Sequences of functions of binomial type

The theory of binomial enumeration leads to sequences of functions of binomial type which are not polynomials. The results of Mullin-Rota for these sequences are developed and a ring structure on the set of sequences is studied.     

Bell polynomials and binomial type sequences

This paper concerns the study of the Bell polynomials and the binomial type sequences. We mainly establish some relations tied to these important concepts. Furthermore, these obtained results are exploited to deduce some interesting relations concerning the Bell polynomials which enable us to obtain some new identities for the Bell polynomials. Our results are illustrated by some comprehensive examples.     

A linear algebra setting for the rota-mullin theory of polynomials of binomial type

The linear algebra and combinatorial aspects of the Rota-Mullin theory of polynomials of binomial type are separated and the former is developed in terms of shift operators on infinite dimensional vector spaces with a view towards application in the calculus of finite differences.     

An exposá of the mullin-rota theory of polynomials of binomial type

The basic theorems of the Mullin-Rota theory of polynomials of binomial type are rederived here by a combination of the classical theory of generating functions and Rota's operator theoretical methods. As a result a certain number of new identities are obtained.     

Central and local limit theorems for the coefficients of polynomials of binomial type

We introduce the problem of establishing a central limit theorem for the coefficients of a sequence of polynomials P_n(x) of binomial type; that is, a sequence P_n(x) satisfying $\text{exp}\text{(xg(u)) = ∑}{}_{\mathrm{n=0}}{}^{\mathrm{\infty }}\text{P}{}_{\mathrm{n}}\text{(x)(}\text{u}{}^{\mathrm{n}}\text{n!}\text{)}$ for some (formal) power series g(u) lacking constant term. We give a complete answer in the case when g(u) is a polynomial, and point out the widest known class of nonpolynomial power series g(u) for which the corresponding central limit theorem is known true. We also give the least restrictive conditions known for the coefficients of P_n(x) which permit passage from a central to a local limit theorem, as well as a simple criterion for the generating function g(u) which assures these conditions on the coefficients of P_n(x). The latter criterion is a new and general result concerning log concavity of doubly indexed sequences of numbers with combinatorial significance. Asymptotic formulas for the coefficients of P_n(x) are developed.     

BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials

We consider 3-parametric polynomialsP ^* (x; q, t, s) which replace theA _n-series interpolation Macdonald polynomialsP ^* (x; q, t) for theBC _n-type root system. For these polynomials we prove an integral representation, a combinatorial formula, Pieri rules, Cauchy identity, and we also show that they do not satisfy any rationalq-difference equation. Ass the polynomialsP ^* (x; q, t, s) becomeP ^* (x; q, t). We also prove a binomial formula for 6-parametric Koornwinder polynomials.     

Sequences of binary irreducible polynomials

In this paper we construct an infinite sequence of binary irreducible polynomials starting from any irreducible polynomial $f_0\in F_2\left[x\right]$. If $f_0$ is of degree $n=2^l?m$, where $m$ is odd and $l$ is a nonnegative integer, after an initial finite sequence of polynomials $f_0,f_1,\dots ,f_s$, with $s\le l+3$, the degree of $f_{i+1}$ is twice the degree of $f_i$ for any $i\ge s$.     

Properties of fractional calculus with respect to a function and Bernstein type polynomials

This paper is divided into two stages. In the first stage, we investigated a new approach for the $\psi$-Riemann–Liouville fractional integral and the Faa di Bruno formula for the $\psi$-Hilfer fractional derivative. In addition, we discussed other properties involving the $\psi$-Hilfer fractional derivative and the $\psi$-Riemann–Liouville fractional integral. In the second stage, Bernstein polynomials involving the $\psi (·)$ function are investigated and the $\psi$-Riemann–Liouville fractional integral and $\psi$-Hilfer fractional derivative from the Bernstein polynomials are evaluated. We also discussed the relationship between the $\psi$-Hilfer fractional derivative with Laguerre polynomials and hypergeometric functions, and a version of the fractional mean value theorem with respect to a function. Motivated by the Bernstein polynomials, the second stage uses the Bernstein polynomials to approximate the solution of a fractional integro-differential equation with Hilfer fractional derivative and concluding with a numerical approach with its respective graph.     

On small fractional parts of polynomials

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We prove that for any real polynomial f(x)∈R[x] the set

     

A fractional q-derivative operator and fractional extensions of some q-orthogonal polynomials

The Ramanujan Journal - A fractional q-derivative operator is introduced and some of its properties have been proved. Next, a fractional q-differential equation of Gauss type is introduced and...     

The fractional parts of a sum of polynomials

LetX=(x ₁,...,x _s ) be a vector ofs real components and , whereP _j (x _j ) are polynomials of exact degree k with real coefficients and without constant terms. The authors extend a result of Davenport and obtain an approximation on f(X) where t means the distance fromt to the nearest integer.     

Sequences of consecutive smooth polynomials over a finite field

Given , we show that there are infinitely many sequences of consecutive -smooth polynomials over a finite field. The number of polynomials in each sequence is approximately .

     

Sequences of polynomials whose zeros lie on fixed lemniscates

For fixed integersk2 we study sequences of polynomialsP _n (z) with the following properties: (i) degP _n ; (ii) the zeros of all theP _n (z) lie on a certain lemniscate withk ₁k foci, one of which is the origin; (iii) theP _n (z) can be cut in such a way that the zeros of the lower part all lie on the unit circle and those of the upper part lie on a lemniscate having the foci in (ii) excluding the origin. Several special cases and examples are considered.