Two Parameters For Ramanujan's Theta-Functions and Their Explicit Values

Some New Explicit Values of Quotients of Ramanujan’s Theta Functions and Continued Fractions

“…Multiplying (25) and (27), we evaluate the value of ℎ 3,3/2 , and dividing (25) by (27) and simplifying, we evaluate the value of ℎ 3,6 . Now the values of ℎ 3,2/3 and ℎ 3,1/6 easily follow from the values of ℎ 3,3/2 and ℎ 3,6 , respectively, and Lemma 7.…”

“…Saikia [8] evaluated ℎ 3, for = 2, 1/2, 4, 1/4, 7, 1/7, 49, and 1/49. Baruah and Saikia [6] evaluated 3, for = 1, 3, 1/3, 9, 1/9, 5, 1/5, 25, 1/25, 7, 1/7, 13, 1/13, 49, and 1/49. As an application to our new values, we evaluate some old and new explicit values of Ramanujan's cubic continued fraction ( ) and a continued fraction of order twelve ( ) which are, respectively, defined by, for | | < 1,…”

Explicit Evaluations of Cubic and Quartic Theta-Functions

“…For values of r 4,7 , r 4,9 , and r 4,49 see [18]. For remaining values we refer to [11] or [17]. We also note that…”

Ramanujan's Cubic Continued Fraction and Ramanujan Type Congruences for a Certain Partition Function

“…See also the elegant results of Heng Huat Chan on the subject [20]. For some wonderful subsequent works on the cubic continued fraction, see [1,10,11,12,15,16,18,21,32]. Note that (4) implies the main result in [19] Theorem 3 (Chan, Theorem 1 in [19]).…”