Related papers: Laurent polynomials, Eulerian numbers, and Bernste…
We provide lower bounds for p-adic valuations of multisums of factorial ratios which satisfy an Ap\'ery-like recurrence relation: these include Ap\'ery, Domb, Franel numbers, the numbers of abelian squares over a finite alphabet, and…
An overlooked formula of E. Lucas for the generalized Bernoulli numbers is proved using generating functions. This is then used to provide a new proof and a new form of a sum involving classical Bernoulli numbers studied by K. Dilcher. The…
Roots of random polynomials have been studied exclusively in both analysis and probability for a long time. A famous result by Ibragimov and Maslova, generalizing earlier fundamental works of Kac and Erdos-Offord, showed that the…
The study of degenerate versions of certain special polynomials and numbers, which was initiated by Carlitz's work on degenerate Euler and degenerate Bernoulli polynomials, has recently seen renewed interest among mathematicians. The aim of…
Hurwitz numbers are the Laurent coefficients of an elliptic function $\wp(u)$ of cyclotomic type, and they are natural generalization of the Bernoulli numbers. This paper gives new generalization of Bernoulli and Hurwitz numbers for higher…
Given a polynomial \[ f(x)=a_0x^n+a_1x^{n-1}+\cdots +a_n \] with positive coefficients $a_k$, and a positive integer $M\leq n$, we define a(n infinite) generalized Hurwitz matrix $H_M(f):=(a_{Mj-i})_{i,j}$. We prove that the polynomial…
New differential-recurrence properties of dual Bernstein polynomials are given which follow from relations between dual Bernstein and orthogonal Hahn and Jacobi polynomials. Using these results, a fourth-order differential equation…
In this paper we introduce the generalization of Multi Poly-Euler polynomials and we investigate some relationship involving Multi Poly-Euler polynomials. Obtaining a closed formula for generalization of Multi Poly-Euler numbers therefore…
We give formulae for the cumulants of complex Wishart (LUE) and inverse Wishart matrices (inverse LUE). Their large-$N$ expansions are generating functions of double (strictly and weakly) monotone Hurwitz numbers which count constrained…
Recently Dritschel proves that any positive multivariate Laurent polynomial can be factorized into a sum of square magnitudes of polynomials. We first give another proof of the Dritschel theorem. Our proof is based on the univariate matrix…
Let $L$ be a linear differential operator with constant coefficients of order $n$ and complex eigenvalues $\lambda_{0},...,\lambda_{n}$. Assume that the set $U_{n}$ of all solutions of the equation $Lf=0$ is closed under complex…
Let A be an n-by-n matrix of real numbers which are weakly decreasing down each column, Z_n = diag(z_1,..., z_n) a diagonal matrix of indeterminates, and J_n the n-by-n matrix of all ones. We prove that per(J_nZ_n+A) is stable in the z_i,…
In the research, with aid of the Fa\`a di Bruno formula, be virtue of several identities for the Bell polynomials of the second kind, with help of two combinatorial identities, by means of the (logarithmically) complete monotonicity of…
[Inserted by J. Maurice Rojas] We give a formula for the number of complex roots of a generic system of two polynomial equations in two unknowns. The formula is completely combinatorial, ultimately depending just on the convex hull of the…
We characterize the polynomials $p_1(t), ... , p_n(t)$ whose Wronskian $W(p_1, ... , p_n)$ is a nonzero constant. Then, we generalize our results to characterize the Laurent polynomials with the same property. Finally, for rational…
We prove quasi-polynomiality for monotone and strictly monotone orbifold Hurwitz numbers. The second enumerative problem is also known as enumeration of a special kind of Grothendieck's dessins d'enfants or $r$-hypermaps. These statements…
The usual product $m\cdot n$ on $\mathbb{Z}$ can be viewed as the sum of $n$ terms of an arithmetic progression whose first term is $a_{1}=m-n+1$ and whose difference is $d=2$. Generalizing this idea, we define new similar product mappings,…
For $m$ an even positive integer and $p$ a prime, we show that the generalized Euler polynomial $E_{mp}^{(mp)}(x)$ is in Eisenstein form with respect to $p$ if and only if $p$ does not divide $m (2^m-1)B_m$. As a consequence, we deduce that…
In this paper we study pairs of polynomials with a given factorization pattern and such that the degree of their difference attains its minimum. We call such pairs of polynomials Davenport--Zannier pairs, or DZ-pairs for short. The paper is…
In this paper, we investigate some properties and identities for degenerate Euler polynomials in connection with degenerate Bernstein polynomials by means of fermionic p-adic integrals on Zp and generating functions. In addition, we study…