Formulas of Ramanujan Involving Lucas Numbers, Pell Numbers, and Bernoulli Numbers

Abstract

Ramanujan’s first paper [11, pages 1–14], [12] was concerned with properties of the Bernoulli numbers B _n . Among his results were some unusual formulas equivalent to “lacunary” recurrences for the Bernoulli numbers; that is, formulas for B _{10n + 2r} (for fixed r) in terms of B _{10k + 2r} (k = 0, 1,··· n − 1). The purpose of the present paper is threefold: (1) Since Ramanujan’s proofs are sketchy, and not always clear, we give detailed proofs of his formulas; (2) We point out how the Lucas numbers occur in Ramanujan’s formulas. The writer believes this relationship between Lucas numbers and Bernoulli numbers is not well-known; (3) Using Ramanujan’s method, we give new lacunary recurrences for the Genocchi numbers G _n = 2(1 − 2ⁿ)B _n . These recurrences involve Lucas numbers and Pell numbers.