No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
Polynomial interpolation and sums of powers of integers
International Journal of Mathematical Education In Science & Technology
Abstract
In this note, we revisit the problem of polynomial interpolation and explicitly construct two polynomials in n of degree k + 1, Pk(n) and Qk(n), such that Pk(n) = Qk(n) = fk(n) for n = 1, 2,… , k, where fk(1), fk(2),… , fk(k) are k arbitrarily chosen (real or complex) values. Then, we focus on the case that fk(n) is given by the sum of powers of the first n positive integers Sk(n) = 1k + 2k + ⋅⋅⋅ + nk, and show that Sk(n) admits the polynomial representations Sk(n) = Pk(n) and Sk(n) = Qk(n) for all n = 1, 2,… , and k ≥ 1, where the first representation involves the Eulerian numbers, and the second one the Stirling numbers of the second kind. Finally, we consider yet another polynomial formula for Sk(n) alternative to the well-known formula of Bernoulli.
... Example 1. As a simple example, we may use formula (2) to calculate the coefficients c p,ℓ for p = 5. As is readily verified, for this case the list of m-tuples satisfying the above conditions is: (0, 0, 0, 0) for ℓ = 0; (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), and (0, 0, 0, 1) for ℓ = 1; (2, 0, 0), (0, 2, 0), (0, 0, 2), (1, 0, 1, 0), (0, 1, 0, 1), and (1, 0, 0, 1) for ℓ = 2; (3, 0), (0, 3), (1, 0, 2), and (2, 0, 1) for ℓ = 3; and 4 for ℓ = 4. Hence, from (2) we get c 5,0 = 120, c 5,1 = 240, c 5,2 = 150, c 5,3 = 30, and c 5,4 = 1. ...
... Proposition 2. For p ≥ 1 and ℓ = 0, 1, . . . , p − 1, let c p,ℓ be the coefficient defined in (2). Then, ML conjecture ⇔ c p,ℓ = (p − ℓ)!S(p, p − ℓ). ...
... In Section 3, we prove that the ML conjecture is true. Specifically, by using the representation of the Stirling numbers of the second kind given in equation (12), we show that the coefficients defined in (2) are indeed identical to c p,ℓ = (p − ℓ)!S(p, p − ℓ) for all p ≥ 1 and ℓ = 0, 1, . . . , p − 1. ...
Recently, Marko and Litvinov conjectured that, for all positive integers $n$ and $p$, the $p$th power of $n$ admits the representation $n^p = \sum_{l=0}^{p-1} (-1)^{l} c_{p,l} F_{n}^{p-l}$, where $F_{n}^{p-l}$ is the $n$th hyper-tetrahedron number of dimension $p-l$ and $c_{p,l}$ denotes the number of $(p-l)$-dimensional facets of the $p$-dimensional simplex $x_{\sigma_1} \geq x_{\sigma_2} \geq \cdots \geq x_{\sigma_p}$ (where $\sigma$ is a permutation of $\{ 1, 2, \ldots, p \}$) formed by cutting the $p$-dimensional cube $0 \leq x_1, x_2, \ldots, x_p \leq n-1$. In this note we show that this conjecture is true for every natural number $p$ if, and only if, $c_{p,l} = (p-l)! S(p, p-l)$, where $S(p,p-l)$ are the Stirling numbers of the second kind. Furthermore, we provide several equivalent formulas expressing the sum of powers $\sum_{i=1}^{n} i^p$, $p =1,2,\ldots$, as a linear combination of figurate numbers.
In this paper, we first focus on the sum of powers of the first n positive odd integers, Tk(n)=1k+3k+5k+⋯+(2n−1)k, and derive in an elementary way a polynomial formula for Tk(n) in terms of a specific type of generalized Stirling numbers. Then we consider the sum of powers of an arbitrary arithmetic progression and obtain the corresponding polynomial formula in terms of the so-called r-Whitney numbers of the second kind. This latter formula produces, in particular, the well-known formula for the sum of powers of the first n natural numbers in terms of the usual Stirling numbers of the second kind. Furthermore, we provide several other alternative formulas for evaluating the sums of powers of arithmetic progressions.
95.47 Sums of powers and Eulerian numbers - Volume 95 Issue 533 - Hung-Ping Tsao
In this article, we use combinatorial reasoning to derive a formula for the sum of the first n squares: $$1^2 + 2^2 + 3^2 + \cdots + n^2.$$ Following this we do the same for the formula for the sum of the first n cubes. Then we look at the problem more generally.
The Eulerian numbers count permutations according to the number of descents.
The two-sided Eulerian numbers count permutations according to number of
descents and the number of descents in the inverse permutation. Here we derive
some results for Eulerian and two-sided Eulerian numbers using an elementary
"balls-in-boxes" approach. We also discuss an open conjecture of Ira Gessel
about the two-sided Eulerian numbers.
There have been many studies of Bernoulli numbers since Jakob Bernoulli first used the numbers to compute sums of powers, 1 p + 2 p + 3 p + ··· + np, where n is any natural number and p is any non-negative integer. By examining patterns of these sums for the first few powers and the relation between their coefficients and Bernoulli numbers, the author hypothesizes and proves a new recursive algorithm for computing Bernoulli numbers, sums of powers, as well as m-ford sums of powers, which enrich the existing literatures of Bernoulli numbers.
Two methods for precalculus students to find exact-fit polynomials.
This article takes a close look at Lagrange and Newton interpolation by examining graphically the component functions of each of these formulas. While interpolation methods are too often considered simply as computational procedures, we demonstrate how the components of the polynomial terms in these formulas provide insight into where these formulas came from, why they work, and how they should be used.
This is a short introduction to the theory of Stirling numbers of the second kind S(m, k) from the point of view of analysis. It is written as an historical survey centered on the representation of these numbers by a certain binomial transform formula. We tell the story of their birth in the book Methodus Differentialis (1730) by James Stirling, and show how they mature in the works of Johann Grünert. The paper demonstrates the usefulness of these numbers in analysis. In particular, they appear in several differentiation and summation formulas. The reader can also see the connection of S(m, k) to Bernoulli numbers, to Euler polynomials, and to power sums.
Formulas presented for the calculation of do not have a closed form; they are in the form of recursive or complex formulas. Here an attempt is made to present a simple formula in which it is only necessary to compute the numerical coefficients in a recursive form, and the coefficients in turn follow a simple pattern (almost similar to Pascal's Triangle). Although the pattern for calculating numerical coefficients based on forming a table is easy, non-recursive formulas are presented to determine the numerical coefficients.
A new method for proving, in an immediate way, many combinatorial identities is presented. The method is based on a simple recursive combinatorial formula involving n + 1 arbitrary real parameters. Moreover, this formula enables one not only to prove, but also generate many different combinatorial identities (not being required to know them a priori), simply by assigning adequate values to those parameters. The identities all flow swiftly from one simple theorem and the presentation is self-contained.
Finding polynomial patterns and Newton interpolation
- Y Yang
- Sp Gordon
Deriving a formula for sums of powers of integers
- Tj Pfaff