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In this article, we construct a general series for 1 π . We indicate that Ramanujan’s 1 π −series are all special cases of this general series and we end the paper with a new class of 1 π −series. Our work is motivated by series recently discovered by Takeshi Sato.
A new infinite product tn was introduced by S. Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujan’s assertions about tn by establishing new connections between the modular j−invariant and Ramanujan’s cubic theory of elliptic functions to alternative bases. We also show that for certain integers n, tn generates the Hilbert… (More)
When we pause to reflect on Ramanujan’s life, we see that there were certain events that seemingly were necessary in order that Ramanujan and his mathematics be brought to posterity. One of these was V. Ramaswamy Aiyer’s founding of the Indian Mathematical Society on 4 April 1907, for had he not launched the Indian Mathematical Society, then the next… (More)
In his lost notebook, Ramanujan offers several results related to the crank, the existence of which was first conjectured by F. J. Dyson and later established by G.E.Andrews and F.G. Garvan. Using an obscure identity found on p. 59 of the lost notebook, we provide uniform proofs of several congruences in the ring of formal power series for the generating… (More)
In this paper, we study the divisibility of the function a(n) defined by ∑ n≥0 a(n)q n := (q; q)−1 ∞ (q ; q2)−1 ∞ . In particular, we prove certain “Ramanujan type congruences” for a(n) modulo powers of 3.
We prove algebraic transformations for the generating series of three Apéry-like sequences. As application, we provide new binomial representations for the sequences. We also illustrate a method that derives three new series for 1/π from a classical Ramanujan’s series. §
Note that for several values of n, Ramanujan did not record the corresponding values of λn. The purpose of this paper is to establish all the values of λn in (1.2), including the ones that are not explicitly stated by Ramanujan, by using the modular jinvariant, modular equations, Kronecker’s limit formula, and an empirical approach. Applications of values… (More)
In this paper, we use the explicit Shimura Reciprocity Law to compute the cubic singular moduli α * n , which are used in the constructions of new rapidly convergent series for 1/π. We also complete a table of values for the class invariant λ n initiated by S. Ramanujan on page 212 of his Lost Notebook.
χ(q) = (−q; q)∞. If n is any postitive rational number and q = exp(−π√n), the two class invariants Gn and gn are defined by (1.1) Gn := 2−1/4q−1/24χ(q) and gn := 2−1/4q−1/24χ(−q). In the notation of H. Weber , Gn =: 2−1/4f( √−n) and gn =: 2−1/4f1( √−n). The term “invariant” is due to Weber. If Q(ω) is the algebraic number field generated by the complex… (More)