A Product of Theta-Functions Analogous to Ramanujan ’ s Remarkable Product of Theta-Functions and Applications

We de�ne a product llkkkkk for any positive real numbers kk and kk involving Ramanujan’s theta-functions φφφφφφ and ψψφφφφ which is analogous to Ramanujan’s remarkable product of theta-functions recorded by Ramanujan (1957) and study its several properties. We prove general theorems for the explicit evaluations of llkkkkk and �nd some explicit values. As application of the product llkkkkk, we also offer explicit formulas for explicit values of Ramanujan’s continued fraction VVφφφφ in terms of llkkkkk and give examples.


Introduction
Ramanujan's theta-functions , , and  are de�ned as where   ∞ ∶= ∏ ∞ =     .�n page 338 of his �rst notebook, Ramanujan [1] de�ned the remarkable product of theta-functions as   =    2  √   2  2√   2     2  2   (3) where  and  are positive real numbers.He then, on pages 338 and 339, offered a list of eighteen particular values of the product   .All these eighteen values are proved by Berndt et al. [2].An account of these can be found in Berndt's book [3].Naika and Dharmendra [4] also established some general theorems for explicit evaluations of the product   and found some new explicit values therefrom.Further results on   can be found in [5,6].In [7], Mahadeva Naika et  ( and found some particular values of .Motivated by the above work, in this paper, we de�ne the product of theta-functions   as where  and  are positive real numbers.We establish several properties of the product   .We prove general formulas for explicit evaluations of   and �nd its explicit values.As application of the product   , we also offer explicit formulas for explicit values of Ramanujan's cubic continued fraction  in terms of   and give examples.
In Section 2, we collect some preliminary results.In Section 3, we prove several properties of the product   .Section 4, is devoted to �nd explicit values of   .Finally in Section 5, we offer explicit formulas for explicit evaluations of continued fraction  in terms of   with examples.
To end this introduction, we de�ne Ramanujan's modular equation.Let   ′ , , and  ′ denote the complete elliptic integrals of the �rst �ind associated with the moduli ,  ′ , , and  ′ , respectively.Suppose that the equality holds for some positive integer .en a modular equation of degree  is a relation between the moduli  and  which is implied by (7).Ramanujan recorded his modular equations in terms of  and , where  =  2 and  =  2 .We say that  has degree  over .By denoting   =  2   , where  = exp− ′ /, ||  , the multiplier  connecting  and  is de�ned by  =   /  .

Some Properties of 𝑙𝑙 𝑘𝑘𝑘𝑘𝑘
In this section, we study some properties of the product   .
eorem 10.For all positive real numbers  and , one has (20) Applying Lemmas 1 and 2 in the denominator of right hand side of (20) and simplifying using eorem 10(ii) and (iii), we complete the proof.
Corollary 13.For all positive real numbers k and n, one has Proof.Setting  =  in eorem 12 and simplifying using eorem 10(ii), we obtain Replacing  by , we complete the proof.eorem 14.Let k, a, b, c, and d be positive real numbers such that  = .en Proof.From the de�nition of   and using  =  for positive real numbers    , and , we deduce that Rearranging the terms in (24) we arrive at the desired result.

Explicit Values of 𝑙𝑙 𝑘𝑘𝑘𝑘𝑘
In this section, we prove general theorems for explicit evaluations of   and �nd some explicit values.First we de�ne Ramanujan's class invariants.Ramanujan's two class invariants   and   which are de�ned by where  = −  2  ∞ and  is a positive rational number.Employing Lemma 5 in (32) it follows that Also, if  has degree  over , then In his notebooks [1] and paper [11], Ramanujan recorded a total of 116 class invariants or monic polynomials satis�ed by them.An account of these can also be found in Berndt's book [3].For further references, see [2,[12][13][14][15][16][17].
eorem 18.One has Proof.Employing Lemmas 3 and 4 in the de�nition of   , we �nd that Again from ( 32) and (33) with    − , we �nd that Combining ( 36) and (37), we complete the proof.
Proof.We use the de�nitions of class invariants   and   in Lemma 9, to complete the proof.

Evaluations of Ramanujan's Cubic Continued Fraction
In this section, we prove two formulas of explicit evaluations Ramanujan's cubic continued fraction () in terms of the product   .We also give examples.
From eorem 24, it is clear that to �nd explicit values of  3 (− −3 ) it is enough to know the explicit values of  3, .For example, noting  3,1 = 1 from eorem 10(i) we evaluate From eorem 25, it is obvious that if we know the values of  9, the corresponding values of (− −√ ) can easily be evaluated.For example, by noting  9,1 = 1 from eorem 10(i) we calculate Similarly, one can �nd explicit values of (− −3 ) by using the value of  9,9 from eorem 21(vi).
ii)  / = /  , (iii)   =   .Proof.�sing the de�nition of   and Lemmas 1 and 2, we easily arrive at (i).Replacing  by / in   and using Lemmas 1 and 2, we �nd that    / =  which completes the proof of (ii).To prove (iii), we interchange  and  in   .By using the de�nitions of , , and   , it can be seen that   has positive real value and that the values of   increases as  increases when   .Proof.�sing the de�nition of   , we �nd that     =  −−  − −    −   − −    −  .
us, by eorem 10(i),     for all    if   .eorem 12.For all positive real numbers k, m, and n, one has Proof.We set    in eorem 18 and use the result  11, to complete the proof.From Corollary 20, it is obvious explicit values of   can easily be determined if the corresponding values of the class invariants   2 are known.We give some examples in the next theorem.Proof.For (i) and (ii), we use the values of  4 and  16 from [18, p. 114-115, eorem 6.2.2(ii) (vi)].For (iii)-(vi), we use the corresponding values of   from [3, p. 189-193].Ramanujan's �chal�i type modular equations of prime degree can also be used to explicit values of the product   .We offer two theorems as examples.Proof.We use the de�nitions of class invariants   and   in Lemma 8, to complete the proof.