Ramanujan's “Lost” Notebook VIII: the entire Rogers–Ramanujan function
نویسندگان
چکیده
منابع مشابه
Ramanujan’s “Lost Notebook” and the Virasoro Algebra
is the Legendre symbol. As a matter of fact we can talk about a single identity (cf. [C2]) because (1.3) can be obtained from (1.2) by applying a Hecke’s result (see, for instance, p. 119 [My]). By now there are several proofs of (1.2) and (1.3) in the literature. The first proof was given by Bailey ([B1] and [B2]) by using the 6Ψ6–summation. For recent proofs see, for instance, [Rg], [C1] and ...
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Recall that Ramanujan’s lost notebook [27] was written in the last year of his life. We can forgive him if, in his insufferable pain, not everything he wrote down was correct. The purpose of this paper is to examine some of these questionable results. In particular, we concentrate on three “identities” found on pages 335 and 336 in [27]. At the close of this paper, we briefly mention some furth...
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In his famous paper, “On certain arithmetical functions”, Ramanujan offers for the first time the Euler product of the Dirichlet series in which the coefficients are given by Ramanujan’s tau-function. In his lost notebook, Ramanujan records further Euler products for L-series attached to modular forms, and, typically, does not record proofs for these claims. In this semi-expository article, for...
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Since the centenary of Ramanujan's birth, a large amount of Ramanujan's mathematics that had been ensconced in his earlier notebooks and in his lost notebook has been brought to light. In particular, in 1987, only one book had been written on Ramanujan's notebooks [2], but now several have appeared [2], [1]. We now have a larger portion of Ramanujan's mathematics to appreciate, evaluate, analyz...
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On two pages in his lost notebook, Ramanujan recorded several theorems involving the modified Bessel function Kν(z). These include Koshliakov’s formula and Guinand’s formula, both connected with the functional equation of nonanalytic Eisenstein series, and both discovered by these authors several years after Ramanujan’s death. Other formulas, including one by K. Soni and two particularly elegan...
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ژورنال
عنوان ژورنال: Advances in Mathematics
سال: 2005
ISSN: 0001-8708
DOI: 10.1016/j.aim.2004.03.012