Related papers: Polynomial Continued Fractions
This paper investigates the length of the repeating decimal part when a fraction is expressed in decimal form. First, it provides a detailed explanation of how to calculate the length of the repeating decimal when the denominator of the…
The generating functions of some sequences of Catalan numbers and Narayana polynomials have simple expansions as continued fractions of Jacobi type. We give an overview of these facts and prove analogous results for q-Narayana polynomials…
Continued fraction expansions provide a well-established bridge between algebraic properties of numbers and combinatorics on words. In this article, we investigate the algebraicity of $p$-adic numbers whose continued fractions arise from…
We highlight some facts about continued fractions of real cubic irrationalities. This may be thought as a small section in a textbook on continued fractions.
We describe various properties of continued fraction expansions of complex numbers in terms of Gaussian integers. Numerous distinct such expansions are possible for a complex number. They can be arrived at through various algorithms, as…
We explain in detail how to accelerate continued fractions (for constants as well as for functions) using the method used by R.~Ap\'ery in his proof of the irrationality of $\zeta(3)$. We show in particular that this can be applied to a…
Ramanujan in his notebook recorded two modular equations involving multiplier with moduli of degrees (1,7) and (1,23). In this paper, we find some new Ramanujan's modular equations involving multiplier with moduli of degrees (3,5) and…
These are the notes of my lectures at the 1996 European Congress of Mathematicians. {} Polynomials appear in mathematics frequently, and we all know from experience that low degree polynomials are easier to deal with than high degree ones.…
In this paper, we study the complexity of p-adic continued fractions of a rational number, which is the p-adic analogue of the theorem of Lame. We calculate the length of Browkin expansion, and the length of Schneider expansion. Also, some…
We find Stieltjes-type and Jacobi-type continued fractions for some "master polynomials" that enumerate permutations, set partitions or perfect matchings with a large (sometimes infinite) number of simultaneous statistics. Our results…
Quadratic irrationals posses a periodic continued fraction expansion. Much less is known about cubic irrationals. We do not even know if the partial quotients are bounded, even though extensive computations suggest they might follow…
In this paper, we introduce a new concept namely degree polynomial for vertices of a simple graph. This notion leads to a concept namely degree polynomial sequence which is stronger than the concept of degree sequence. After obtaining the…
We investigate which polynomials can possibly occur as factors in the denominators of rational solutions of a given partial linear difference equation (PLDE). Two kinds of polynomials are to be distinguished, we call them /periodic/ and…
We study Ramanujan's cubic continued fraction and explicit evaluations of theta-functions
Our overall goal is to unify and extend some results in the literature related to the approximation of generating functions of finite and infinite sequences over a field by rational functions. In our approach, numerators play a significant…
In this paper, we obtain several new factorization results for certain classes of polynomials having integer coefficients. In doing so, we use the information about prime factorization of the value taken up by such polynomials and their…
In the present work, we investigate real numbers whose sequence of partial quotients enjoys some combinatorial properties involving the notion of palindrome. We provide three new transendence criteria, that apply to a broad class of…
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 real vector space combined with an inverse for vectors is sufficient to define a vector continued fraction whose parameters consist of vector shifts and changes of scale. The choice of sign for different components of the vector inverse…
We use the method of generating functions to find the limit of a $q$-continued fraction, with 4 parameters, as a ratio of certain $q$-series. We then use this result to give new proofs of several known continued fraction identities,…