Related papers: The Balkans Continued Fraction
We give a new proof of the $k$-fold convolution of the Catalan numbers. This is done by enumerating a certain class of polygonal dissections called $k$-in-$n$ dissections. Furthermore, we give a formula for the average number of cycles in a…
We provide a context around a conjectured closed form for the Hankel transform of linear combinations of consecutive pairs of Catalan numbers. This generalizes the formula for the Hankel transforms of the shifted Catalan numbers and the…
This paper is a sequel to our previous work in which we found a combinatorial realization of continued fractions as quotients of the number of perfect matchings of snake graphs. We show how this realization reflects the convergents of the…
In this paper, we investigate the monotone property of the continued fractions $G(m,\lambda)$ as a function of $m$ and $\lambda$. In particular, we obtain new inequality for the relative continued fractions.
We derive continued fractions for partition generating functions, utilizing both Euler's techniques and Ramanujan's techniques. Although our results are for integer partitions there is scope to extend this work to vector partitions,…
In this paper, we apply results on number systems based on continued fraction expansions to modular arithmetic. We provide two new algorithms in order to compute modular multiplication and modular division. The presented algorithms are…
We explore a novel link between two seemingly disparate mathematical concepts: Egyptian fractions and fractals. By examining the decomposition of rationals into sums of distinct unit fractions, a practice rooted in ancient Egyptian…
We introduce a class of sequences, defined by means of partial Bell polynomials, that contains a basis for the space of linear recurrence sequences with constant coefficients as well as other well-known sequences like Catalan and Motzkin.…
Multidimensional continued fractions generalize classical continued fractions with the aim of providing periodic representations of algebraic irrationalities by means of integer sequences. However, there does not exist any algorithm that…
We discuss continued fractions on real quadratic number fields of class number 1. If the field has the property of being 2-stage euclidean, a generalization of the euclidean algorithm can be used to compute these continued fractions.…
In this paper we study a continuity of the "values" of modular functions at the real quadratic numbers which are defined in terms of their cycle integrals along the associated closed geodesics. Our main theorem reveals a more finer…
Let $h$: $x\mapsto \frac{ax+b}{cx+d} $ be the nondegenerate M\"{o}bius transformation with integer entries. We get a bound of the continued fraction of $h(x)$ by the upper and lower bound of continued fraction of $x$, which extends a result…
A new algebraic object is introduced - recurrent fractions, which is an n-dimensional generalization of continued fractions. It is used to describe an algorithm for rational approximations of algebraic irrational numbers. Some…
We construct a class of quadratic irrationals having continued fractions of period $n\geq2$ with "small" partial quotients for which certain integer multiples have continued fractions of period $1$, $2$ or $4$ with "large" partial…
We give an extensive list of parametrized families of polynomial continued fractions of smallest possible degrees for $\pi^2$ and $\zeta(3)$, and mention similar results for other constants.
The main aim of this paper is to further develop the multiple-correction method that formulated in our previous works~\cite{CXY, Cao}. As its applications, we establish a kind of hybrid-type finite continued fraction approximations related…
We derive closed-form expressions for several new classes of Hurwitzian- and Tasoevian continued fractions, including $[0;\overline{p-1,1,u(a+2nb)-1,p-1,1,v(a+(2n+1)b)-1 }\,\,]_{n=0}^\infty$, $[0; \overline{c + d m^{n}}]_{n=1}^{\infty}$ and…
In practice many problems related to space/time fractional equations depend on fractional parameters. But these fractional parameters are not known a priori in modelling problems. Hence continuity of the solutions with respect to these…
It is well known that if $0.a_1a_2a_3\dots$ is the base-$b$ expansion of a number normal to base-$b$, then the numbers $0.a_ka_{m+k}a_{2m+k}\dots$ for $m\ge 2$, $k\ge 1$ are all normal to base-$b$ as well. In contrast, given a continued…
We present several continued fraction algorithms, each of which gives an eventually periodic expansion for every quadratic element of ${\mathbb Q}_p$ over ${\mathbb Q}$ and gives a finite expansion for every rational number. We also give,…