Some Modular Relation on Analogous of Ramanujan’s Remarkable Product of Theta-Function

B. N. Dharmendra*, M. C. Mahesh Kumar and P. Nagendra* abstract: In this article, we derive new modular relations on Ramanujan’s product of theta-functions φ(q) and f(−q), which is analogous to Ramanujan’s remarkable product of theta-functions and their explicit evaluations.

On page 338 in his first notebook [18], [3] Ramanujan defines  . (1.5) He then, on pages 338 and 339, offers a list of eighteen particular values. All these eighteen values have been established by Berndt, Chan and Zhang [4]. M. S. Mahadeva Naika and B. N. Dharmendra [10], 2 B. N. Dharmendra, M. C. Mahesh Kumar and P. Nagendra also established some general theorems for explicit evaluations of the product of a m,n and found some new explicit values from it. Further results on a m,n can be found by Mahadeva Naika, Dharmendra and K. Shivashankar [12], and Mahadeva Naika and M. C. Mahesh Kumar [13]. Recently Nipen Saikia [16] established new properties of a m,n . In [15], Mahadeva Naika et al. defined the product . (1.6) They established general theorems for explicit evaluation of b m,n and obtained some particular values. Mahadeva Naika et al. [14] established general formulas for explicit values of Ramanujan's cubic continued fraction V (q) in terms of the products a m,n and b m,n defined above, where and found some particular values of V (q) In [5], B. N. Dharmendra defined product of theta-fuctions d m,n as where m and n are positive real numbers. He established several properties of the product d m,n and proved general formulas for explicit evaluations of d m,n and their explicit values. Let K, K ′ , L and L ′ denote the complete elliptic integrals of the first kind associated with the moduli k, k ′ := √ 1 − k 2 , l and l ′ := √ 1 − l 2 respectively, where 0 < k, l < 1. For a fixed positive integer n, suppose that Then a modular equation of degree n is a relation between k and l induced by (1.5). Following Ramanujan, set α = k 2 and β = l 2 . Then we say β is of degree n over α. Define Moreover, if q = e −π √ n m and β has degree n over α, then The main purpose of this paper is to obtain several general theorems for the explicit evaluations of analogous of Ramanujan's product of theta-function of d m,n and also some new explicit evaluations from it. Some Modular Relation on Analogous of Ramanujan's Remarkable Product of Theta-Function 3

Preliminary Results
In this section, we collect several identities which are useful in proving our main results.  Theorem 4.5] If n is any rational,

Modular Relation Between d 3,n and d 3,k 2 n
In this section, we obtain some modular relation between d 3,n and d 3,k 2 n .
, then can be written in terms of r := d 3,n as Employing the above equation Here s = d 3,4n and by examining the behavior of the above factors near q = 0, we can find a neighborhood about the origin, where the first factor is zero; whereas other factor is not zero in this neighborhood. By the Identity Theorem second factor vanishes identically. This completes the proof. .

Modular Relation Between d 5,n and d 5,k 2 n
In this section, we obtain some modular relation between d 5,n and d 5,k 2 n . , then  Here s = d 5,4n and by examining the behavior of the above factors near q = 0, we can find a neighborhood about the origin, where the first factor is zero; whereas other factors are not zero in this neighborhood. By the Identity Theorem second factor vanishes identically. This completes the proof.
where t := d 5,49 . The above equation can be written as, (4.14) where T := t + 1 t . Solving in the above equation, we get (4.11).