相关论文: Fibonacci q-gaussian sequences

The q-binomial coefficients are the polynomial cousins of the traditional binomial coefficients, and a number of identities for binomial coefficients can be translated into this polynomial setting. For instance, the familiar vanishing of…

The Fibonacci sequence is obtained as weighted sum along the rows in the Pascal triangle by choosing a periodic up-and-down pattern of weights from the set $\{-1,-\frac{1}{2},0, \frac{1}{2}, 1\}$. A graphical illustration of this identity…

In this paper we state some conjectures about q-Fibonacci polynomials which for q=1 reduce to well-known results about Fibonacci numbers and Fibonacci polynomials.

We derive a general recurrence relation for squares of Fibonacci-like numbers. Various properties are developed, including double binomial summation identites.

In this note, we present a simple summation formula for $k$-bonacci numbers. The derivation consists in obtaining the generating function of such numbers, and noting that its evaluation at a particular value yields a formula generalizing a…

In part 1 of this paper some linear weighted generalized Fibonacci number summation identities were derived using the fact that the Fibonacci number is the residue of a rational function. In this part, using the same method, some quadratic…

The Fibonacci number is the residue of a rational function, from which follows that Fibonacci number summation identities can be derived with the integral representation method, a method also used to derive combinatorial identities. A…

In this article we present a new recurrence formula for a finite sum involving the Fibonacci sequence. Furthermore, we state an algorithm to compute the sum of a power series related to Fibonacci series, without the use of term-by-term…

We investigate paths in Bernoulli's triangles, and derive several relations linking the partial sums of binomial coefficients to the Fibonacci numbers.

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…

We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.

We give a generalization of the Pascal triangle called the quasi s-Pascal triangle where the sum of the elements crossing the diagonal rays produce the s-bonacci sequence. For this, consider a lattice path in the plane whose step set is {L…

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 give a combinatorial proof of a formula giving the partial sums of the $k$-bonacci sequence as alternating sums of powers of two multiplied by binomial coefficients. As a corollary we obtain a formula for the $k$-bonacci numbers.

In this short note, we establish some identities containing sums of binomials with coefficients satisfying third order linear recursive relations. As a result and in particular, we obtain general forms of earlier identities involving…

We derive some q-analogs of Euler-Cassini-type identities and of recurrence formulas for powers of Fibonacci polynomials.

Let $(F_n)_{n\ge 1}$ be the Fibonacci sequence. Define $P(F_n): = (\sum_{i=1}^n F_i)_{n\ge 1}$; that is, the function $P$ gives the sequence of partial sums of $(F_n)$. In this paper, we first give an identity involving $P^k(F_n)$, which is…

From two q-summation formulas we deduce certain series expansion formulas involving the q-gamma function. With these formulas we can give q-analogues of series expansions for certain constants.

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…

Let (F_n^{(k)})_{n\geq -(k-2)} be the k-generalized Fibonacci sequence, defined as the linear recurrence sequence whose first k terms are \(0, 0, \ldots, 0, 1\), and whose subsequent terms are determined by the sum of the preceding k terms.…