On a Ramanujan’s Eisenstein series identity of level fifteen

On pages 255–256 of his second notebook, Ramanujan recorded an Eisenstein series identity of level 15 without offering a proof. Previously, Berndt (Ramanujan’s Notebooks: Part III ( 1991 ) (New York: Springer)) proved this identity using the theory of modular forms. In this paper, we give an elementary proof of this identity. In the process, we also give an elementary proof of three Ramanujan’s P - Q identities of level 15. Further, using the P - Q identities we prove four Ramanujan type Eisenstein series of level 15 due to Cooper and Ye ( Trans. Am. Math. Soc. 368 ( 2016 ) 7883–7910), where they have proved using the theory of modular forms.