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1.
In the “Lost” note book, Ramanujan had stated a large number of results regarding evaluation of his continued fraction for certain values of τ. It is shown that all these results and many more have their source in the Kronecker limit formula.  相似文献   

2.
In a previous paper, we showed the existence of an uncountable set of points on the unit circle at which the Rogers-Ramanujan continued fraction does not converge to a finite value. In this present paper, we generalise this result to a wider class of q-continued fractions, a class which includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fractions. We show, for each q-continued fraction, G(q), in this class, that there is an uncountable set of points, Y G , on the unit circle such that if y ? Y G then G(y) does not converge to a finite value. We discuss the implications of our theorems for the convergence of other q-continued fractions, for example the Göllnitz-Gordon continued fraction, on the unit circle.  相似文献   

3.
Let zC be imaginary quadratic in the upper half plane. Then the Rogers-Ramanujan continued fraction evaluated at q = e iz is contained in a class field of Q(z). Ramanujan showed that for certain values of z, one can write these continued fractions as nested radicals. We use the Shimura reciprocity law to obtain such nested radicals whenever z is imaginary quadratic. 2000 Mathematics Subject Classification Primary—11Y65; Secondary—11Y40  相似文献   

4.
Denote by pn/qn,n=1,2,3,…pn/qn,n=1,2,3,, the sequence of continued fraction convergents of the real irrational number xx. Define the sequence of approximation coefficients by θn:=qn|qnx−pn|,n=1,2,3,…θn:=qn|qnxpn|,n=1,2,3,. A laborious way of determining the mean value of the sequence |θn+1θn1|,n=2,3,…|θn+1θn1|,n=2,3,, is simplified. The method involved also serves for showing that for almost all xx the pattern θn1<θn<θn+1θn1<θn<θn+1 occurs with the same asymptotic frequency as the pattern θn+1<θn<θn1θn+1<θn<θn1, namely 0.12109?0.12109?. All the four other patterns have the same asymptotic frequency 0.18945?0.18945?. The constants are explicitly given.  相似文献   

5.
We present an idea on how Ramanujan found some of his beautiful continued fraction identities. Or more to the point: why he chose the ones he wrote down among all possible identities.   相似文献   

6.
In his first and second letters to Hardy, Ramanujan made several assertions about the Rogers-Ramanujan continued fraction . In order to prove some of these claims, G. N. Watson established two important theorems about that he found in Ramanujan's notebooks. In his lost notebook, after stating a version of the quintuple product identity, Ramanujan offers three theta function identities, two of which contain as special cases the celebrated two theorems of Ramanujan proved by Watson. Using addition formulas, the quintuple product identity, and a new general product formula for theta functions, we prove these three identities of Ramanujan from his lost notebooks.

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7.
In this note we establish continued fraction developments for the ratios of the basic hypergeometric function2ϕ1(a,b;c;x) with several of its contiguous functions. We thus generalize and give a unified approach to establishing several continued fraction identities including those of Srinivasa Ramanujan.  相似文献   

8.
We prove that the Ramanujan AGM fraction diverges if |a|=|b| with a 2b 2. Thereby we prove two conjectures posed by J. Borwein and R. Crandall. We also demonstrate a method for accelerating the convergence of this continued fraction when it converges.   相似文献   

9.
For any formal Laurent series with coefficients cn lying in some given finite field, let x=[a0(x);a1(x),a2(x),…] be its continued fraction expansion. It is known that, with respect to the Haar measure, almost surely, the sum of degrees of partial quotients grows linearly. In this note, we quantify the exceptional sets of points with faster growth orders than linear ones by their Hausdorff dimension, which covers an earlier result by J. Wu.  相似文献   

10.
We study the metrical properties of a class of continued fraction-like mappings of the unit interval, each of which is defined as the fractional part of a Möbius transformation taking the endpoints of the interval to zero and infinity.

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11.
A generalized Bailey pair, which contains several special cases considered by Bailey (Proc. London Math. Soc. (2), 50, 421–435 (1949)), is derived and used to find a number of new Rogers-Ramanujan type identities. Consideration of associated q-difference equations points to a connection with a mild extension of Gordon’s combinatorial generalization of the Rogers-Ramanujan identities (Amer. J. Math., 83, 393–399 (1961)). This, in turn, allows the formulation of natural combinatorial interpretations of many of the identities in Slater’s list (Proc. London Math. Soc. (2) 54, 147–167 (1952)), as well as the new identities presented here. A list of 26 new double sum–product Rogers-Ramanujan type identities are included as an Appendix. 2000 Mathematics Subject Classification Primary—11B65; Secondary—11P81, 05A19, 39A13  相似文献   

12.
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 of the AGM continued fraction is equivalent to an analysis of the convergence of certain difference equations and the stability of dynamical systems. Using the matrix analytical tools developed in 2004, we determine the convergence properties of deterministic difference equations and so divergence of their corresponding continued fractions. Russell Luke’s work was supported in part by a postdoctoral fellowship from the Pacific Institute for the Mathematical Sciences at Simon Fraser University.  相似文献   

13.
《Indagationes Mathematicae》2022,33(6):1189-1220
This paper investigates the quadratic irrationals that arise as periodic points of the Gauss type shift associated to the odd continued fraction expansion. It is shown that these numbers, which we call O-reduced, when ordered by the length of the associated closed primitive geodesic on some modular surface Γ?H, are equidistributed with respect to the Lebesgue absolutely continuous invariant probability measure of the Odd Gauss shift.  相似文献   

14.
We investigate for which real numbers α the series (4) converges, and prove that, even though it converges almost everywhere in the sense of Lebesgue to a periodic, with a period 1, odd function in L2([0,1]), it is divergent at uncountably many points, the set of which is dense in [0,1]. Finally, we find the Fourier expansion of the function defined by the series (4).  相似文献   

15.
16.
《Quaestiones Mathematicae》2013,36(3):437-448
Abstract

The connection between cutting sequences of a directed geodesic in the tessellated hyperbolic plane ?2, the modular group Γ = PSL(2, ?) and the simple continued fractions of an end point w of the geodesic have been established by Series [13]. In this paper we represent the simple continued fractions of w ∈ ? and the “L” and “R” codes of the cutting sequence in terms of modular and extended modular transformations. We will define a T 0-path on a graph whose vertices are the set of Farey triangles, as the equivalent of the cutting sequence. The relationship between the directed geodesic with end point w on ?, the Farey tessellation and the simple continued fraction expansion of w ∈ ? then follows easily as a consequence of this redefinition. Finite, infinite and periodic simple continued fractions are subsequently examined in this light.  相似文献   

17.
Although it is difficult to differentiate analytic functions defined by continued fractions, it is relatively easy in some cases to determine uniform bounds on such derivatives by perceiving the continued fraction as an infinite composition of linear fractional transformations and applying an infinite chain rule for differentiation.  相似文献   

18.
For let be the continued fraction expansion of . Write


We construct some numbers 's with


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19.
In this paper, we study suborbital graphs for congruence subgroup Γ0(n) of the modular group Γ to have hyperbolic paths of minimal lengths. It turns out that these graphs give rise to a special continued fraction which is a special case of very famous fraction coming out from Pringsheim’s theorem.  相似文献   

20.
The dynamics of the Gauss Map suggests a way to compare the convergence to a real number ζ ε(0,l) of a continued fraction and the divergence of the orbit of ζ Of particular interest is the comparison of the rate of convergence to ζ of its simple continued fraction and the rate of divergence by the Gauss Map of the orbit of ζ for all irrational numbers in (0,l). We state and prove sharp inequalities for the convergence of the sequence of rational convergents of an irrational number ζ. We show that the product of the rate of convergence of the continued fraction of ζ and the rate of divergence by the Gauss Map of the orbit of ζ equals 1.  相似文献   

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