Related papers: Certain Binomial Sums with recursive coefficients
The main result of this paper is to show that all binomial identities are orderable. This is a natural statement in the combinatorial theory of finite sets, which can also be applied in distributed computing to derive new strong bounds on…
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.
We prove certain identities involving Euler and Bernoulli polynomials that can be treated as recurrences. We use these and also other known identities to indicate connection of Euler and Bernoulli numbers with entries of inverses of certain…
We propose and recursively prove polynomial identities which imply Capparelli's partition theorems. We also find perfect companions to the results of Andrews, and Alladi, Andrews and Gordon involving $q$-trinomial coefficients. We follow…
One of the most popular and studied recursive series is the Fibonacci sequence. It is challenging to see how Fibonacci numbers can be used to generate other recursive sequences. In our article, we describe some families of integer…
Divisibility sequences are defined by the property that their elements divide each other whenever their indices do. The divisibility sequences that also satisfy a linear recurrence, like the Fibonacci numbers, are generated by polynomials…
In this paper, with the help of trinomial coefficients we study some arithmetic properties of certain determiants involving reciprocals of binary quadratic forms over finite fields.
We define generalized bivariate polynomials, from which upon specification of initial conditions the bivariate Fibonacci and Lucas polynomials are obtained. Using essentially a matrix approach we derive identities and inequalities that in…
We recast homogeneous linear recurrence sequences with fixed coefficients in terms of partial Bell polynomials, and use their properties to obtain various combinatorial identities and multifold convolution formulas. Our approach relies on a…
In this paper we present a family of identities for recursive sequences arising from a second order recurrence relation, that gives instances of Zeckendorf representation. We prove these results using a special case of an universal property…
We explicitly determine the Ap\'ery limits for the sums of powers of binomial coefficients. As an application, we prove a weak version of Franel's conjecture on the order of the recurrences for these sequences. Namely, we prove the…
We derive several identities for arbitrary homogeneous second order recurrence sequences with constant coefficients. The results are then applied to present a unified study of six well known integer sequences, namely the Fibonacci sequence,…
Given a linear recurrence of the form $c_n=a_1c_{n-1}+\cdots+a_j c_{n-j}$, it is well-known that $c_n=\sum_{r}p_r(n)r^n$, where the sum is taken over the set of characteristic roots and each $p_r(n)$ is some polynomial. We give a closed…
Given two infinite sequences with known binomial transforms, we compute the binomial transform of the product sequence. Various identities are obtained and numerous examples are given involving sequences of special numbers: Harmonic…
A class of determinants is introduced. Different kind of mathematical objects, such as Fibonacci, Lucas, Tchebychev, Hermite, Laguerre, Legendre polynomials, sums and covergents are represented as determinants from this class. A closed…
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…
The binomial coefficients and Catalan triangle numbers appear as weight multiplicities of the finite-dimensional simple Lie algebras and affine Kac--Moody algebras. We prove that any binomial coefficient can be written as weighted sums…
$q$-Analogues of the coefficients of $x^a$ in the expansion of $\prod_{j=1}^N (1+x+...+x^j)^{L_j}$ are proposed. Useful properties, such as recursion relations, symmetries and limiting theorems of the ``$q$-supernomial coefficients'' are…
In this paper, we study some properties of associated sequences of special polynomials. From the properties of associated sequences of polynomials, we derive some interesting identities of special polynomials.
In this note we present examples of cumulative connection constants included new fibonomial ones. All examples posses combinatorial interpretation.