Hurwitzian Continued Fractions Containing a Repeated Constant and An Arithmetic Progression
نویسندگان
چکیده
منابع مشابه
Hurwitzian Continued Fractions Containing a Repeated Constant and An Arithmetic Progression
We prove an explicit formula for infinitely many convergents of Hurwitzian continued fractions that repeat several copies of the same constant and elements of one arithmetic progression, in a quasi-periodic fashion. The proof involves combinatorics and formal Laurent series. Using very little analysis we can express their limits in terms of (modified) Bessel functions and Fibonacci polynomials....
متن کاملSome New Families of Tasoevian- and Hurwitzian Continued Fractions
We derive closed-form expressions for several new classes of Hurwitzianand Tasoevian continued fractions, including [0; p− 1, 1, u(a + 2nb)− 1, p− 1, 1, v(a + (2n + 1)b)− 1 ]n=0, [0; c + dmn]n=1 and [0; eun, fvn] ∞ n=1. One of the constructions used to produce some of these continued fractions can be iterated to produce both Hurwitzianand Tasoevian continued fractions of arbitrary long quasi-pe...
متن کاملIrrationality Measures for Continued Fractions with Arithmetic Functions
Let f(n) or the base-2 logarithm of f(n) be either d(n) (the divisor function), σ(n) (the divisor-sum function), φ(n) (the Euler totient function), ω(n) (the number of distinct prime factors of n) or Ω(n) (the total number of prime factors of n). We present good lower bounds for ∣ ∣M N − α ∣ ∣ in terms of N , where α = [0; f(1), f(2), . . .].
متن کاملPeriodic Continued Fractions And
We investigate when an algebraic function of the form φ(λ) = −B(λ)+ √ R(λ) A(λ) , where R(λ) is a polynomial of odd degree N = 2g + 1 with coefficients in C, can be written as a periodic α-fraction of the form
متن کاملContinued Fractions and Gaps
Given a continued fraction, we construct a certain function that is discontinuous at every rational number p/q. We call this discontinuity the “gap”. We then try to characterize the gap sizes, and find, to the first order, the size is 1/q2, and that, for higher orders, the gap appears to be perfectly ’randomly’ distributed, in that it is Cauchy-dense on the unit square, and thus, this function ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: SIAM Journal on Discrete Mathematics
سال: 2014
ISSN: 0895-4801,1095-7146
DOI: 10.1137/130926092