Search: MSC category 33C05 ( Classical hypergeometric functions, ${}_2F_1$ )
 On Russell-Type Modular Equations In this paper, we revisit Russell-type modular equations, a collection of modular equations first studied systematically by R.~Russell in 1887. We give a proof of Russell's main theorem and indicate the relations between such equations and the constructions of Hilbert class fields of imaginary quadratic fields. Motivated by Russell's theorem, we state and prove its cubic analogue which allows us to construct Russell-type modular equations in the theory of signature~$3$. Categories:33D10, 33C05, 11F11