An essay on continued fractions

@article{Euler2005AnEO,
  title={An essay on continued fractions},
  author={Leonhard Euler and Myra F. Wyman and Bostwick F. Wyman},
  journal={Mathematical systems theory},
  year={2005},
  volume={18},
  pages={295-328}
}
English translation of the paper: "De Fractionibus Continuis Dissertatio" by Leonhard Euler. 
Quadratic irrational integers with partly prescribed continued fraction expansion
We generalise remarks of Euler and of Perron by explaining how to detail all quadratic irrational integers for which the symmetric part of the period of their continued fraction expansion commences
Partition-theoretic Abelian theorems
Using a theorem of Frobenius filtered through partition generating function techniques, we prove partition-theoretic and $q$-series Abelian theorems, yielding analogues of Abel's convergence theorem
Some formulas for determinants of tridiagonal matrices in terms of finite generalized continued fractions
In the paper, by virtue of induction and properties of determinants, the authors discover explicit and recurrent formulas of evaluations for determinants of general tridiagonal matrices in terms of
Non-commutative double-sided continued fractions
We study double-sided continued fractions whose coefficients are non-commuting symbols. We work within the formal approach of the Mal'cev-Neumann series and free division rings. We start with
On the Irrationality and Transcendence of Rational Powers of e
A number that can’t be expressed as the ratio of two integers is called an irrational number. Euler and Lambert were the first mathematicians to prove the irrationality and transcendence of e. Since
On a Desert Island with Unit Sticks, Continued Fractions and Lagrange
GLY 4866, Computational Geology, provides an opportunity, welcomed by our faculty, to teach quantitative literacy to geology majors at USF. The course continues to evolve although the second author
How Euler Did It
The picture that caught my eye was the squarish-looking spiral below. It was part of the Summarium of [E275], "Notes on a certain passage of Descartes for looking at the quadrature of the circle."
Eulerian series, zeta functions and the arithmetic of partitions
In this Ph.D. dissertation (2018, Emory University) we prove theorems at the intersection of the additive and multiplicative branches of number theory, bringing together ideas from partition theory,
...
...

References

SHOWING 1-10 OF 31 REFERENCES
Continued Fractions
The study of continued fractions is an ancient part of elementary Number Theory. It was studied by Leonhard Euler in the 18-th century. Actually, a remarkable paper from him was translated from Latin
A Treatise on the Theory of Bessel Functions
THE memoir in which Bessel, the astronomer, examined in detail the functions which now bear his name was published in 1824, and was the outcome of his earlier researches concerning the expression of
Geometry of the Algebraic Riccati Equation, Part II
We prove that the set of real symmetric solutions of the algebraic Riccati equation is isomorphic to the algebraic variety of invariant subspaces of a related $n \times n$ matrix. By characterizing
On the partial realization problem
Geometry of the Algebraic Riccati Equation, Part I
We synthesize and generalize the two principal methods for classifying the set of real symmetric solutions of the algebraic Riccati equation (ARE) to obtain a result which combines the advantages of
...
...