Related papers: Binomial convolutions for rational power series

In this paper we initiate the study of products and sums divisible by central binomial coefficients. We show that 2(2n+1)binom(2n,n)| binom(6n,3n)binom(3n,n) for every n=1,2,3,... Also, for any nonnegative integers $k$ and $n$ we have…

An algebra denoted $m\mathfrak{H}$ with three generators is introduced and shown to admit embeddings of the Hahn algebra and the rational Hahn algebra. It has a real version of the deformed Jordan plane as a subalgebra whose connection with…

A positive linear recurrence sequence is of the form $H_{n+1} = c_1 H_n + \cdots + c_L H_{n+1-L}$ with each $c_i \ge 0$ and $c_1 c_L > 0$, with appropriately chosen initial conditions. There is a notion of a legal decomposition (roughly,…

Let $F_n(k)$ be the generalized Fibonacci number defined by (with $F_i(k)$ abbreviated to $F_i$): $F_n = F_{n-1} + F_{n-2} + \dots + F_{n-k}$, for $n \geq k$, and the initial values $(F_0,F_1,...,F_{k-1})$. Let $B_n(k,j)$ be $F_n(k)$ with…

We consider Problem 6.94 posed in the book Concrete Mathematics by Graham, Knuth, and Patashnik, and solve it by using bivariate exponential generating functions. The family of recurrence relations considered in the problem contains many…

In this paper, we derive formal general formulas for noncommutative exponentiation and the exponential function, while also revisiting an unrecognized, and yet powerful theorem. These tools are subsequently applied to derive counterparts…

For each positive integer n greater than or equal to 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n=2 case is equivalent to the standard continued fraction algorithm. For n=3, it reduces to…

A conjecture is given that, if true, could lead to an algorithm for computing definite sums of rational functions.

We consider finite iterated generalized harmonic sums weighted by the binomial $\binom{2k}{k}$ in numerators and denominators. A large class of these functions emerges in the calculation of massive Feynman diagrams with local operator…

Power series in which the summand satisfies a linear recurrence relation with polynomial coefficients are shown to be the solution of a linear differential or algebraic equation. Solving the associated differential or algebraic equation…

The integral representation of the Hadamard product of two functions is used to prove several Euler-type series transformation formulas. As applications we obtain three binomial identities involving harmonic numbers and an identity for the…

In this study, we apply "r" times the binomial transform to k-Lucas sequence. Also, the Binet formula, summation, generating function of this transform are found using recurrence relation. Finally, we give the properties of iterated…

The powers of generating functions and its properties are analyzed. A new class of functions is introduced, based on the application of compositions of an integer $n$, called composita. The methods for obtaining reciprocal and reverse…

We derive modular parametrizations for certain infinite series whose summands involve central binomial coefficients and higher-order harmonic numbers. When the rates of convergence are certain rational numbers, modularity allows us to…

We describe a combinatorial approach for investigating properties of rational numbers. The overall approach rests on structural bijections between rational numbers and familiar combinatorial objects, namely rooted trees. We emphasize that…

Wu showed that certain sums of products of binomial coefficients modulo 2 are given by the run length transforms of several famous linear recurrence sequences, such as the positive integers, the Fibonacci numbers, the extended Lucas…

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…

We show generic existence of power series a with complex coefficients a_n, such that the sequence of partial sums of a new power series where its coefficients b_n are functions of a_0, a_1, ..., a_n approximate every polynomial uniformly on…

We examine the rationality conjecture which states that (a) the formal power series $\sum_{r\ge 1} \tc_{r+1}(X)\cdot x^r$ represents a rational function of $x$ with a single pole of order 2 at $x=1$ and (b) the leading coefficient of the…

We investigate a class of combinatorial sums involving reciprocals of central binomial coefficients , employing generating functions as the primary solution technique to formulate and analyze series involving the Catalan's constant. Using a…