arXiv subjects
Publications and source records attributed to Örs Rebák.
The primary purpose of this paper is to provide a survey of properties, values, identities, and generalizations of the Rogers--Ramanujan continued fraction, which is closely related to the Rogers--Ramanujan identities. Many of these results are found in Ramanujan's first two letters to Hardy, Ramanujan's notebooks, and his lost notebook. Short historical accounts are provided for both the notebooks and lost notebook.
On page 206 in his lost notebook, Ramanujan recorded a seventh degree identity for his theta function $φ(q)$. We give an analogous ninth degree identity. We also provide an application of an entry from his second notebook on a cubic equation and an interpretation with theta functions for some of his trigonometric identities. Lastly, we calculate five examples for $φ(e^{-π\sqrt{n}})$.
On page 206 in his lost notebook, Ramanujan recorded an incomplete septic theta function identity. Motivated by the completion of this identity by the second author, we offer cubic and quintic analogues. Using the theory generated by these two analogues and Ramanujan's class invariants, we provide many evaluations for Ramanujan's most prominent theta function, $φ(q)$ in his notation.
On page 206 in his lost notebook, Ramanujan recorded the following enigmatic identity for his theta function $φ(q)$: \begin{equation*} φ(e^{-7π\sqrt{7}}) = 7^{-3/4}φ(e^{-π\sqrt{7}})\big\{1 + (\quad)^{2/7} + (\quad)^{2/7} + (\quad)^{2/7}\big\}. \end{equation*} We give the three missing terms. In addition, we calculate the class invariant $G_{343}$ and further special values of $φ(e^{-nπ}),$ for $n = 7, 21, 35,$ and $49$.
The Euler product for the Landau--Ramanujan constant could have motivated a curious identity by Ramanujan that appears in his notebooks two times. This observation involves a square root and the first four prime numbers of the form $4n+3$, i.e., $3, 7, 11, 19$. Berndt asks whether Ramanujan's identity is an isolated result, or if there are other identities of this type. With this work we would like to give a possible answer to Berndt's question.
This paper provides a survey of particular values of Ramanujan's theta function $φ(q)=\sum_{n=-\infty}^{\infty}q^{n^2}$, when $q=e^{-π\sqrt{n}}$, where $n$ is a positive rational number. First, descriptions of the tools used to evaluate theta functions are given. Second, classical values are briefly discussed. Third, certain values due to Ramanujan and later authors are given. Fourth, the methods that are used to determine these values are described. Lastly, an incomplete evaluation found in Ramanujan's lost notebook, but now completed and proved, is discussed with a sketch of its proof.
Efficiency is a core concept of multi-objective optimization problems and multi-attribute decision making. In the case of pairwise comparison matrices a weight vector is called efficient if the approximations of the elements of the pairwise comparison matrix made by the ratios of the weights cannot be improved in any position without making it worse in some other position. A pairwise comparison matrix is called double perturbed if it can be made consistent by altering two elements and their reciprocals. The most frequently used weighting method, the eigenvector method is analyzed in the paper, and it is shown that it produces an efficient weight vector for double perturbed pairwise comparison matrices.