Poly-central factorial sequences and poly-central-Bell polynomials

A determinantal form for Δ h -Appell sequences is proposed and general properties are obtained by using elementary linear algebra tools. As particular cases of Δ h -Appell sequences the sequence of Bernoulli polynomials of second kind and the one of Boole polynomials are considered. A general linear interpolation problem, which generalizes the classical interpolation problem on equidistant points, is proposed. The solution of this problem is expressed by a basis of Δ h -Appell polynomials. Numerical examples which justify theoretical results on the interpolation problem are given.

In this paper, we observe the behavior of complex roots of the generalized Euler polynomials En(x) of the second kind, using numerical investigation. By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the generalized Euler polynomials En(x) of the second kind. Finally, we give a table for the solutions of the generalized Euler polynomials En(x) of the second kind.

<abstract><p>Bell polynomials are widely applied in many problems arising from physics and engineering. The aim of this paper is to introduce new types of special polynomials and numbers, namely Bell polynomials and numbers of the second kind and poly-Bell polynomials and numbers of the second kind, and to derive their explicit expressions, recurrence relations and some identities involving those polynomials and numbers. We also consider degenerate versions of those polynomials and numbers, namely degenerate Bell polynomials and numbers of the second kind and degenerate poly-Bell polynomials and numbers of the second kind, and deduce their similar results.</p></abstract>

The aim of this paper is to study the complete and incomplete degenerate Bell polynomials, which are degenerate versions of the complete and incomplete Bell polynomials, and to derive some properties and identities for those polynomials. In particular, we introduce some new polynomials associated with the incomplete degenerate Bell polynomials. In fact, they are the coefficients of the reciprocal of the power series given by 1 plus the one appearing as the exponent of the generating function of the complete degenerate Bell polynomials.

The main objective in this paper is to study on special numbers and polynomials that contain finite sums of powers of binomial coefficients. By using generating function methods, some formulas and relations related to these numbers and the Apostol-Bernoulli and Apostol-Euler numbers of negative higher order, the Bernoulli and Euler numbers, the Stirling type numbers, the combinatorial numbers, the Bell polynomials, the Fubini type polynomials, and the Legendre polynomials are presented. Moreover, asymptotic expressions of the finite sums of powers of binomial coefficients for these numbers are given. Some numeric values of these asymptotic expressions are illustrated by the tables. Finally, some inequalities for these numbers are given.

This paper studies increasing trees on n labeled vertices, in which labels increase from the root to the leaves. It is known that the number of binary increasing trees coincides with the number of alternating permutations (Euler numbers). Riordan obtained explicit formulas for the numbers of ternary and quaternary trees. This article derives a general formula for the number of m-ary increasing trees for any m. The main result is expressed in terms of the degree-chromatic polynomial of the complete graph and Bell polynomials. It is shown how the corresponding generating function is related to the inversion problem and how combinatorial methods, including the lemma on coefficients of the multiplicative inverse function and the Lagrange inversion formula, can be used to compute the coefficients. A connection is also established between the values of the degree-chromatic polynomial at λ = −1 and the numbers of special permutations studied by Gessel.

Our aim is to construct and compute efficient generating functions enumerating the k-ary Lyndon words having prime number length which arise in many branches of mathematics and computer science. We prove that these generating functions coincide with the Apostol–Bernoulli numbers and their interpolation functions and obtain other forms of these generating functions including not only the Frobenius–Euler numbers, but also the Fubini type numbers. Moreover, we derive some identities, relations and combinatorial sums including the numbers of the k-ary Lyndon words, the Bernoulli numbers and polynomials, the Stirling numbers and falling factorials. Using these generating functions and recurrence relation for the Apostol–Bernoulli numbers, we give two algorithms to compute these generating functions. Using these algorithms, we compute some infinite series formulas including the number of the k-ary Lyndon words on some special classes of primes with the purpose of providing some numerical evaluations about these generating functions. In addition, we approximate these generating functions by the rational functions of the Apostol–Bernoulli numbers to show that the complexity of the aforementioned algorithms may be decreased by means of approximation method which are illustrated by some numerical evaluations with their plots for varying prime numbers. Finally, using Bell polynomials (i.e., exponential functions) approach to the numbers of Lyndon words, we construct the exponential generating functions for the numbers of Lyndon words. Finally, we define a new family of special numbers related to these special words and investigate some of their fundamental properties.

Motivated by time series analysis, we consider the problem of solving the recurrence relation u n+1 = v n+1 +u n⊗v n for n ≠ 0 and u n, given the sequence v n. A solution is given as a Bell polynomial. When v n can be written as a weighted sum of nth powers, then the solution u n also takes this form. Mathematical subject classification: 33E99.

It is well known that if $\Phi _n (x)$ is the $n$th cyclotomic polynomial, then there is a factorization $x^n - 1 = \prod \Phi _d (x)$, where the product is taken over the divisors $d$ of $n$. Thus, one can obtain, by Möbius inversion, a product formula for each $\Phi _n (x)$ in terms of the various factors $x^d - 1$. The purpose of this note is two-fold. First, we show that the above factorization implies a similar factorization for the minimal polynomials of the algebraic numbers $\cos (2\pi / n)$, where $n$ is a positive integer. Secondly, we give an explicit formula for the minimal polynomials of $\cos (2\pi / p)$, where $p$ is prime.

Combinatorial and analytic properties of the n-dimensional Hermite polynomials

In the present paper, we recall a formula, due to L. Comtet [2], for the nth order derivative of a function with the exponential inner function. We apply the formula to get addition formulas for the Stirling numbers of the second kind. Then we obtain explicit formulas for the generalized Euler polynomials, the generalized Euler numbers and the Bell polynomials.

We consider a class of strongly q-log-convex polynomials based on a triangular recurrence relation with linear coefficients, and we show that the Bell polynomials, the Bessel polynomials, the Ramanujan polynomials and the Dowling polynomials are strongly q-log-convex. We also prove that the Bessel transformation preserves log-convexity.

This evaluation proves a conjecture in [9, p. 370] and solves the problem of finding explicitly the number of rational points (mod/?) on the surface z = (x + l)(y + ΐ)(x + y), a problem some algebraic geometers had worked on without success. Character sum analogues of the important formulas for orthogonal polynomials are potentially as useful as those for hypergeometric series, so a systematic study should be made. Indeed, many character sums studied in the literature are analogues of special functions, e.g., the generalized Kloosterman sum (see (2.5), Theorem 2.6, and, say, [10], [21A, p. 253]). In this paper, the focus is on analogues of Hermite polynomials, namely Hermite character sums HN(x) defined in (2.1). Each of the theorems in §4 is an analogue over finite fields of a classical formula stated just above it. The classical formulas are stated without conditions of validity; such conditions are often unrelated to the unpredictable conditions of validity for the finite field formulas. It is not always possible to give proofs of the finite field formulas which parallel classical proofs. This is because no satisfactory analogues of limits, first derivatives, logarithms, and three term recurrence relations are known. It would be of great importance to find a unified approach which simultaneously explains formulas for orthogonal polynomials and the analogues over finite fields. Perhaps this will be accomplished by connecting the polynomials with Lie groups having counterparts over finite fields.

In this paper, the authors give a simple review of closed-form, explicit, and recursive formulas and related results for the nth derivative of the power-exponential function xx, establish two closed-form and explicit formulas for partial Bell polynomials at some specific arguments, and present several new closed-form and explicit formulas for the nth derivative of the power-exponential function xx and for related functions and integer sequences.

Application of a composition of generating functions for obtaining explicit formulas of polynomials