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1.
The Ramanujan Journal - Ramanujan recorded four reciprocity formulas for the Rogers–Ramanujan continued fractions. Two reciprocity formulas each are also associated with the... 相似文献
2.
Eduard Y. Lerner 《Functional Analysis and Other Mathematics》2010,3(1):75-83
In this paper we answer certain questions posed by V.I. Arnold, namely, we study periods of continued fractions for solutions of quadratic equations in the form x 2+px=q with integer p and q, p 2+q 2≤R 2. We prove a weak variant of Arnold conjectures about the Gauss–Kuzmin statistics with R→∞. 相似文献
3.
Chadwick Gugg 《Journal of Number Theory》2012,132(7):1519-1553
In this paper, we prove modular identities involving cubes of the Rogers–Ramanujan functions. Applications are given to proving relations for the Rogers–Ramanujan continued fraction. Some of our identities are new. We establish analogous results for the Ramanujan–Göllnitz–Gordon functions and the Ramanujan–Göllnitz–Gordon continued fraction. Finally, we offer applications to the theory of partitions. 相似文献
4.
To compare continued fraction digits with the denominators of the corresponding approximants we introduce the arithmetic-geometric
scaling. We will completely determine its multifractal spectrum by means of a number-theoretical free-energy function and
show that the Hausdorff dimension of sets consisting of irrationals with the same scaling exponent coincides with the Legendre
transform of this free-energy function. Furthermore, we identify the asymptotic of the local behaviour of the spectrum at
the right boundary point and discuss a connection to the set of irrationals with continued-fraction digits exceeding a given
number which tends to infinity. 相似文献
5.
6.
Krishnaswami Alladi 《The Ramanujan Journal》2009,20(3):329-339
We will interpret a partial theta identity in Ramanujan’s Lost Notebook as a weighted partition theorem involving partitions into distinct parts with smallest part odd. A special case of this yields a new result on the parity of the number of parts in such partitions, comparable to Euler’s pentagonal numbers theorem. We will provide simple and novel proofs of the weighted partition theorem and the special case. Our proof leads to a companion to Ramanujan’s partial theta identity which we will explain combinatorially. 相似文献
7.
We undertake a thorough investigation of the moments of Ramanujan?s alternative elliptic integrals and of related hypergeometric functions. Along the way we are able to give some surprising closed forms for Catalan-related constants and various new hypergeometric identities. 相似文献
8.
Michael D. Hirschhorn 《The Ramanujan Journal》2011,24(1):85-92
We present elementary proofs, using only Jacobi’s triple product identity, of four identities of Ramanujan and eight identities of Hirschhorn relating to the 2-dissection and the 4-dissection of Ramanujan’s continued fraction and its reciprocal, and of two identities from Ramanujan’s famous list of forty. 相似文献
9.
《Journal of Computational and Applied Mathematics》2001,132(2):467-477
We characterize the sequences {zn} of complex numbers which are sequences of approximants of continued fractions K(an/bn) with |an|+1⩽|bn|, and study some of their properties. In particular we give truncation error bounds for such continued fractions. 相似文献
10.
Stefan Paszkowski 《Numerical Algorithms》1992,2(2):155-170
A new method of convergence acceleration is proposed for continued fractions of Poincaré's type 1. Each step of the method (and not only the first one, as in the Hautot method [1]) is based on an asymptotic behaviour of continued fraction tails. A theorem is proved detailing properties of the method in six cases considered here. Results of numerical tests for all Hautot's examples confirm a good performance of the method. 相似文献
11.
The Ramanujan Journal - In this paper, we give some extensions for Ramanujan’s circular summation formulas with the mixed products of two Jacobi’s theta functions. As applications, we... 相似文献
12.
The Ramanujan Journal - We show that if $$E/\mathbb {Q}$$ is an elliptic curve with a rational p-torsion for $$p=2$$ or 3, then there is a congruence relation between Ramanujan’s tau function... 相似文献
13.
Olga Kushel 《The Ramanujan Journal》2013,32(1):109-124
In this paper, we study the continued fraction y(s,r) which satisfies the equation y(s,r)y(s+2r,r)=(s+1)(s+2r?1) for $r > \frac{1}{2}$ . This continued fraction is a generalization of the Brouncker’s continued fraction b(s). We extend the formulas for the first and the second logarithmic derivatives of b(s) to the case of y(s,r). The asymptotic series for y(s,r) at ∞ are also studied. The generalizations of some Ramanujan’s formulas are presented. 相似文献
14.
By guessing the relative quantities and proving the recursive relation, we present some continued fraction expansions of the Rogers–Ramanujan type. Meanwhile, we also give some J-fraction expansions for the q-tangent and q-cotangent functions. 相似文献
15.
Mean-value theorems and extensions of the Elliott-Daboussi theorem on additive arithmetic semigroups
Wen-Bin Zhang 《The Ramanujan Journal》2008,15(1):47-75
We present more general forms of the mean-value theorems established before for multiplicative functions on additive arithmetic
semigroups and prove, on the basis of these new theorems, extensions of the Elliott-Daboussi theorem. Let
be an additive arithmetic semigroup with a generating set ℘ of primes p. Assume that the number G(m) of elements a in
with “degree” ∂(a)=m satisfies
with constants q>1, ρ
1<ρ
2<⋅⋅⋅<ρ
r
=ρ, ρ≥1, γ>1+ρ. For the main result, let α,τ,η be positive constants such that α>1,τ
ρ≥1, and τ
α
ρ≥1. Then for a multiplicative function f(a) on
the following two conditions (A) and (B) are equivalent. These are (A) All four series
converge and
and (B) The order τ
ρ mean-value
exists with m
f
≠0 and the limit
exists with M
v
(α)>0.
相似文献
16.
A. V. Tsygvintsev 《The Ramanujan Journal》2008,15(3):407-413
We consider the limit periodic continued fractions of Stieltjes type
appearing as Schur–Wall g-fraction representations of certain analytic self maps of the unit disc |w|<1, w∈ℂ. We make precise the convergence behavior and prove the general convergence [2, p. 564] of these continued fractions at
Runckel’s points [6] of the singular line (1,+∞). It is shown that in some cases the convergence holds in the classical sense.
As a result we provide an interesting example of convergence relevant to one result found in the Ramanujan’s notebook [1,
pp. 38–39].
Dedicated to Sacha B. 相似文献
17.
A. V. Bykovskaya 《Mathematical Notes》2012,92(3-4):312-326
A multidimensional geometric analog of Lagrange’s theorem on continued fractions is proposed. The multidimensional generalization of the geometric interpretation of a continued fraction uses the notion of a Klein polyhedron, that is, the convex hull of the set of nonzero points in the lattice ? n contained inside some n-dimensional simplicial cone with vertex at the origin. A criterion for the semiperiodicity of the boundary of a Klein polyhedron is obtained, and a statement about the nonempty intersection of the boundaries of the Klein polyhedra corresponding to a given simplicial cone and to a certain modification of this cone is proved. 相似文献
18.
19.
Hei-Chi Chan 《The Ramanujan Journal》2010,21(2):173-180
In this paper, we give a new proof of two identities involving Ramanujan’s cubic continued fraction. These identities are the key ingredients to an analog of Ramanujan’s “Most Beautiful Identity” discovered recently. 相似文献
20.
Arnaldo Nogueira 《Journal d'Analyse Mathématique》2001,85(1):1-41
A central result in the metric theory of continued fractions, the Borel—Bernstein Theorem gives statistical information on
the rate of increase of the partial quotients. We introduce a geometrical interpretation of the continued fraction algorithm;
then, using this set-up, we generalize it to higher dimensions. In this manner, we can define known multidimensional algorithms
such as Jacobi—Perron, Poincaré, Brun, Rauzy induction process for interval exchange transformations, etc. For the standard
continued fractions, partial quotients become return times in the geometrical approach. The same definition holds for the
multidimensional case. We prove that the Borel—Bernstein Theorem holds for recurrent multidimensional continued fraction algorithms.
Supported by a grant from the CNP
q
-Brazil, 301456/80, and FINEP/CNP
q
/MCT 41.96.0923.00 (PRONEX). 相似文献