We find some new explicit values of the parameter
For
In his notebook [
In the sequel of the previous work, in this paper we find some new explicit values of the parameters
In Section
We end the introduction by defining Ramanujan's modular equation. The complete elliptic integral of the first kind
If we set
For positive real numbers
For positive real numbers
One has
One has
If
If
In this section, we prove two new identities for theta-function
If
Transcribing
Equivalently,
Squaring (
If
Transcribing
Equivalently,
Squaring (
In this section, we find some new values of
The values of
One has
where
For (i) and (ii), setting
For proofs of (iii) and (iv), we set
Equivalently,
To prove (v) and (vi), applying the definition of
For proofs of (vii) and (viii), setting
Invoking (
In this section, we find some new values of the parameter
One has
Setting
The proofs of Theorems
One has
where
Proof of Theorem
One has
where
To prove Theorem
One has
We employ Theorem
One has
where a and b are given in Theorem
Proof follows from Theorem
In this section, we use the new values of the parameters
The parameter
One has
For example, employing the value of
Next, the parameter
For any positive real number n, one has
From Theorem
Similarly, we can evaluate new values of
The author is thankful to the University Grants Commission, New Delhi, India for partially supporting the research work under the Grant no. F. No. 41-1394/2012(SR).