Related papers: Squares of Tribonacci numbers
In this paper, we obtain a generalization of an identity due to Carlitz on Bernoulli polynomials. Then we use this generalized formula to derive two symmetric identities which reduce to some known identities on Bernoulli polynomials and…
We present other proofs, generalizations and analogues of the identities concerning multiple Dirichlet series by Tahmi and Derbal (2022). As applications, we obtain asymptotic formulas with remainder terms for certain related sums.
The summation formula within pascalian triangle resulting in the fibonacci sequence is extended to the $q$-binomial coefficients $q$-gaussian triangles.
We state and prove a general summation identity. The identity is then applied to derive various summation formulas involving the generalized harmonic numbers and related quantities. Interesting results, mostly new, are obtained for both…
We prove a curious identity for the Bernoulli numbers.
Many kinds of convolution identities have been considered about several numbers, including Bernoulli, Euler, Genocchi, Cauchy, Stirling, and Fibonacci numbers. The well-known basic result about Bernoulli numbers is due to Euler. The…
Fibonacci sequence, generated by summing the preceding two terms, is a classical sequence renowned for its elegant properties. In this paper, leveraging properties of generalized Fibonacci sequences and formulas for consecutive sums of…
In this study, we apply the binomial transforms to Tribonacci and Tribonacci-Lucas sequences. Also, the Binet formulas, summations, generating functions of these transforms are found using recurrence relations. Finally, we illustrate the…
The equation commonly known as Sury's identity is a deceptively simple summation formula that connects the Lucas numbers, Fibonacci numbers, and powers of two. Many authors have given extensions and generalizations over the years; in this…
Sury's 2014 proof of an identity for Fibonacci and Lucas numbers (Identity 236 of Benjamin and Quinn's 2003 book: {\em Proofs that count: The art of combinatorial proof}) has excited a lot of comment. We give an alternate, telescoping,…
We extend a result of I. J. Good and prove more symmetry properties of sums involving generalized Fibonacci numbers
In this paper, we prove identities for members of the k-generalized Fibonacci sequence with negative indices and we apply these identities to deduce an exact formula for its zero-multiplicity.
We prove some separation results for the roots of the generalized Fibonacci polynomials and their absolute values
We give alternative proof of Melham's and Howard's identities on generalised Fibonacci numbers
We present a, hopefully, elementary mathematical treatment of the computational aspects of congruent numbers, such that an amateur could understand the problem and perform their own calculations.
Motivated by the works of Andrews-Merca and Guo-Zeng, we establish some truncated identities of Gauss by using some summation formulas from the works of Zhi-Guo Liu. These give three new expansions for partial sums of Gauss' triangular…
We derive some Fibonacci and Lucas identities which contain inverse binomial coefficients. Extension of the results to the general Horadam sequence is possible, in some cases.
We prove several congruences for trinomial coefficients.
We provide elementary proof of several congruences involving single sum and multisums of binomial coefficients.
In this note, we obtain some identities for the generalized Fibonacci polynomial by using the Q(x) matrix. These identities including the Cassini identity and Honsberger formula can be applied to some polynomial sequences, such as Fibonacci…