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Register a new userArithmetic properties of generalized Fibonacci sequences
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Authors
Soohyun Park
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The generalized Fibonacci sequences are sequences ${f_n}$ which satisfy the recurrence $f_n(s, t) = sf_{n - 1}(s, t) + tf_{n - 2}(s, t)$ ($s, t in mathbb{Z}$) with initial conditions $f_0(s, t) = 0$ and $f_1(s, t) = 1$. In a recent paper, Amdeberhan, Chen, Moll, and Sagan considered some arithmetic properites of the generalized Fibonacci sequence. Specifically, they considered the behavior of analogues of the $p$-adic valuation and the Riemann zeta function. In this paper, we resolve some conjectures which they raised relating to these topics. We also consider the rank modulo $n$ in more depth and find an interpretation of the rank in terms of the order of an element in the multiplicative group of a finite field when $n$ is an odd prime. Finally, we study the distribution of the rank over different values of $s$ when $t = -1$ and suggest directions for further study involving the rank modulo prime powers of generalized Fibonacci sequences.
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We study the generalized random Fibonacci sequences defined by their first nonnegative terms and for $nge 1$, $F_{n+2} = lambda F_{n+1} pm F_{n}$ (linear case) and $widetilde F_{n+2} = |lambda widetilde F_{n+1} pm widetilde F_{n}|$ (non-linear case), where each $pm$ sign is independent and either $+$ with probability $p$ or $-$ with probability $1-p$ ($0<ple 1$). Our main result is that, when $lambda$ is of the form $lambda_k = 2cos (pi/k)$ for some integer $kge 3$, the exponential growth of $F_n$ for $0<ple 1$, and of $widetilde F_{n}$ for $1/k < ple 1$, is almost surely positive and given by $$ int_0^infty log x d u_{k, rho} (x), $$ where $rho$ is an explicit function of $p$ depending on the case we consider, taking values in $[0, 1]$, and $ u_{k, rho}$ is an explicit probability distribution on $RR_+$ defined inductively on generalized Stern-Brocot intervals. We also provide an integral formula for $0<ple 1$ in the easier case $lambdage 2$. Finally, we study the variations of the exponent as a function of $p$.
We study higher-dimensional interlacing Fibonacci sequences, generated via both Chebyshev type functions and $m$-dimensional recurrence relations. For each integer $m$, there exist both rational and integ
In this paper we study two functions $F(x)$ and $J(x)$, originally found by Herglotz in 1923 and later rediscovered and used by one of the authors in connection with the Kronecker limit formula for real quadratic fields. We discuss many interesting properties of these functions, including special values at rational or quadratic irrational arguments as rational linear combinations of dilogarithms and products of logarithms, functional equations coming from Hecke operators, and connections with Starks conjecture. We also discuss connections with 1-cocycles for the modular group $mathrm{PSL}(2,mathbb{Z})$.
The Fibonacci sequence is a sequence of numbers that has been studied for hundreds of years. In this paper, we introduce the new sequence S_{k,n} with initial conditions S_{k,0} = 2b and S_{k,1} = bk + a, which is generated by the recurrence relation S_{k,n} = kS_{k,n-1} +S{k,n-2} for n >= 2, where a, b, k are real numbers. Using the sequence S_{k,n}, we introduce and prove some special identities. Also, we deal with the circulant and skew circulant matrices for the sequence S_{k,n}.
Carmichael showed for sufficiently large $L$, that $F_L$ has at least one prime divisor that is $pm 1({rm mod}, L)$. For a given $F_L$, we will show that a product of distinct odd prime divisors with that congruence condition is a Fibonacci pseudoprime. Such pseudoprimes can be used in an attempt, here unsuccessful, to find an example of a Baillie-PSW pseudoprime, i.e. an odd Fibonacci pseudoprime that is congruent to $pm 2({rm mod}, 5)$ and is also a base-2 pseudoprime.
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