arXiv subjects
Publications and source records attributed to Taras Goy.
We evaluate some new three parameter families of finite reciprocal sums involving Horadam numbers. We will also be able to state the results for the infinite sums. Some Fibonacci and Lucas sums will be presented as examples.
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.
We establish some new combinatorial identities involving Euler polynomials and balancing (Lucas-balancing) polynomials. The derivations use elementary techniques and are based on functional equations for the respective generating functions. From these polynomial relations, we deduce interesting identities with Fibonacci and Lucas numbers, and Euler numbers. The results must be regarded as companion results to some Fibonacci-Bernoulli identities, which we derived in our previous paper.
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).
Using generating functions, we derive many identities involving balancing and Lucas-balancing polynomials. By relating these polynomials to Chebyshev polynomials of the first and second kind, and Fibonacci and Lucas numbers, we offer some presumably new combinatorial identities involving these famous sequences.
The purpose of this article is to present closed forms for various types of infinite series involving Fibonacci (Lucas) numbers and the Riemann zeta function at integer arguments.
In this paper, we evaluate determinants of some families of Toeplitz-Hessenberg matrices having tribonacci number entries. These determinant formulas may also be expressed equivalently as identities that involve sums of products of multinomial coefficients and tribonacci numbers. In particular, we establish a connection between the tribonacci and the Fibonacci and Padovan sequences via Toeplitz-Hessenberg determinants. We then obtain, by combinatorial arguments, extensions of our determinant formulas in terms of generalized tribonacci sequences satisfying an r-th order recurrence of a more general form with the appropriate initial conditions, where r>2 is arbitrary.