We give new sufficient conditions for a sequence of polynomials to have only real zeros based on the method of interlacing zeros. As applications we derive several well-known facts, including the reality of zeros of orthogonal polynomials, matching polynomials, Narayana polynomials and Eulerian polynomials. We also settle certain conjectures of Stahl on genus polynomials by proving them for cer...

Using random variables as motivation, this paper presents an exposition of formalisms developed in [RT1, RT2] for the classical umbral calculus. A variety of examples are presented, culminating in several descriptions of sequences of binomial type in terms of umbral polynomials.

In this paper, we consider the problem of enumerating all maximal motifs in an input string for the class of repeated motifs with wild cards. A maximal motif is such a representative motif that is not properly contained in any larger motifs with the same location lists. Although the enumeration problem for maximal motifs with wild cards has been studied in (Parida et al., CPM’01), (Pisanti et a...

We show that for any countable amenable group action, along Folner sequences have $c>1$ a two sided $c$-tempered tail, one universal estimate the probability there are $n$ fluctuations in ergodic averages of $L^{\infty}$ functions, and this gives exponential decay $n$. Any two-sided sequence can be thinned out to satisfy above property, particular, amenble admits such sequence. This extends res...

Let p be a prime and let φ ∈ Zp[x1, x2, . . . , xp] be a symmetric polynomial, where Zp is the field of p elements. A sequence T in Zp of length p is called a φ-zero sequence if φ(T ) = 0; a sequence in Zp is called a φ-zero free sequence if it does not contain any φ-zero subsequence. Define g(φ, Zp) to be the smallest integer l such that every sequence in Zp of length l contains a φ-zero seque...

The paper solves the enumeration of set partitions in which nondecreasing sequence parts , is contained a degree-2 polynomial . This generalization number into arithmetic progressions. We also study problem dividing whose differences between consecutive are integers. In particular, we focus on sum triangular numbers.

In this paper, we study the ε-lemniscate of the characteristic polynomial in relation to the pseudospectrum of the associated matrix. It is natural to investigate this question because these two sets can be seen as generalizations of eigenvalues. The question of numerical determination of the ε-lemniscate raises the problem of computing the characteristic polynomial p. We can express the coeffi...

Let K be an algebraically closed field of characteristic zero and let f ∈ K[x]. The m-th cyclic resultant of f is rm = Res(f, x m − 1). A generic monic polynomial is determined by its full sequence of cyclic resultants; however, the known techniques proving this result give no effective computational bounds. We prove that a generic monic polynomial of degree d is determined by its first 2d+1 cy...

Many researchers have been working on recurrence relation which is an important topic not only in mathematics but also physics, economics and various applications computer science. There are many useful results sequence there main problem to find any term of we need all previous terms sequence. were theorems obtained relations. In this paper given special identity for generalized kth order rela...