Additional close links between balancing and Lucas-balancing polynomials

Positive Integer Solutions of Certain Diophantine Equations Involving Lucas-Balancing numbers

In this article the Diophantine equations of the form $x^{2}-8C_{n}xy+ 16y^{2}=\pm2^{r}$, $x^{4}-8C_{n}x^{2}y+ 16y^{2}=\pm2^{r}$ and $x^{2}-8C_{n}xy^{2}+ 16y^{4}=\pm2^{r}$ are taken into consideration for the investigations of existence of any positive integer solutions. Furthermore, the solutions are manifested in terms of the balancing, Pell and Lucas-Pell numbers.

More Fibonacci-Bernoulli relations with and without balancing polynomials

We continue our study on relationships between Bernoulli polynomials and balancing (Lucas-balancing) polynomials. From these polynomial relations, we deduce new combinatorial identities with Fibonacci (Lucas) and Bernoulli numbers. Moreover, we prove a special identity involving Bernoulli polynomials and Fibonacci numbers in arithmetic progression. Special cases and some corollaries will highlight interesting aspects of our findings. Our results complement and generalize these of Frontczak (2019).

Balancing polynomials, Fibonacci numbers and some new series for $π$

We evaluate some types of infinite series with balancing and Lucas-balancing polynomials in closed form. These evaluations will lead to some new curious series for $π$ involving Fibonacci and Lucas numbers. Our findings complement those of Castellanos from 1986 and 1989.

Appearance of Balancing and related number sequences in steady state probabilities of some Markov chains

Balancing and Lucas-balancing numbers are solutions of a Diophantine equation and satisfy a second order homogeneous recurrence relation. Interestingly, these numbers can be seen as numerators and denominators in the steady state probabilities of a class of transition probability matrices of Markov chains. An identity relating the balancing numbers and the silver ratio can be obtained as a byproduct.

Pell和相伴Pell辫序列作为$k$个连续Pell、平衡及相关数之和的最大公约数

Pell and associated Pell braid sequences as GCDs of sums of $k$ consecutive Pell, balancing, and related numbers

研究了六个著名序列（Pell、相伴Pell、平衡及相关数）中$k$个连续项之和的最大公约数，给出了依赖$k$的闭式，并发现其可表示为Pell和相伴Pell辫序列，同时探讨了平方项之和的GCD问题。

对称函数的新组合恒等式

The new combinatorial identities of symmetric functions

本文利用指数生成函数等初等方法，研究了与对称函数相关的二项式卷积及Bernoulli、Euler和Genocchi数与多项式的卷积和，推导并建立了二元多项式及相关特殊多项式的多个新组合恒等式。

Chords of an ellipse, Lucas polynomials, and cubic equations

A beautiful theorem of Thomas Price links the Fibonacci numbers and the Lucas polynomials to the plane geometry of an ellipse, generalizing a classic problem about circles. We give a brief history of the circle problem, an account of Price's ellipse proof, and a reorganized proof, with some new ideas, designed to situate the result within a dense web of connections to classical mathematics. It is inspired by Cardano's solution of the cubic equation and Newton's theorem on power sums, and yields an interpretation of generalized Lucas polynomials in terms of the theory of symmetric polynomials. …

Chebyshev-Fibonacci polynomial relations using generating functions

The main object of the paper is to reveal connections between Chebyshev polynomials of the first and second kinds and Fibonacci polynomials introduced by Catalan. This is achieved by relating the respective (ordinary and exponential) generating functions to each other. As a consequence, we also establish new combinatorial identities for balancing polynomials and Fibonacci (Lucas) numbers.

$k$-Fibonacci 与 $k$-Lucas 数的 Hölder 不等式

$k$-Fibonacci and $k$-Lucas numbers with the Hölder inequality

本文将Hölder与Cauchy不等式及其逆形式的幂和不等式链统一推广到k-Fibonacci与k-Lucas数，利用广义恒等式及交叉恒等式建立连接两族的Cauchy-Schwarz型不等式，并在k=1时得到普通Fibonacci与Lucas数间的显式初等不等式及相应细化。