Related papers: Dilogarithm identities after Bridgeman

By expressing Fibonacci and Lucas numbers in terms of the powers of the golden ratio, $\alpha=(1+\sqrt 5)/2$ and its inverse, $\beta=-1/\alpha=(1-\sqrt 5)/2$, a multitude of Fibonacci and Lucas identities have been developed in the…

Angular parts of certain solvable models are studied. We find that an extension of this class may be based on suitable trigonometric identities. The new exactly solvable Hamiltonians are shown to describe interesting two- and three-particle…

We use $q$-binomial theorem to prove three new polynomial identities involving $q$-trinomial coefficients. We then use summation formulas for the $q$-trinomial coefficients to convert our identities into another set of three polynomial…

Using a straightforward elementary approach, we derive numerous infinite arctangent summation formulas involving Fibonacci and Lucas numbers. While most of the results obtained are new, a couple of celebrated results appear as particular…

We develop closed form expressions for various finite binomial Fibonacci and Lucas sums depending on the modulo 5 nature of the upper summation limit. Our expressions are inferred from some trigonometric identities.

We combinatorially prove Tetranacci, Tetranacci-Fibonacci, and additional identities using only squares and dominoes on a hexagonal double-strip. Some of these are new proofs of old identities, and others we believe have never been seen…

We consider $m$-th order linear recurrences that can be thought of as generalizations of the Lucas sequence. We exploit some interplay with matrices that again can be considered generalizations of the Fibonacci matrix. We introduce the…

Using elementary methods, we establish old and new relations between binomial coefficients, Fibonacci numbers, Lucas numbers, and more.

We propound the thesis that there is a limitation to the number of possible structures which are axiomatically endowed with identities involving operations. In the case of algebras with a binary operation satisfying a formally reducible (to…

Motivated by the optimal continued fractions studied independently by Selenius and Bosma, we define and introduce algorithms producing superoptimal continued fraction expansions of irrationals. The convergents of these expansions…

We derive closed form expressions for finite and infinite Fibonacci-Lucas sums having products of Fibonacci or Lucas numbers in the denominator of the summand. Our results generalize and extend those obtained by pioneer Brother Alfred…

In this note, infinite series involving Fibonacci and Lucas numbers are derived by employing formulae similar to that which Roger Ap\'ery utilized in his seminal paper proving the irrationality of $\zeta(3)$.

The $d$-Fibonacci digraphs $F(d,k)$, introduced here, have the number of vertices following generalized Fibonacci-like sequences. They can be defined both as digraphs on alphabets and as iterated line digraphs. Here we study some of their…

We evaluate various binomial sums involving the powers of Fibonacci and Lucas numbers.

We derive a new Fibonacci identity. This single identity subsumes important known identities such as those of Catalan, Ruggles, Halton and others, as well as standard general identities found in the books by Vajda, Koshy and others. We also…

Most well-known multidimensional continued fractions, including the M\"{o}nkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by…

Special kinds of continued fractions have been proved to converge to transcendental real numbers by means of the celebrated Subspace Theorem. In this paper we study the analogous $p$--adic problem. More specifically, we deal with Browkin…

We give combinatorial descriptions of the terms occurring in continuants of general continued fractions that diverge to three limits. Equating these with the usual combinatorial descriptions due to Euler, Sylvester, and Minding induces…

We consider the (A_n,A_1) Y-system arising in the Thermodynamic Bethe Ansatz. We prove the periodicity of solutions of this Y-system conjectured by Al.B. Zamolodchikov, and the dilogarithm identities conjectured by F. Gliozzi and R. Tateo.

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…