Irreducibility of Bernoulli Polynomials of Higher Order

1962 ◽  
Vol 14 ◽  
pp. 565-567 ◽  
Author(s):  
P. J. McCarthy
Keyword(s):  
Positive Integer ◽  
Bernoulli Number ◽  
Bernoulli Polynomials ◽  
Constant Term ◽  
Rational Numbers ◽  
Higher Order ◽  
Image Position

The Bernoulli polynomials of order k, where k is a positive integer, are defined byBm(k)(x) is a polynomial of degree m with rational coefficients, and the constant term of Bm(k)(x) is the mth Bernoulli number of order k, Bm(k). In a previous paper (3) we obtained some conditions, in terms of k and m, which imply that Bm(k)(x) is irreducible (all references to irreducibility will be with respect to the field of rational numbers). In particular, we obtained the following two results.

scholarly journals GENERALIZED CONTINUED FRACTION EXPANSIONS WITH CONSTANT PARTIAL DENOMINATORS

2018 ◽  
Vol 107 (02) ◽  
pp. 272-288
Author(s):  
TOPI TÖRMÄ
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Rational Numbers ◽  
Positive Real ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Continued Fraction Expansions ◽  
Positive Integers

We study generalized continued fraction expansions of the form $$\begin{eqnarray}\frac{a_{1}}{N}\frac{}{+}\frac{a_{2}}{N}\frac{}{+}\frac{a_{3}}{N}\frac{}{+}\frac{}{\cdots },\end{eqnarray}$$ where $N$ is a fixed positive integer and the partial numerators $a_{i}$ are positive integers for all $i$ . We call these expansions $\operatorname{dn}_{N}$ expansions and show that every positive real number has infinitely many $\operatorname{dn}_{N}$ expansions for each $N$ . In particular, we study the $\operatorname{dn}_{N}$ expansions of rational numbers and quadratic irrationals. Finally, we show that every positive real number has, for each $N$ , a $\operatorname{dn}_{N}$ expansion with bounded partial numerators.

scholarly journals Sums of Products Involving Power Sums of φ(n) Integers

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Jitender Singh
Keyword(s):  
Closed Form ◽  
Complex Variable ◽  
Bernoulli Polynomials ◽  
Rational Numbers ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Power Sums ◽  
Power Sum ◽  
Mobius Function ◽  
Sums Of Products

A sequence of rational numbers as a generalization of the sequence of Bernoulli numbers is introduced. Sums of products involving the terms of this generalized sequence are then obtained using an application of Faà di Bruno's formula. These sums of products are analogous to the higher order Bernoulli numbers and are used to develop the closed form expressions for the sums of products involving the power sums Ψk(x,n):=∑d|n‍μ(d)dkSkx/d,  n∈ℤ+ which are defined via the Möbius function μ and the usual power sum Sk(x) of a real or complex variable x. The power sum Sk(x) is expressible in terms of the well-known Bernoulli polynomials by Sk(x):=(Bk+1(x+1)-Bk+1(1))/(k+1).

The range of the (n + 1)th moment for distributions on [0, 1]

1967 ◽  
Vol 4 (03) ◽  
pp. 543-552 ◽  
Author(s):  
Morris Skibinsky
Keyword(s):  
Positive Integer ◽  
Convex Hull ◽  
Higher Order ◽  
Unit Interval ◽  
Space Curve ◽  
Order Moment ◽  
Probability Measures ◽  
Image Position ◽  
Moment Spaces

Let p denote the class of all probability measures defined on the Borel subsets of the unit interval I = [0, 1]. For each positive integer n, take Mn is convex, closed, bounded, and n-dimensional; the convex hull of the space curve {(t,t2, …, tn ): 0 ≦ t ≦ 1}; e.g., see Theorems 7.2, 7.3 of [1]. At each point (c1, C2, …, cn ) of Mn , define Note that v −, v + depend only on C1, C2, …, Cn− 1; Vm only on cn ; We shall as notational convenience dictates and as will be apparent from the context regard v ± n as functions on Mn− 1 or on higher order moment spaces and also regard Vn as a function on moment spaces of order higher than n.

The range of the (n + 1)th moment for distributions on [0, 1]

1967 ◽  
Vol 4 (3) ◽  
pp. 543-552 ◽  
Author(s):  
Morris Skibinsky
Keyword(s):  
Positive Integer ◽  
Convex Hull ◽  
Higher Order ◽  
Unit Interval ◽  
Space Curve ◽  
Order Moment ◽  
Probability Measures ◽  
Image Position ◽  
Moment Spaces

Let p denote the class of all probability measures defined on the Borel subsets of the unit interval I = [0, 1]. For each positive integer n, take Mn is convex, closed, bounded, and n-dimensional; the convex hull of the space curve {(t,t2, …, tn): 0 ≦ t ≦ 1}; e.g., see Theorems 7.2, 7.3 of [1]. At each point (c1, C2, …, cn) of Mn, define Note that v−, v+ depend only on C1, C2, …, Cn− 1; Vm only on cn; We shall as notational convenience dictates and as will be apparent from the context regard v±n as functions on Mn− 1 or on higher order moment spaces and also regard Vn as a function on moment spaces of order higher than n.

Effects of Higher-Order Laue Zone reflections on structure images

1993 ◽  
Vol 51 ◽  
pp. 450-451
Author(s):  
H. S. Kim ◽  
S. S. Sheinin
Keyword(s):  
Wave Vector ◽  
Theoretical Calculations ◽  
Reciprocal Lattice ◽  
Image Plane ◽  
Bloch Wave ◽  
Dynamical Theory ◽  
Higher Order ◽  
Lattice Image ◽  
Lattice Vector ◽  
Image Position

The importance of image simulation in interpreting experimental lattice images is well established. Normally, in carrying out the required theoretical calculations, only zero order Laue zone reflections are taken into account. In this paper we assess the conditions for which this procedure is valid and indicate circumstances in which higher order Laue zone reflections may be important. Our work is based on an analysis of the requirements for obtaining structure images i.e. images directly related to the projected potential. In the considerations to follow, the Bloch wave formulation of the dynamical theory has been used.The intensity in a lattice image can be obtained from the total wave function at the image plane is given by: where ϕg(z) is the diffracted beam amplitide given by In these equations,the z direction is perpendicular to the entrance surface, g is a reciprocal lattice vector, the Cg(i) are Fourier coefficients in the expression for a Bloch wave, b(i), X(i) is the Bloch wave excitation coefficient, ϒ(i)=k(i)-K, k(i) is a Bloch wave vector, K is the electron wave vector after correction for the mean inner potential of the crystal, T(q) and D(q) are the transfer function and damping function respectively, q is a scattering vector and the summation is over i=l,N where N is the number of beams taken into account.

scholarly journals A New Class of Higher-Order Hypergeometric Bernoulli Polynomials Associated with Lagrange–Hermite Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 648
Author(s):  
Ghulam Muhiuddin ◽  
Waseem Ahmad Khan ◽  
Ugur Duran ◽  
Deena Al-Kadi
Keyword(s):  
Generating Function ◽  
Functional Equations ◽  
Laguerre Polynomials ◽  
Hermite Polynomials ◽  
Bernoulli Polynomials ◽  
Higher Order ◽  
New Class

The purpose of this paper is to construct a unified generating function involving the families of the higher-order hypergeometric Bernoulli polynomials and Lagrange–Hermite polynomials. Using the generating function and their functional equations, we investigate some properties of these polynomials. Moreover, we derive several connected formulas and relations including the Miller–Lee polynomials, the Laguerre polynomials, and the Lagrange Hermite–Miller–Lee polynomials.

scholarly journals Two-Variable Type 2 Poly-Fubini Polynomials

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 281
Author(s):  
Ghulam Muhiuddin ◽  
Waseem Ahmad Khan ◽  
Ugur Duran
Keyword(s):  
Integral Representation ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Higher Order ◽  
Variable Type ◽  
Summation Formulas

In the present work, a new extension of the two-variable Fubini polynomials is introduced by means of the polyexponential function, which is called the two-variable type 2 poly-Fubini polynomials. Then, some useful relations including the Stirling numbers of the second and the first kinds, the usual Fubini polynomials, and the higher-order Bernoulli polynomials are derived. Also, some summation formulas and an integral representation for type 2 poly-Fubini polynomials are investigated. Moreover, two-variable unipoly-Fubini polynomials are introduced utilizing the unipoly function, and diverse properties involving integral and derivative properties are attained. Furthermore, some relationships covering the two-variable unipoly-Fubini polynomials, the Stirling numbers of the second and the first kinds, and the Daehee polynomials are acquired.

scholarly journals Partitioning the positive integers with higher order recurrences

1991 ◽  
Vol 14 (3) ◽  
pp. 457-462 ◽  
Author(s):  
Clark Kimberling
Keyword(s):  
Positive Integer ◽  
Positive Root ◽  
Irrational Number ◽  
Higher Order ◽  
Typical Result ◽  
Algebraic Integers ◽  
Positive Integers

Associated with any irrational numberα>1and the functiong(n)=[αn+12]is an array{s(i,j)}of positive integers defined inductively as follows:s(1,1)=1,s(1,j)=g(s(1,j−1))for allj≥2,s(i,1)=the least positive integer not amongs(h,j)forh≤i−1fori≥2, ands(i,j)=g(s(i,j−1))forj≥2. This work considers algebraic integersαof degree≥3for which the rows of the arrays(i,j)partition the set of positive integers. Such an array is called a Stolarsky array. A typical result is the following (Corollary 2): ifαis the positive root ofxk−xk−1−…−x−1fork≥3, thens(i,j)is a Stolarsky array.

scholarly journals The Divisibility of Divisor Functions

1961 ◽  
Vol 5 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Positive Integer ◽  
Divisor Functions ◽  
Image Position ◽  
Positive Integers

For any positive integers n and v letwhere d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

The Distribution Of Totatives

1955 ◽  
Vol 7 ◽  
pp. 347-357 ◽  
Author(s):  
D. H. Lehmer
Keyword(s):  
Positive Integer ◽  
Prime Factors ◽  
Image Position

This paper is concerned with the numbers which are relatively prime to a given positive integerwhere the p's are the distinct prime factors of n. Since these numbers recur periodically with period n, it suffices to study the ϕ(n) numbers ≤n and relatively prime to n.

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