We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler’s Telescoping Lemma, we give alternate proofs of all the key summation theorems for terminating Hypergeometric Series and Basic Hypergeometric Series, including the terminating Binomial Theorem, the Ch...

We obtain some new identities for the generalized Fibonacci polynomial by a approach, namely, Q(x) matrix.
 These including Cassini type identity and Honsberger formula can be applied to polynomial
 sequences such as polynomials, Lucas Pell Pell-Lucas polynomials so on, which
 generalize previous results in references.

In this note we consider two classes of polynomials Un and Vn. These polynomials are special cases of Un,m and Vn,m (see [2]), respectively. Also, Un and Vn are generalized Fibonacci and Lucas polynomials. In fact, in this paper we study the polynomials Un,3 and Vn,3, together with their k−derivative sequences U (k) n and V (k) n . Some interesting identities are proved in the paper, for Un, Vn...

The Fibonacci numbers and the Pell numbers can be interpreted as the number of tilings of a (1 × n)-board by colored squares and dominoes. We explore the tilings of (2 × n)-boards by colored squares and dominoes. We develop a recurrence relation and prove several combinatorial identities in the style of recent work by Benjamin and Quinn. We also give a bijection between these (2 × n)-tilings an...

0 [10, 7]. 1 [10, 7]. 13 [4]. 53 [10]. 8 [10]. Addendum [10]. allied [2]. Amer [10]. conformal [3]. converse [9]. Determinants [10, 7, 1]. dimensional [1]. diophantine [5]. elements [10, 7]. equation [5]. Fermat [9]. Fibonacci [2]. Higher [1]. identities [2]. II [9]. inequality [6]. isoperimetric [6]. J [10]. late [10]. linear [5]. mapping [3]. Math [10]. Monthly [10]. MR [10]. Never [10]. note...

We study the equation Fb = Fx(1)+F−x(3)+F−x(4)+ · · ·+F−x(m) with m ≥ 3, 0 < x(1) < b, and 0 < x(3) < x(4) < · · · < x(m). This equation naturally arises in the generalization of several problems that have appeared in The Fibonacci Quarterly in the problem sections. This equation also has intrinsic interest in its own right. The main theorem the Accident theorem–states, that under very mild con...

We obtain some new formulas for the Fibonacci and Lucas p-numbers, by using the symmetric infinite matrix method. We also give some results for the Fibonacci and Lucas p-numbers by means of the binomial inverse pairing.

5 2 is the golden mean. Surprisingly, no one appears to have investigated the distribution of the number of summands; our main result is that this converges to a Gaussian as n → ∞. Moreover, such a result holds not just for the Fibonacci numbers but many other problems, such as linear recurrence relation with non-negative integer coefficients (which is a generalization of base B expansions of n...

for n ≥ 2, it can be shown that fn counts the number of ways to tile a 1 × n board with squares and dominoes. This interpretation of the Fibonacci numbers admits clever counting proofs of many Fibonacci identities. When we consider more complex combinatorial objects, we find that simply counting the number of tilings is not quite enough, and that we must consider instead weighted tilings. For e...

We establish some identities relating two sequences that are, as explained, related to the Tribonacci sequence. One of these sequences bears the same resemblance to the Tribonacci sequence as the Lucas sequence does to the Fibonacci sequence. Defining a matrix that we call Tribomatrix, which extends the Fibonacci matrix, we see that the other sequence is related to the sum of the determinants o...