii) Replacing q to −q in (4.1), using Theorem 7.2(iii) [3] and on simplifying, we arrive at (ii).
(iii) Replacing q to −q in (4.1), then by using Theorem 2.2 (i) and (iii) [4] and on simplification, we arrive at (iii).
(iv) Replacing q to −q in (4.1), then by using Theorem 2.2 (ii) and (iv) [4] and on simplification, we arrive at (iv).
Acknowledgements
The authors are extremely grateful to the referee for suggestions which have significantly improved the presentation of this work. The authors are thankful to Dr. K. R. Vasuki, Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysuru, for his valuable suggestions. The research of the first author is partially supported under Extra Mural Research Funding by SERB, a statutory body of DST, Government of India [File No. EMR/2016/001601].
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