Related papers: Generating Functions, Weighted and Non-Weighted Su…
In this note, we explore two families of sequences associated to a suitable integer sequence: the gap-sum sequence and the gap-product sequence. These are the sums and the products of consecutive numbers not in the original sequence. We…
In this paper, we investigate an example of summation of non-logarithmic singularities of a specific type in a two-dimensional non-linear sigma model. As a result of the study, we obtained an explicit formula, which, upon formal expansion…
In this paper, we investigate the existence of $C^n$, $n\in \mathbb{N}^+$, solutions for a class of second-order iterative functional equations involving iterates of the unknown function and a nonlinear term. Applying the Fiber Contraction…
Working in combinatorial model $\mathrm{W_{co}}(d)$, $d=1,2,\dots$, of P\'olya's random walker in $\mathbb{Z}^d$, we prove two theorems on recurrence to a vertex. We obtain an effective version of the first theorem if $d=2$. Using a…
We focus on writing closed forms of generating functions for the number of partitions with gap conditions as double sums starting from a combinatorial construction. Some examples of the sets of partitions with gap conditions to be discussed…
We establish that the sequences formed by logarithms and by "fractional" powers of integers, as well as the sequence of prime numbers, are non-holonomic, thereby answering three open problems of Gerhold [Electronic Journal of Combinatorics…
A weight function which $q$-generalizes the ground state wave function of the multi-component Calogero-Sutherland quantum many body system is introduced. Conjectures, and some proofs in special cases, are given for a constant term identity…
We construct new integral representations for transformations of the ordinary generating function for a sequence, $\langle f_n \rangle$, into the form of a generating function that enumerates the corresponding "square series" generating…
For an arbitrary homogeneous linear recurrence sequence of order d with constant coefficients, we derive recurrence relations for all subsequences with indices in arithmetic progression. The coefficients of these recurrences are given…
Using a regularization by putting the system in finite volume, we develop a novel approach to form factor perturbation theory for nonintegrable models described as perturbations of integrable ones. This permits to go beyond first order in…
We evaluate some new three parameter families of finite reciprocal sums involving Horadam numbers. We will also be able to state the results for the infinite sums. Some Fibonacci and Lucas sums will be presented as examples.
In this paper we study substitutions and some of their associated generating functions. This association takes aperiodicity to transcendence, and vice-versa. These generating functions have a recursive structure arising from the…
Non-linear recurrences which generate integers in a surprising way have been studied by many people. Typically people study recurrences that are linear in the highest order term. In this paper I consider what happens when the recurrence is…
We obtain explicit factored closed-form expressions for Fibonacci and Lucas sums of the form \mbox{$\sum_{k = 1}^n {F_{2rk}^3 }$} and \mbox{$\sum_{k = 1}^n {L_{2rk}^3 }$}, where $r$~and~$n$ are integers.
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…
Given an operad P with a finite Groebner basis of relations, we study the generating functions for the dimensions of its graded components P(n). Under moderate assumptions on the relations we prove that the exponential generating function…
A fermionic representation is given for all the quantities entering in the generating function approach to weighted Hurwitz numbers and topological recursion. This includes: KP and 2D Toda $\tau$-functions of hypergeometric type, which…
We evaluate various binomial sums involving the powers of Fibonacci and Lucas numbers.
Our aim in this work is to give explicit formula of the linear processes solution of autoregressive time series AR(2) with hint of generating functions theory by using the Horadam numbers and polynomials.
In this paper, we construct two binary linear codes associated with multi-dimensional and $m -$multiple power Kloosterman sums (for any fixed $m$) over the finite field $\mathbb{F}_{q}$. Here $q$ is a power of two. The former codes are dual…