AN INNOVATIVE STUDY ON SUM OF POWERS OF INTEGERS

The generalization of sum of integral powers of first n-natural numbers has been an interesting problem among the researchers in Analytical Number Theory for decades. This research article mainly focuses on the derivation of generalized result of this sum. More explicit formula has been derived in order to get the sum of any arbitrary integral powers of first n-natural numbers. Furthermore by using the fundamental principles of Combinatorics and Linear Algebra an attempt has been made to answer an interesting question namely: Is the sum of integral powers of natural numbers a unique polynomial? As a result it is depicted that this sum always equals a unique polynomial over natural numbers. Moreover some properties of the coefficients of this polynomial are derived.More importantly a recurrence relation which can give the formulas for sum of any positive integral powers of first n-natural numbers has been proposed and it is strongly believed that this recurrence relation is the most significant thing in this entire discussion

Polynomial expressions of p-ary auction functions

One of the common ways to design secure multi-party computation is twofold: to realize secure fundamental operations and to decompose a target function to be securely computed into them. In the setting of fully homomorphic encryption, as well as some kinds of secret sharing, the fundamental operations are additions and multiplications in the base field such as the field {\mathbb{F}_{2}} with two elements. Then the second decomposition part, which we study in this paper, is (in theory) equivalent to expressing the target function as a polynomial. It is known that any function over the finite prime field {\mathbb{F}_{p}} has a unique polynomial expression of degree at most {p-1} with respect to each input variable; however, there has been little study done concerning such minimal-degree polynomial expressions for practical functions. This paper aims at triggering intensive studies on this subject, by focusing on polynomial expressions of some auction-related functions such as the maximum/minimum and the index of the maximum/minimum value among input values.

A NOTE ON THE PERIODICITY OF ENTIRE FUNCTIONS

We give some sufficient conditions for the periodicity of entire functions based on a conjecture of C. C. Yang, using the concepts of value sharing, unique polynomial of entire functions and Picard exceptional value.

On the divergence of Hermite-Fejér type interpolation with equidistant nodes

If f(x) is defined on [−1, 1], let H¯1 n(f, x) denote the Lagrange interpolation polynomial of degree n (or less) for f which agrees with f at the n+1 equally spaced points xk, n = −1 + (2k)/n (0 ≤ k ≤ n). A famous example due to S. Bernstein shows that even for the simple function h(x) = │x│, the sequence H¯1 n (h, x) diverges as n → ∞ for each x in 0 < │x│ < 1. A generalisation of Lagrange interpolation is the Hermite-Fejér interpolation polynomial H¯mn (f, x), which is the unique polynomial of degree no greater than m(n + 1) – 1 which satisfies (f, Xk, n) = δo, pf(xk, n) (0 ≤ p ≤ m − 1, 0 ≤ k ≤ n). In general terms, if m is an even number, the polynomials H¯mn(f, x) seem to possess better convergence properties than the H¯1 n (f, x). Nevertheless, D.L. Berman was able to show that for g(x) ≡ x, the sequence H¯2n(g, x) diverges as n → ∞ for each x in 0 < │x│. In this paper we extend Berman's result by showing that for any even m, H¯mn(g, x) diverges as n → ∞ for each x in 0 < │x│ < 1. Further, we are able to obtain an estimate for the error │H¯mn(g, x) – g(x)│.

On the non-existence of some interpolatory polynomials

Here we prove that ifxk,k=1,2,…,n+2are the zeros of(1−x2)Tn(x)whereTn(x)is the Tchebycheff polynomial of first kind of degreen,αj,βj,j=1,2,…,n+2andγj,j=1,2,…,n+1are any real numbers there does not exist a unique polynomialQ3n+3(x)of degree≤3n+3satisfying the conditions:Q3n+3(xj)=αj,Q3n+3(xj)=βj,j=1,2,…,n+2andQ‴3n+3(xj)=γj,j=2,3,…,n+1. Similar result is also obtained by choosing the roots of(1−x2)Pn(x)as the nodes of interpolation wherePn(x)is the Legendre polynomial of degreen.

On interpolation polynomials of the Hermite-Fejér type II

Given a ŕeal-valued function f on [−1, 1], n ∈ N, and the following partition of [−1, 1[:there exists a unique polynomial R4n−1(f; x) of degree not exceeding 4n − 1 such thatand, for j = 1, 2 and 3,

A note of equivalence classes of matrices over a finite field

LetFqm×mdenote the algebra ofm×mmatrices over the finite fieldFqofqelements, and letΩdenote a group of permutations ofFq. It is well known that eachϕϵΩcan be represented uniquely by a polynomialϕ(x)ϵFq[x]of degree less thanq; thus, the groupΩnaturally determines a relation∼onFqm×mas follows: ifA,BϵFqm×mthenA∼Bifϕ(A)=Bfor someϕϵΩ. Hereϕ(A)is to be interpreted as substitution into the unique polynomial of degree<qwhich representsϕ.In an earlier paper by the second author [1], it is assumed that the relation∼is an equivalence relation and, based on this assumption, various properties of the relation are derived. However, ifm≥2, the relation∼is not an equivalence relation onFqm×m. It is the purpose of this paper to point out the above erroneous assumption, and to discuss two ways in which hypotheses of the earlier paper can be modified so that the results derived there are valid.

Next-To-Interpolatory Approximation on Sets with Multiplicities

It is known that given a set X of m (⩾n) distinct real numbers and a real-valued function f denned on X, there exists a unique polynomial pn-1,f,x of degree n — 1 or less which approximates best to f(x) on X, that is, which minimizes the deviation δ = δ(f, p) defined by the αth-power metric (α < 1) with positive weights, or by the positively weighted maximum of |f — p| on X; these deviations shall be denoted by δα and δβ. The polynomial pn-1,f,x has the property that f — pn-1,f,x has at least n strong sign changes; in other words, there are at least n + 1 points in X where the difference takes alternatingly positive and negative values.