On Some Modular Equations and Their Applications I

Abstract

Abstract. We derive several modular equations and present their proofs based on concise algebraic computations. In addition, we establish explicit relations and formulas for some parameterizations for the theta functions ϕ and ψ and show some applications of the modular equations to evaluations of the cubic continued fraction and the theta function ψ.

On Evaluations of the Cubic Continued Fraction by a Modular Equation of Degree 9

“…For (i), letting n = 1 in (3.2), putting the value of [8,Theorem 4.1], solving for h 1/4 , and using the fact that h 1/4 has a positive value less than 1, we complete the proof.…”

“…We next recall a modular equation of degree 9 given in [8] it follows that √ 3 a 0 < 1. Now assume that √ 3 a k < 1 for some nonnegative integer k. Then, by (3.2),…”

On Some Modular Equations and Their Applications Ii

“…For (i), letting n = 1 in (3.2) and putting the value of l ′ 3,1 = (2 + √ 3 ) 1/4 from Theorem 4.3(i) in [9], we find that…”

“…In this paper, we further derive some more modular equations of degrees 3 and 9 for the theta functions ϕ and ψ and present their concise proofs based on algebraic computations as in [9]. Furthermore, we find explicit relations and formulas for the corresponding parameterizations, evaluate some numerical values of h k,n , h ′ k,n , l k,n , and l ′ k,n for some positive real numbers k and n by employing the relations and formulas established earlier, and evaluate some numerical values of the cubic continued fraction.…”

“…In [7,8,10], several new modular equations for the theta functions were derived, some explicit relations and formulas for the parameterizations were offered, and some values of the parameterizations were determined. Moreover, in [9], some new modular equations of degrees 3 and 9 for the theta functions ϕ and ψ were derived in order to establish explicit relations and formulas for the parameterizations for h k,n , h ′ k,n , l k,n , and l ′ k,n and show some applications of those modular equations to evaluations of the cubic continued fraction.…”