Related papers: Summing series using recurrence relations
We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.
In this paper, we study sequences of positive numbers preserving summability. In particular, the open set property for such a family of sequences is shown. Several classes of sequences preserving summability, including polynomials, sums of…
We study two types of series over a real alternative $^*$-algebra $A$. The first type are series of the form $\sum_{n} (x-y)^{\punto n}a_n$, where $a_n$ and $y$ belong to $A$ and $(x-y)^{\punto n}$ denotes the $n$--th power of $x-y$ w.r.t.\…
Motivated by recent results on the Waring problem for polynomial rings and representation of monomial as sum of powers of linear forms, we consider the problem of presenting monomials of degree kd as sums of k-th powers of forms of degree…
In this paper, we consider sums of values of degenerate falling factorials and give a probabilistic proof of a recurrence relation for them. This may be viewed as a degenerate version of the recent probabilistic proofs on sums of powers of…
We establish the mapping relations between analytic functions and periodic functions using the abstract operators $\cos(h\partial_x)$ and $\sin(h\partial_x)$, including the mapping relations between power series and trigonometric series,…
We construct arithmetic terms representing the partial sums of binomial coefficients, and we extend these results to obtain arithmetic terms representing the multisections of binomial coefficient sums. We also introduce an arithmetic term…
It is described how the Hermite-Pad\'e polynomials corresponding to an algebraic approximant for a power series may be used to predict coefficients of the power series that have not been used to compute the Hermite-Pad\'e polynomials. A…
Using a simple recurrence relation we give a new method to compute Jones polynomials of closed braids: we find a general expansion formula and a rational generating function for Jones polynomials. The method is used to estimate degree of…
This text is about the mathematical use of certain divergent power series. The first part is an introduction to 1-summability. The definitions rely on the formal Borel transform and the Laplace transform along an arbitrary direction of the…
We generalize the solution of linear recurrence relations from fields to central division algebras, adapting the standard tools of companion matrices and characteristic polynomials to the non-commutative setting. We then solve linear…
The recursive relation starts to appear by putting a function $y(x)=\sum_{n=0}^{\infty }d_n x^n$ into a linear ordinary differential equation (ODE). There can be $d$-term of sequences in the recurrence relation of a power series where…
Derivative-matching approximations are constructed as power series built from functions. The method assumes the knowledge of special values of the Bell polynomials of the second kind, for which we refer to the literature. The presented…
A novel method, connecting the space of solutions of a linear differential equation, of arbitrary order, to the space of monomials, is used for exploring the algebraic structure of the solution space. Apart from yielding new expressions for…
The form of the coefficients of power series expressions corresponding to solutions of Fuchsian differential equations (or their associated degenerated confluent forms) with n regular singular points is determined by solving the…
We outline a new algorithm to solve coupled systems of differential equations in one continuous variable $x$ (resp. coupled difference equations in one discrete variable $N$) depending on a small parameter $\epsilon$: given such a system…
This paper gives new explicit formulas for sums of powers of integers and their reciprocals.
In this paper, we present a new algorithm for computing the linear recurrence relations of multi-dimensional sequences. Existing algorithms for computing these relations arise in computational algebra and include constructing structured…
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.
Finding integer solutions to norm form equations is a classical Diophantine problem. Using the units of the associated coefficient ring, we can produce sequences of solutions to these equations. It is known that these solutions can be…