Related papers: Recurrence for powers of recursive sequence II
In this paper, we find a new recurrence formula fo the Euler zeta functions.
We obtain solutions to the recursive sequences of the form $$x_{n + 1} = \frac{x_{n - 3}x_{n }}{x_{n - 2}(a_n + b_nx_{n -3}x_{n})}$$ where $a_n$ and $b_n$ are arbitrary sequences of real numbers, and the initial values are gives as;…
In this note we provide a simple formula of general term of recurrent sequence.
In this article we present a new recurrence formula for a finite sum involving the Fibonacci sequence. Furthermore, we state an algorithm to compute the sum of a power series related to Fibonacci series, without the use of term-by-term…
We derive the necessary and sufficient condition, for a given Polynomial Recurrence Sequence to converge to a given target rational K. By converge, we mean that the Nth term of the sequence, is equal to K, as N tends to positive infinity.…
Number sequences defined by a linear recursion relation are studied by means of generating functions. Indices of the terms in the recursion relation have arbitrary differenses. In addition to formulas for the nth term an algorithm is…
Roughly speaking, a recurrence relation is nested if it contains a subexpression of the form ... A(...A(...)...). Many nested recurrence relations occur in the literature, and determining their behavior seems to be quite difficult and…
In this work, we investigate $H$-recurrence which is defined by $H(n) = H(n-H(n-2)) + H(n-H(n-3))$ thanks to a recent approach to certain recurrences such as Conway and Hofstadter $Q$ recursions.
In this article one builds a class of recursive sets, one establishes properties of these sets, and one proposes applications.
We derive some q-analogs of Euler-Cassini-type identities and of recurrence formulas for powers of Fibonacci polynomials.
We introduce a necessary and sufficient criterion for determining the existence and the values of ratio limits of complex sequences generated by arbitrary linear recurrences.
We present some Euler-type recurrences for the partition function $p(n)$.
Here, we give upper and lower bounds on the count of positive integers $n\le x$ dividing the $n$th term of a nondegenerate linearly recurrent sequence with simple roots.
We propose a notion of iterating functions $f:X^{k}\rightarrow X$ in a way that represents recurrence relations of the form $a_{n+k}=f(a_{n},a_{n+1},...,a_{n+k-1})$. We define a function as $n$-involutory when its $n$th iterate is the…
Recursive formulas are derived for the number of solutions of linear and quadratic Diophantine equations with positive coefficients. This result is further extended to general non-linear additive Diophantine equations. It is shown that all…
A recurrence relations for sums of powers of complex functions can be written as a system of linear equation AX=B. Using properties of determinant and Cramer's rule for solving systems of linear equation, this paper presents an absolutely…
This paper examines the recursive sequence of polynomials $p_n(x)$, defined by $p_0(x) = x^2 - 2$ and $p_n(x) = p_{n-1}(x)^2 - 2$ for $n \geq 1$. It describes the field-theoretic motivations behind this sequence, derives a recursive formula…
We derive the P-finite recurrences for classes of sequences with ordinary generating function containing roots of polynomials. The focus is on establishing the D-finite differential equations such that the familiar steps of reducing their…
We study the expressive power of polynomial recursive sequences, a nonlinear extension of the well-known class of linear recursive sequences. These sequences arise naturally in the study of nonlinear extensions of weighted automata, where…
The central idea of this article is to present a systematic approach to construct some recurrence relations for the solutions of the second-order linear difference equation of hypergeometric-type defined on the quadratic-type lattices. We…