The Ramanujan Journal
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On index divisors and monogenity of certain number fields defined by $$x^{12}+ax^m+b$$
The Ramanujan Journal - - 2024
On modular equations of degree 25
The Ramanujan Journal - Tập 56 - Trang 743-752 - 2020
On page 237–238 of his second notebook, Ramanujan recorded five modular equations of composite degree 25. Berndt proved all these using the method of parametrization. He also expressed that his proofs undoubtedly often stray from the path followed by Ramanujan. The purpose of this paper is to give direct proofs to four of the five modular equations using the identities known to Ramanujan.
Progress on Olga Taussky-Todd’s circulant problem
The Ramanujan Journal - Tập 28 - Trang 45-60 - 2012
Determining the possible values of integer circulant determinants is an open problem proposed by Taussky-Todd. Recent interest in this question comes from studying the Lehmer constant of finite cyclic groups. By refining the approach by Laquer and Newman we contribute to the circulant determinant problem in the case that the order is a power of two.
Dynamics of a Ramanujan-type continued fraction with cyclic coefficients
The Ramanujan Journal - Tập 16 - Trang 285-304 - 2008
We study several generalizations of the AGM continued fraction of Ramanujan inspired by a series of recent articles in which the validity of the AGM relation and the domain of convergence of the continued fraction were determined for certain complex parameters (Borwein et al., Exp. Math. 13, 275–286, 2004, Ramanujan J., in press, 2004; Borwein and Crandall, Exp. Math. 12, 287–296, 2004). A study o......
Congruences for cubic partition pairs modulo powers of 3
The Ramanujan Journal - Tập 46 - Trang 563-578 - 2017
Let b(n) be the number of cubic partition pairs of n. We establish three congruences for b(n) modulo arbitrary powers of 3. Meanwhile, we prove some conjectures proposed by the first author.
Some new asymptotic expansions of certain partial theta functions
The Ramanujan Journal - Tập 34 - Trang 443-448 - 2013
In (Proc. Am. Math. Soc. 139(11):3779–3788, 2011), B.C. Berndt and B. Kim established asymptotic expansions for a class of partial theta functions, generalizing a result by S. Ramanujan. In this paper, we also prove some asymptotic expansions of such functions which are different from the results in (Berndt and Kim in Proc. Am. Math. Soc. 139(11):3779–3788, 2011).
A fractional q-derivative operator and fractional extensions of some q-orthogonal polynomials
The Ramanujan Journal - Tập 54 - Trang 29-41 - 2019
A fractional q-derivative operator is introduced and some of its properties have been proved. Next, a fractional q-differential equation of Gauss type is introduced and solved by means of a power series method. Finally, q-extensions of some classical q-orthogonal polynomials are introduced and some of the main properties of the newly defined functions are given.
Stirling modular forms and special values of multiple cotangent functions
The Ramanujan Journal - Tập 28 - Trang 409-422 - 2012
We give a new proof of the modularity of the Dedekind eta function using a transformation formula of Stirling modular forms. Moreover, we study special values of multiple cotangent functions. Then we obtain some relations between special values of multiple cotangent functions.
On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane
The Ramanujan Journal - Tập 16 Số 1 - Trang 105-129 - 2008
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