Ramanujan’s Notebooks

Abstract

Pt. 1. Magic squares. Sums related to the harmonic series or the inverse tangent function. Combinatorial analysis and series inversions. Iterates of the exponential function and an ingenious formal technique. Eluerian polynomials and numbers, Bernoulli numbers, and the Riemann zeta-function. Ramanujan's theory of divergent series. Sums of powers, Bernoulli numbers, and the gamma function. Analogues of the gamma function. Infinite series identities, transformations, and evaluations. Ramanujan's quarterly reports -- pt. 2. Hypergeometric series, I. Hypergeometric series, II. Continued fractions. Integrals and asymptotic expansions. Infinite series. Asymptotic expansions and modular forms -- pt. 3. q-series and theta-functions. Fundamental properties of elliptic functions. The Jacobian elliptic functions. Modular equations of degrees 3, 5, and 7 and associated theta-function identities. Modular equations of higher and composite degrees. Eisenstein series