New analogues of Clausenʼs identities arising from the theory of modular forms

Around 1828, T. Clausen discovered that the square of certain hypergeometric function can be expressed as a hypergeometric function. Special cases of Clausenʼs identities were later used by S. Ramanujan in his derivation of 17 series for 1/π. Since then, there were several attempts to find new analogues of Clausenʼs identities with the hope to derive new classes of series for 1/π. Unfortunately, none were successful. In this article, we will present three new analogues of Clausenʼs identities. Their discovery is motivated by the study of relations between modular forms of weight 2 and modular functions associated with modular groups of genus 0.