scholarly journals Some cut-generating functions for second-order conic sets

2017 ◽  
Vol 24 ◽  
pp. 51-65 ◽  
Author(s):  
Asteroide Santana ◽  
Santanu S. Dey
Keyword(s):  
Generating Functions ◽  
Second Order

scholarly journals General Linear Recurrence Sequences and Their Convolution Formulas

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 132 ◽  
Author(s):  
Paolo Emilio Ricci ◽  
Pierpaolo Natalini
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Second Order ◽  
General Linear ◽  
Linear Recurrence ◽  
Order Linear ◽  
Matrix Polynomials ◽  
Recurrence Sequences ◽  
Straightforward Extension

We extend a technique recently introduced by Chen Zhuoyu and Qi Lan in order to find convolution formulas for second order linear recurrence polynomials generated by 1 1 + a t + b t 2 x . The case of generating functions containing parameters, even in the numerator is considered. Convolution formulas and general recurrence relations are derived. Many illustrative examples and a straightforward extension to the case of matrix polynomials are shown.

Fibonacci and Lucas polynomials

1981 ◽  
Vol 90 (3) ◽  
pp. 385-387 ◽  
Author(s):  
B. G. S. Doman ◽  
J. K. Williams
Keyword(s):  
Difference Equation ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Linear Difference Equation ◽  
Second Order ◽  
Lucas Numbers ◽  
Order Linear ◽  
Lucas Polynomials ◽  
Fibonacci And Lucas Numbers

The Fibonacci and Lucas polynomials Fn(z) and Ln(z) are denned. These reduce to the familiar Fibonacci and Lucas numbers when z = 1. The polynomials are shown to satisfy a second order linear difference equation. Generating functions are derived, and also various simple identities, and relations with hypergeometric functions, Gegenbauer and Chebyshev polynomials.

scholarly journals Mersenne-Horadam identities using generating functions

2020 ◽  
Vol 12 (1) ◽  
pp. 34-45
Author(s):  
R. Frontczak ◽  
T. Goy
Keyword(s):  
Generating Functions ◽  
Second Order ◽  
Linear Recurrence ◽  
Order Linear ◽  
Balancing Numbers ◽  
Mersenne Numbers

The main object of the present paper is to reveal connections between Mersenne numbers $M_n=2^n-1$ and generalized Fibonacci (i.e., Horadam) numbers $w_n$ defined by a second order linear recurrence $w_n=pw_{n-1}+qw_{n-2}$, $n\geq 2$, with $w_0=a$ and $w_1=b$, where $a$, $b$, $p>0$ and $q\ne0$ are integers. This is achieved by relating the respective (ordinary and exponential) generating functions to each other. Several explicit examples involving Fibonacci, Lucas, Pell, Jacobsthal and balancing numbers are stated to highlight the results.

scholarly journals On Generating Functions

1930 ◽  
Vol 2 (2) ◽  
pp. 71-82 ◽  
Author(s):  
W. L. Ferrar
Keyword(s):  
Differential Equation ◽  
Recurrence Relation ◽  
Generating Function ◽  
Generating Functions ◽  
Order Differential Equation ◽  
Second Order ◽  
Second Order Differential Equation ◽  
Image Position ◽  
Three Term Recurrence Relation ◽  
Sequence Of Functions

It is well known that the polynomial in x,has the following properties:—(A) it is the coefficient of tn in the expansion of (1–2xt+t2)–½;(B) it satisfies the three-term recurrence relation(C) it is the solution of the second order differential equation(D) the sequence Pn(x) is orthogonal for the interval (— 1, 1),i.e. whenSeveral other familiar polynomials, e.g., those of Laguerre Hermite, Tschebyscheff, have properties similar to some or all of the above. The aim of the present paper is to examine whether, given a sequence of functions (polynomials or not) which has one of these properties, the others follow from it : in other words we propose to examine the inter-relation of the four properties. Actually we relate each property to the generating function.

Hamiltonian Mechanics for Functionals Involving Second-Order Derivatives

2002 ◽  
Vol 69 (6) ◽  
pp. 749-754 ◽  
Author(s):  
B. Tabarrok ◽  
C. M. Leech
Keyword(s):  
Differential Equations ◽  
Generating Functions ◽  
Jacobi Equation ◽  
Equations Of Motion ◽  
Second Order ◽  
Hamiltonian Mechanics ◽  
Energy Functionals ◽  
First Order ◽  
Independent Variable ◽  
Hamilton Jacobi Equation

Hamilton’s principle was developed for the modeling of dynamic systems in which time is the principal independent variable and the resulting equations of motion are second-order differential equations. This principle uses kinetic energy which is functionally dependent on first-order time derivatives, and potential energy, and has been extended to include virtual work. In this paper, a variant of Hamiltonian mechanics for systems whose motion is governed by fourth-order differential equations is developed and is illustrated by an example invoking the flexural analysis of beams. The variational formulations previously associated with Newton’s second-order equations of motion have been generalized to encompass problems governed by energy functionals involving second-order derivatives. The canonical equations associated with functionals with second order derivatives emerge as four first-order equations in each variable. The transformations of these equations to a new system wherein the generalized variables and momenta appear as constants, can be obtained through several different forms of generating functions. The generating functions are obtained as solutions of the Hamilton-Jacobi equation. This theory is illustrated by application to an example from beam theory the solution recovered using a technique for solving nonseparable forms of the Hamilton-Jacobi equation. Finally whereas classical variational mechanics uses time as the primary independent variable, here the theory is extended to include static mechanics problems in which the primary independent variable is spatial.

scholarly journals Monadic Second-Order Classes of Forests with a Monadic Second-Order 0-1 Law

2012 ◽  
Vol Vol. 14 no. 1 (Automata, Logic and Semantics) ◽  
Author(s):  
Jason P. Bell ◽  
Stanley N. Burris ◽  
Karen A. Yeats
Keyword(s):  
Explicit Expression ◽  
Generating Function ◽  
Generating Functions ◽  
Second Order ◽  
Radius Of Convergence ◽  
Structure Theory ◽  
Strengthened Version ◽  
International Audience

Automata, Logic and Semantics International audience Let T be a monadic-second order class of finite trees, and let T(x) be its (ordinary) generating function, with radius of convergence rho. If rho >= 1 then T has an explicit specification (without using recursion) in terms of the operations of union, sum, stack, and the multiset operators n and (>= n). Using this, one has an explicit expression for T(x) in terms of the initial functions x and x . (1 - x(n))(-1), the operations of addition and multiplication, and the Polya exponentiation operators E-n, E-(>= n). Let F be a monadic-second order class of finite forests, and let F (x) = Sigma(n) integral(n)x(n) be its (ordinary) generating function. Suppose F is closed under extraction of component trees and sums of forests. Using the above-mentioned structure theory for the class T of trees in F, Compton's theory of 0-1 laws, and a significantly strengthened version of 2003 results of Bell and Burris on generating functions, we show that F has a monadic second-order 0-1 law iff the radius of convergence of F (x) is 1 iff the radius of convergence of T (x) is >= 1.

scholarly journals Some problems of Analytic number theory IV

2002 ◽  
Vol Volume 25 ◽  
Author(s):  
R Balasubramanian ◽  
K Ramachandra
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Asymptotic Formula ◽  
Kernel Function ◽  
Generating Functions ◽  
Analytic Number Theory ◽  
Second Order ◽  
Analytic Number ◽  
International Audience ◽  
Error Terms

International audience In the present paper, we use Ramachandra's kernel function of the second order, namely ${\rm Exp} ((\sin z)^2)$, which has some advantages over the earlier kernel ${\rm Exp} (z^{4a+2})$ where $a$ is a positive integer. As an outcome of the new kernel we are able to handle $\Omega$-theorems for error terms in the asymptotic formula for the summatory function of the coefficients of generating functions of the ${\rm Exp}(\zeta(s)), {\rm Exp\,Exp}(\zeta(s))$ and also of the type ${\rm Exp\,Exp}((\zeta(s))^{\frac{1}{2}})$.

scholarly journals On Third-Order Bronze Fibonacci Numbers

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2606
Author(s):  
Mücahit Akbiyik ◽  
Jeta Alo
Keyword(s):  
Generating Functions ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Second Order ◽  
Matrix Representations ◽  
Third Order ◽  
The Third ◽  
Encryption And Decryption ◽  
Fibonacci Sequences

In this study, we firstly obtain De Moivre-type identities for the second-order Bronze Fibonacci sequences. Next, we construct and define the third-order Bronze Fibonacci, third-order Bronze Lucas and modified third-order Bronze Fibonacci sequences. Then, we define the generalized third-order Bronze Fibonacci sequence and calculate the De Moivre-type identities for these sequences. Moreover, we find the generating functions, Binet’s formulas, Cassini’s identities and matrix representations of these sequences and examine some interesting identities related to the third-order Bronze Fibonacci sequences. Finally, we present an encryption and decryption application that uses our obtained results and we present an illustrative example.

Disappearance Voltage Determinations Using a Convergent Beam

1971 ◽  
Vol 29 ◽  
pp. 184-185
Author(s):  
W. L. Bell
Keyword(s):  
Second Order ◽  
Objective Method ◽  
Uniform Thickness ◽  
Rocking Curve ◽  
Crystal Perfection ◽  
Kikuchi Line ◽  
Central Maximum ◽  
Diffraction Patterns ◽  
Convergent Beam ◽  
Beam Technique

Disappearance voltages for second order reflections can be determined experimentally in a variety of ways. The more subjective methods, such as Kikuchi line disappearance and bend contour imaging, involve comparing a series of diffraction patterns or micrographs taken at intervals throughout the disappearance range and selecting that voltage which gives the strongest disappearance effect. The estimated accuracies of these methods are both to within 10 kV, or about 2-4%, of the true disappearance voltage, which is quite sufficient for using these voltages in further calculations. However, it is the necessity of determining this information by comparisons of exposed plates rather than while operating the microscope that detracts from the immediate usefulness of these methods if there is reason to perform experiments at an unknown disappearance voltage.The convergent beam technique for determining the disappearance voltage has been found to be a highly objective method when it is applicable, i.e. when reasonable crystal perfection exists and an area of uniform thickness can be found. The criterion for determining this voltage is that the central maximum disappear from the rocking curve for the second order spot.

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