Related papers: A note on Farey fractions with denominators in ari…
We enumerate the connected graphs that contain a number of edges growing linearly with respect to the number of vertices. So far, only the first term of the asymptotics and a bound on the error were known. Using analytic combinatorics, ie…
Several continued fraction expansions for $e$ have been produced by an automated conjecture generator (ACG) called \emph{The Ramanujan Machine}. Some of these were already known, some have recently been proved and some remain unproven.…
Discrete random probability measures are a key ingredient of Bayesian nonparametric inferential procedures. A sample generates ties with positive probability and a fundamental object of both theoretical and applied interest is the…
In this paper we introduce a link between geometry of ordinary continued fractions and trajectories of points that moves according to the second Kepler law. We expand geometric interpretation of ordinary continued fractions to the case of…
A novel asymptotic representation of the analytic solutions to a family of singularly perturbed $q-$difference-differential equations in the complex domain is obtained. Such asymptotic relation shows two different levels associated to the…
Continued fractions are linked to Stern's diatomic sequence 0,1,1,2,1,3,2,3,1,4,... (given by the recursion relation a_2n=a_n and a_{2n+1} = a_n + a_{n+1}, where a_0=0 and a_1=1), which has long been known. Using a particular…
We study the asymptotics of iterates of the transfer operator for non-uniformly hyperbolic $\alpha$-Farey maps. We provide a family of observables which are Riemann integrable, locally constant and of bounded variation, and for which the…
For subordinators with positive drift we extend recent results on the structure of the potential measures and the renewal densities. Applying Fourier analysis a new representation of the potential densities is derived from which we deduce…
In this paper, we introduce the polynomial continued fraction, a close relative of the well-known simple continued fraction expansions which are widely used in number theory and in general. While they may not possess all the intriguing…
Computer algebra algorithms are developed for evaluating the coefficients in Airy-type asymptotic expansions that are obtained from integrals with a large parameter. The coefficients are defined from recursive schemes obtained from…
We devise a three-parameter random search strategy to obtain accurate estimates of the large-coupling amplitude and exponent of an observable from its divergent Taylor expansion, known to some desired order. The endeavor exploits the power…
In this note, we give a detailed proof of an asymptotic for averages of coefficients of a class of degree three $L$-functions which can be factorized as a product of a degree one and a degree two $L$-functions. We emphasize that we can…
The purpose of this paper is to study the asymptotic behavior of the weighted least square estimators of the unknown parameters of random coefficient bifurcating autoregressive processes. Under suitable assumptions on the immigration and…
In [arXiv:2212.04969], the authors stated some conjectures on the variance of certain sums of the divisor function $d_k(n)$ over number fields, which were inspired by analogous results over function fields proven in [arXiv:2107.01437].…
Recent advances in the periodic orbit theory of stochastically perturbed systems have permitted a calculation of the escape rate of a noisy chaotic map to order 64 in the noise strength. Comparison with the usual asymptotic expansions…
We analyze the Hermite polynomials $H_{n}(x)$ and their zeros asymptotically, as $n\to\infty.$ We obtain asymptotic approximations from the differential-difference equation which they satisfy, using the ray method. We give numerical…
We study fractions associated to Ford circles which are extracted by means continuous curves. We show that the extracted fractions have similar properties to Farey sequences, like the Farey sum, and we prove that every ordered sequence that…
Symmetrical subdivisions in the space of Jager Pairs for continued fractions-like expansions will provide us with bounds on their difference. Results will also apply to the classical regular and backwards continued fractions expansions,…
In this work, we deal with extreme value theory in the context of continued fractions using techniques from probability theory, ergodic theory and real analysis. We give an upper bound for the rate of convergence in the Doeblin-Iosifescu…
We investigate evolution equations for anomalous diffusion employing fractional derivatives in space and time. Linkage between the space-time variables leads to a new type of fractional derivative operator. Fractional diffusion equations…