I present and discuss an extremely simple algorithm for expanding a formal power series as a continued fraction. This algorithm, which goes back to Euler (1746) and Viscovatov (1805), deserves to be ...better known. I also discuss the connection of this algorithm with the work of Gauss (1812), Stieltjes (1889), Rogers (1907) and Ramanujan, and a combinatorial interpretation based on the work of Flajolet (1980).
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We give a new class of multidimensional p-adic continued fraction algorithms. We propose an algorithm in the class for which we can expect that the multidimensional p-adic version of Lagrange's ...Theorem will hold.
By using Sulanke-Xin continued fractions method, Xin proposed a recursion system to solve the Somos 4 Hankel determinant conjecture. We find Xin's recursion system indeed give a sufficient condition ...for (α,β) Somos 4 sequences. This allows us to prove 4 conjectures of Barry on (α,β) Somos 4 sequences in a unified way.
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Define a map ν on the rationals by
and extend it over the irrationals by setting
. We prove that, under the iterates of this map, all rational points return to themselves with period
for some
; some ...quadratic irrational points eventually get to rational points; other irrational points do not eventually return to themselves but return periodically to their
-neighborhoods, however small
is.
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The Euler numbers occur in the Taylor expansion of \tan (x)+\sec (x). Since Stieltjes, continued fractions and Hankel determinants of the even Euler numbers, on the one hand, of the odd Euler ...numbers, on the other hand, have been widely studied separately. However, no Hankel determinants of the (mixed) Euler numbers have been obtained. The reason for that is that some Hankel determinants of the Euler numbers are null. This implies that the Jacobi continued fraction of the Euler numbers does not exist. In the present paper, this obstacle is bypassed by using the Hankel continued fraction, instead of the J-fraction. Consequently, an explicit formula for the Hankel determinants of the Euler numbers is being derived, as well as a full list of Hankel continued fractions and Hankel determinants involving Euler numbers. Finally, a new q-analog of the Euler numbers E_n(q) based on our continued fraction is proposed. We obtain an explicit formula for E_n(-1) and prove a conjecture by R. J. Mathar on these numbers.
Continued fraction and quasi-reciprocal continued fraction expansions of the generating function of Bernoulli numbers have been obtained. The convergence and uniform convergence of continued fraction ...expansions have been proved. Representations of the generating function of Bernoulli polynomials in the form of the product of three continued fractions, as well as the product of three quasi-reciprocal continued fractions, have been found.
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For any equation complex roots occur in pairs. For finding a root of the equation in continued fractions algorithms are available. But to get complex roots in continued fraction no such procedure so ...far we have. Here, in this paper if i has a representation in continued fractions we try to find its conjugate i in terms of continued fractions. This will be useful in finding complex roots of a quadratic equations in continued fraction.
For β>1 and x∈0,1), let kn(x) represent the exact number of digits in the Oppenheim continued fraction expansions of x given by the first n digits in the β-expansion of x. In this paper, we obtain ...the relations between kn(x) and n as limsupn→∞lnn=0, where ln=sup{k≥0:ϵn+j⁎(1)=0,1≤j≤k} and 1=∑n=1∞ϵn⁎(1)βn is the infinite β-expansion of 1.
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The paper deals with the problem of approximation of functions of several variables by branched continued fractions. We study the correspondence between formal multiple power series and the so-called ..."multidimensional $S$-fraction with independent variables". As a result, the necessary and sufficient conditions for the expansion of the formal multiple power series into the corresponding multidimensional $S$-fraction with independent variables have been established. Several numerical experiments show the efficiency, power and feasibility of using the branched continued fractions in order to numerically approximate certain functions of several variables from their formal multiple power series.
The Sylvester–Kac matrix (also known as the Clement matrix) and its generalizations are of interest to researchers in diverse fields. A new parameterization of the matrix has recently been presented ...with closed forms for the eigenvalues and Oste and Van der Jeugt (2017) have proposed their family of matrices as a source of test problems for numerical eigensolvers. In this article we extend their generalization by adding a further two parameters to the matrix definition and, for this new extension, we obtain closed forms for the eigenvalues, determinant and, the left and right eigenvectors. We show that, for certain values of the free parameters and relatively low order, these matrices may become very ill-conditioned w.r.t. both eigenvalues and inversion. In light of this we re-assess the numerical results presented by Oste and Van der Jeugt (2017) and propose a possible new testing role for parameterized Clement matrices.
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