On Explicit Evaluation of Ratio’s of Theta Function Which is Analogous to Ramanujan’s Function am,n

In this article, Ramanujan defined am,n [3], B. N. Dharmendra and S. Vasanth Kumar defined Em,n [5] for any positive real numbers m and n involving Ramanujan’s product of theta-functions. We established new relation between am,n and Em,n and explicit evaluations of Em,n.

Three special cases of f (a, b) are defined as follows: On page 338 in his first notebook [11], Ramanujan defines a m,n = ne , (1.5) where m and n are positive real numbers.
In [3], on pages 337 -338, Ramanujan has listed eighteen particular values. Berndt, Chan and Zhang [4] have been established all these values. For some general theorems and explicit evaluation on a m,n one can refer [6,7,8,10]. Following the above definition [9], Mahadeva Naika et al. defined a new function b m,n and in [5], B. N. Dharmendra and S. Vasanth Kumar defined the Ramanujan theta function E m,n . They established new properties of b m,n and E m,n and find its explicit values.
In [9], defined the theta function . (1.6) In [5], B. N. Dharmendra and S. Vasanth Kumar defined the Ramanujan theta function . (1.7) The main purpose of this paper to be establish new relation between a m,n and E m,n and explicit evaluation of E m,n .

Preliminary Results
In this section, we tend to collect many identities that square measure helpful in proving our main results.
Lemma 2.1. [6] If m is any positive rational, then we have, If n is any positive rational, and Q := f (q)
If m is any positive rational, then we have, If n is any positive rational, ; q := e −π √ n 5 (2.10) . (2.11) then we have, We have, a m,n = a n,m and E m,n = E n,m .
3. Modular relation between a m,n and E m,n Theorem 3.1. If x := E m,3 and y := a m,3 then Proof. From Lemma (2.1), we obtain P 4 := 9 − 9y + 3 9y 2 − 14y + 9 2y , where, y := a 2 m,3 . Employing the above equation (3.2) in Lemma (2.2), we obtain By examining the behavior of the above factors near q = 0, we can find a neighborhood about the origin, where the second factor is zero; whereas another factor is not zero in this neighborhood. By the Identity Theorem second factor vanishes identically. This completes the proof. Proof. From Lemma (2.3), we obtain P 2 := 5 − 5y + 25y 2 − 30y + 25 2y . (3.5) Employing the above equation (3.5) in Lemma (2.4), we get By examining the behavior of the above term near q = 0. This completes the proof.
Proof. In Ramanujan notebook Part V [3] he recorded many values of a 3,n . In particularly, he recorded for n = 3, 5, 7, 9, 11, 15, 19, 31, 59. Then, M. S. Mahadeva Naika , B. N. Dharmendra and K. Shivashankar [7] also evaluated the values of a 3,n for n = 2, 35, 55. Noting all these values of n, we have established the values for E 3,n If n = 3 then, we find in [3], a 3,3 = 1 √ 3 , substituting this value in (3.3) we obtain an equation and solving for x we get the desired result.
i.e.,   Proof. In [3] Ramanujan has recorded many values of a 5,n for n = 9, 11, 13, 29. Then Similarly we can obtain for remaining values of n which is mentioned in the above table2. ✷ Conclusion: Finally in this article we established new relation between a m,n and E m,n and explicit evaluations of E 3,n and E 5,n by setting particular values to n, similarly we can also obtain for other values of m.