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 give a direct and intuitive proof (via sliding some columns up and down) of the following interesting fact: if we write out the Chebyshev polynomials in a chart and take the sums of coefficients along certain diagonals, we obtain the Fibonaccis, the Tribonaccis, the Tetranaccis, and so on.
Analytic summation of series involving higher-order derivatives of Chebyshev polynomials of the second kind and their applications to convolved linear recurrent sequences
It is shown that some q-analogues of the Fibonacci and Lucas polynomials lead to q-analogues of the Chebyshev polynomials which retain most of their elementary properties.
The moments of the Lucas polynomials and of the Chebyshev polynomials of the first kind are (multiples of) central binomial coefficients and the moments of the Fibonacci polynomials and of the Chebyshev polynomials of the second kind are Catalan numbers. In this survey paper we present some generalizations of these results together with various q-analogues.
In this paper we present two families of Fibonacci-Lucas identities, with the Sury's identity being the best known representative of one of the families. While these results can be proved by means of the basic identity relating Fibonacci and Lucas sequences we also provide a bijective proof. Both families are then treated by generating functions.
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 number of weighted generalized Fibonacci number summation identities are derived this way. From these identities some infinite series, generating functions and convolution identities are obtained. In addition, some weighted generalized Fibonacci number summation identities with binomial coefficients are derived. Many examples of both types of summation identities…
The generating function for the sequence A166861 in the OEIS (Euler transform of Fibonacci numbers) is Product_{k>0} 1/(1-x^k)^F(k), where F(k) are the Fibonacci numbers. This paper analyzes the more general generating function U(x) = Product_{k>0} 1/(1-x^k)^F(k+z), where z is a nonnegative integer, which provides asymptotics for the sequences A166861 (z=0), A200544 (z=1) and A260787 (z=2) in the OEIS.
We prove that $|Av_n(231,312,1432)|$, $|Av_n(312,321,1342)|$ $|Av_n(231,312,4321,21543)|$, and $ |Av_n(321,231,4123,21534)|$, are all equal to $F_{n+1} - 1$ where $F_n$ is the $n$-th Fibonacci number using the convention $F_0 = F_1 = 1$ and $Av_n(S)$ is the set of all permutations of length $n$ that avoid all of the patterns in the set $S$. To do this, we characterize the structures of the permutations in these sets in terms of Fibonacci permutations. Then, we further quantify the structures using statistics such as inversion number and a statistic that measures the length of Fibonacci subsequ…
It is demonstrated how an explicit expression of the (partial) sum of Tetranacci numbers can be found and proved using generating functions and the Hadamard product. We also provide a Binet-type formula for generalized Fibonacci numbers, by explicitly factoring the denominator of their generating functions.
We continue our study on relationships between Fibonacci (Lucas) numbers and Bernoulli numbers and polynomials. The derivations of our results are based on functional equations for the respective generating functions, which in our case are combinations of hyperbolic functions. Special cases and some corollaries will highlight interesting aspects of our findings.
We study a family of finite--dimensional representations of the hyperspecial parabolic subalgebra of the twisted affine Lie algebra of type $\tt A_2^{(2)}$. We prove that these modules admit a decreasing filtration whose sections are isomorphic to stable Demazure modules in an integrable highest weight module of sufficiently large level. In particular, we show that any stable level $m'$ Demazure module admits a filtration by level $m$ Demazure modules for all $m\ge m'$. We define the graded and weighted generating functions which encode the multiplicity of a given Demazure module and establish…
In this paper we first generalize the numerical recurrence relation given by Hosoya to polynomials. Using this generalization we construct a Hosoya-like triangle for polynomials, where its entries are products of generalized Fibonacci polynomials (GFP). Examples of GFP are: Fibonacci polynomials, Chebyshev polynomials, Morgan-Voyce polynomials, Lucas polynomials, Pell polynomials, Fermat polynomials, Jacobsthal polynomials, Vieta polynomials and other familiar sequences of polynomials. For every choice of a GFP we obtain a triangular array of polynomials. In this paper we extend the star of Da…
We prove various formulas which express exponential generating functions counting permutations by the peak number, valley number, double ascent number, and double descent number statistics in terms of the exponential generating function for Chebyshev polynomials, as well as cyclic analogues of these formulas for derangements. We give several applications of these results, including formulas for the $(-1)$-evaluation of some of these distributions. Our proofs are combinatorial and involve the use of monomino-domino tilings, the modified Foata-Strehl action (a.k.a. valley-hopping), and a cyclic …
A filter lattice is a distributive lattice formed by all filters of a poset in the anti-inclusion order. We study the combinatorial properties of the Hasse diagrams of filter lattices of certain posets, so called Fibonacci-like cubes, in this paper. Several enumerative polynomials, e.g.\ rank generating function, cube polynomials and degree sequence polynomials are obtained. Some of these results relate to Fibonacci sequence and Padovan sequence.
Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of partitions into odd parts and the generating function for the number of odd divisors.
A \Def{composition} of a positive integer $n$ is a $k$-tuple $(ł_1, ł_2, \dots, ł_k) \in \Z_{> 0}^k$ such that $n = ł_1 + ł_2 + \dots + ł_k$. Our goal is to enumerate those compositions whose parts $ł_1, ł_2, \dots, ł_k$ avoid a fixed arithmetic sequence. When this sequence is given by the even integers (i.e., all parts of the compositions must be odd), it is well known that the number of compositions is given by the Fibonacci sequence. A much more recent theorem says that when the parts are required to avoid all multiples of a given integer $k$, the resulting compositions are counted by a seq…
We show that the only orthogonal polynomials with a generating function of the form $F(x z - αz^2)$ are the ultraspherical, Hermite, and Chebyshev polynomials of the first kind. For special $F$ for which this is the case, we then finish the classification of orthogonal polynomials with more general generating functions $F(x w(z) - R(z))$.
The paper investigates relationship between algebraic expressions and graphs. We consider a digraph called a Fibonacci graph which gives a generic example of non-series-parallel graphs. Our intention in this paper is to simplify the expressions of Fibonacci graphs and eventually find their shortest representations. With that end in view, we describe the number of methods for generating Fibonacci graph expressions and carry out their comparative analysis.
By using definition of Golden derivative, corresponding Golden exponential function and Fibonomial coefficients, we introduce generating functions for Bernoulli-Fibonacci polynomials and related numbers. Properties of these polynomials and numbers are studied in parallel with usual Bernoulli counterparts. Fibonacci numbers and Golden ratio are intrinsically involved in formulas obtained.
In chemistry, Cyvin-Gutman enumerates Kekulé numbers for certain benzenoids and record it as $A050446$ on OEIS. This number is exactly the two variable array $T(n,m)$ defined by the recursion $T(n, m) = T(n, m-1) + \sum^{\lfloor\frac{n-1}{2}\rfloor}_{k=0} T(2k, m-1)T(n-1-2k, m)$, where $T(n,0)=T(0,m)=1$ for all nonnegative integers $m,n$. Interestingly, this number also appeared in the context of weighted graphs, graph polytopes, magic labellings, and unit primitive matrices, studied by different authors. Several interesting conjectures were made on the OEIS. These conjectures are related to b…
Recently, a new class of words, denoted by L_n, was shown to be in bijection with a subset of the Dyck paths of length 2n having cardinality given by the (n-1)-st Catalan number. Here, we consider statistics on L_n recording the number of occurrences of a letter i. In the cases i = 0 and i = 1, we are able to determine explicit expressions for the number of members of L_n containing a given number of zeros or ones, which generalizes the prior result. To do so, we make use of recurrences to derive a functional equation satisfied by the generating function, which we solve by a new method employi…
We show that the only polynomial sets with a generating function of the form F (xt -- R(t)) and satisfying a three-term recursion relation are the monomial set and the rescaled ultraspherical, Hermite, and Chebyshev polynomials of the first kind.
The paper investigates relationship between algebraic expressions and graphs. We consider a digraph called a Fibonacci graph which gives a generic example of non-series-parallel graphs. Our intention in this paper is to simplify the expressions of Fibonacci graphs and eventually find their shortest representations. With that end in view, we describe the optimal decomposition method for generating Fibonacci graph expressions that is conjectured to provide these representations. Proof (or disproof) of this conjecture is presented as an open problem.
We give multiple proofs of two formulas concerning the enumeration of permutations avoiding a monotone consecutive pattern with a certain value for the inverse peak number or inverse left peak number statistic. The enumeration in both cases is given by a sequence related to Fibonacci numbers. We also show that there is exactly one permutation whose inverse peak number is zero among all permutations with any fixed descent composition, and we give a few elementary consequences of this fact. Our proofs involve generating functions, symmetric functions, regular expressions, and monomino-domino til…
We study the level $m$--Demazure flag of a level $\ell$--Demazure module for $\frak{sl}_2[t]$. We define the generating series $A_n^{\ell \rightarrow m}(x,q)$ which encodes the $q$--multiplicity of the level $m$ Demazure module of weight $n$. We establish two recursive formulae for these functions. We show that the specialization to $q=1$ is a rational function involving the Chebyshev polynomials. We give a closed form for $A_n^{\ell \rightarrow \ell+1}(x,q)$ and prove that it is given by a rational function. In the case when $m=\ell+1$ and $\ell=1,2$, we relate the generating series to partia…
Let $V(k)$ denote the waiting time, the number of trials needed to get a consecutive $k$ ones. We propose recurrence algorithms for the probability distribution function (pdf) and the probability generating function (pgf) of $V(k)$ in sequences of independent and Markov dependent Bernoulli trials using generalized Fibonacci sequences of order $k$. Maximum likelihood estimation (MLE) methods for the probability distributions are presented in both cases with simulation examples.
Let $f_1=1,f_2=2$ and $f_i=f_{i-1}+f_{i-2}$ for $i>2$ be the sequence of Fibonacci numbers. Let $Φ_h(n)$ be the quantity of partitions of natural number $n$ into $h$ different Fibonacci numbers. In terms of Zeckendorf partition of $n$ I deduce a formula for the function $Φ(n;t):=\sum_{h\geq 1}Φ_h(n)t^h$, and use it to analyze the functions $F(n):=Φ(n;1)$ and $χ(n):=Φ(n;-1)$. I obtain the least upper bound for $F(n)$ when $f_i-1\<n\<f_{i+1}-1$. It implies that $F(n)\<\sqrt{n+1}$ for any natural $n$. I prove also that $|χ(n)|\<1$, and $\mathop{\lim}\limits_{N\to\infty}\frac{1}{N} \left(χ^2(1)+χ^…
Riordan arrays, denoted by pairs of generating functions (g(z), f(z)), are infinite lower-triangular matrices that are used as combinatorial tools. In this paper, we present Riordan and stochastic Riordan arrays that have connections to the Fibonacci and modified Lucas numbers. Then, we present some pseudo-involutions in the Riordan group that are based on constructions starting with a certain generating function g(z). We also present a theorem that shows how to construct pseudo-involutions in the Riordan group starting with a certain generating function f(z) whose additive inverse has composi…
