scholarly journals Perrin’s bivariate and complex polynomials

2021 ◽  
Vol 27 (2) ◽  
pp. 70-78
Author(s):  
Renata Passos Machado Vieira ◽  
Milena Carolina dos Santos Mangueira ◽  
Francisco Regis Vieira Alves ◽  
Paula Maria Machado Cruz Catarino
Keyword(s):  
Generating Function ◽  
Complex Polynomials ◽  
Bivariate Polynomials ◽  
Binet Formula ◽  
Generating Matrix ◽  
Definition Of

In this article, a study is carried out around the Perrin sequence, these numbers marked by their applicability and similarity with Padovan’s numbers. With that, we will present the recurrence for Perrin’s polynomials and also the definition of Perrin’s complex bivariate polynomials. From this, the recurrence of these numbers, their generating function, generating matrix and Binet formula are defined.

System of thermodynamic quantities in the McMillan-Mayer solution theory

1985 ◽  
Vol 50 (4) ◽  
pp. 791-798 ◽  
Author(s):  
Vilém Kodýtek
Keyword(s):  
Free Energy ◽  
Generating Function ◽  
Simple Form ◽  
Unit Volume ◽  
Thermodynamic Quantities ◽  
Solution Theory ◽  
Pure Solvent ◽  
Second Derivatives ◽  
Definition Of ◽  
Derivatives Of

The McMillan-Mayer (MM) free energy per unit volume of solution AMM, is employed as a generating function of the MM system of thermodynamic quantities for solutions in the state of osmotic equilibrium with pure solvent. This system can be defined by replacing the quantities G, T, P, and m in the definition of the Lewis-Randall (LR) system by AMM, T, P0, and c (P0 being the pure solvent pressure). Following this way the LR to MM conversion relations for the first derivatives of the free energy are obtained in a simple form. New relations are derived for its second derivatives.

On the symplectically invariant variational principle and generating functions

1990 ◽  
Vol 431 (1883) ◽  
pp. 403-417 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Variational Principle ◽  
Generating Function ◽  
Generating Functions ◽  
Geometric Approach ◽  
Canonical Transformations ◽  
Weyl Transform ◽  
Analogous Relation ◽  
Definition Of ◽  
Symplectic Invariance

The generating function for canonical transformations derived by Marinov has the important property of symplectic invariance (i. e. under linear canonical transformations). However, a more geometric approach to the rederivation of this function from the variational principle reveals that it is not free from caustic singularities after all. These singularities can be avoided without breaking the symplectic invariance by the definition of a complementary generating function bearing an analogous relation to the Woodward ambiguity function in telecommunications theory as that tying Marinov’s function to the Wigner function and the Weyl transform in quantum mechanics. Marinov’s function is specially apt to describe canonical transformations close to the identity, but breaks down for reflections through a point in phase space, easily described by the new generating function.

scholarly journals THE MOMENTS OF MINKOWSKI QUESTION MARK FUNCTION: THE DYADIC PERIOD FUNCTION

2009 ◽  
Vol 52 (1) ◽  
pp. 41-64 ◽  
Author(s):  
GIEDRIUS ALKAUSKAS
Keyword(s):  
Integral Equation ◽  
Generating Function ◽  
Period Function ◽  
Question Mark ◽  
Real Distribution ◽  
Exponential Generating Function ◽  
Wave Forms ◽  
Schmidt Operator ◽  
Definition Of ◽  
Minkowski Question Mark Function

AbstractThe Minkowski question mark function ?(x) arises as a real distribution of rationals in the Farey tree. We examine the generating function of moments of ?(x). It appears that the generating function is a direct dyadic analogue of period functions for Maass wave forms and it is defined in the cut plane \ (1, ∞). The exponential generating function satisfies an integral equation with kernel being the Bessel function. The solution of this integral equation leads to the definition of dyadic eigenfunctions, arising from a certain Hilbert–Schmidt operator. Finally, we describe p-adic distribution of rationals in the Stern–Brocot tree. Surprisingly, the Eisenstein series G2(z) does manifest in both real and p-adic cases.

scholarly journals A New Generalization of Jacobsthal Lucas Numbers (Bi-Periodic Jacobsthal Lucas Sequence)

2020 ◽  
pp. 1-13 ◽  
Author(s):  
Sukran Uygun ◽  
Evans Owusu
Keyword(s):  
Generating Function ◽  
Initial Conditions ◽  
Convergence Property ◽  
Lucas Numbers ◽  
Lucas Sequence ◽  
Summation Formulas ◽  
Binet Formula

In this study, we bring into light a new generalization of the Jacobsthal Lucas numbers, which shall also be called the bi-periodic Jacobsthal Lucas sequence as   with initial conditions $$\ \hat{c}_{0}=2,\ \hat{c}_{1}=a.$$ The Binet formula as well as the generating function for this sequence are given. The convergence property of the consecutive terms of this sequence is examined after which the well known Cassini, Catalan and the D'ocagne identities as well as some related summation formulas are also given.

scholarly journals Generalized Gaussian Jacobsthal and Generalized Gaussian Jacobsthal Lucas Polynomial

2019 ◽  
Vol 8 (6S3) ◽  
pp. 708-712
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Binet Formula ◽  
Q Matrix

The aim of the present paper is to present a generalization of Gaussian Jacobsthal polynomial and Gaussian Jacobsthal Lucas polynomial. Present paper extends the work of Asci and Gurel [3]. Some important generalizations of Generating function, Binet formula, Explicit formula, Q matrix and determinantal representations of these polynomials are also produced.

scholarly journals Construction of the Type 2 Poly-Frobenius-Genocchi Polynomials with Their Certain Applications

2020 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz ◽  
Serkan Araci
Keyword(s):  
Linear Combination ◽  
Generating Function ◽  
Bernoulli Polynomials ◽  
Stirling Numbers ◽  
Genocchi Polynomials ◽  
Definition Of ◽  
Special Case

Motivated by the definition of the type 2 poly-Bernoulli polynomials introduced by Kim-Kim, in the present paper, we consider a class of new generating function for the Frobenius-Genocchi polynomials, called the type 2 poly-Frobenius-Genocchi polynomials, by means of the polyexponential function. Then, we derive some useful relations and properties. We show that the type 2 poly-Frobenius-Genocchi polynomias equal a linear combination of the classical Frobenius-Genocchi polynomials and Stirling numbers of the first kind. In a special case, we give a relation between the type 2 poly-Frobenius-Genocchi polynomials and Bernoulli polynomials of order k. Moreover, inspired by the definition of the unipoly-Bernoulli polynomials introduced by Kim-Kim, we introduce the unipoly-Frobenius-Genocchi polynomials by means of unipoly function and give multifarious properties including derivative and integral properties. Furthermore, we provide a correlation between the unipoly-Frobenius-Genocchi polynomials and the classical Frobenius-Genocchi polynomials.

scholarly journals ON THE BINOMIAL TRANSFORMS OF THE HORADAM QUATERNION SEQUENCES

2020 ◽  
Author(s):  
Faruk Kaplan ◽  
Arzu Özkoç Öztürk
Keyword(s):  
Recurrence Relation ◽  
General Formula ◽  
Generating Function ◽  
Second Order ◽  
Binet Formula ◽  
New Type ◽  
Binomial Transform

The main object of the present paper is to consider the binomial transforms for Horadam quaternion sequences. We gave new formulas for recurrence relation, generating function, Binet formula and some basic identities for the binomial sequence of Horadam quaternions. Working with Horadam quaternions, we have found the most general formula that includes all binomial transforms with recurrence relation from the second order. In the last part, we determined the recurrence relation for this new type of quaternion by working with the iterated binomial transform, which is a dierent type of binomial transform.

scholarly journals Matrix Representation of Bi-Periodic Jacobsthal Sequence

2020 ◽  
pp. 1-12
Author(s):  
Sukran Uygun ◽  
Evans Owusu
Keyword(s):  
Generating Function ◽  
Initial Conditions ◽  
Matrix Representation ◽  
General Term ◽  
Identity Matrix ◽  
Summation Formulas ◽  
Matrix Sequence ◽  
Binet Formula ◽  
The Matrix ◽  
Generalized Matrix

In this paper, we bring into light the matrix representation of bi-periodic Jacobsthal sequence, which we shall call the bi-periodic Jacobsthal matrix sequence. We dene it as with initial conditions J0 = I identity matrix, . We obtained the nth general term of this new matrix sequence. By studying the properties of this new matrix sequence, the well-known Cassini or Simpson's formula was obtained. We then proceed to find its generating function as well as the Binet formula. Some new properties and two summation formulas for this new generalized matrix sequence were also given.

scholarly journals Induction Models on $\mathbb{N}$

10.29007/kvp3 ◽  
2020 ◽  
Author(s):  
A. Dileep ◽  
Kuldeep S. Meel ◽  
Ammar F. Sabili
Keyword(s):  
Computer Science ◽  
Generating Function ◽  
Generating Functions ◽  
Mathematical Induction ◽  
Base Case ◽  
And Mathematics ◽  
Definition Of ◽  
Induction Model

Mathematical induction is a fundamental tool in computer science and mathematics. Henkin [12] initiated the study of formalization of mathematical induction restricted to the setting when the base case B is set to singleton set containing 0 and a unary generating function S. The usage of mathematical induction often involves wider set of base cases and k−ary generating functions with different structural restrictions. While subsequent studies have shown several Induction Models to be equivalent, there does not exist precise logical characterization of reduction and equivalence among different Induction Models. In this paper, we generalize the definition of Induction Model and demonstrate existence and construction of S for given B and vice versa. We then provide a formal characterization of the reduction among different Induction Models that can allow proofs in one Induction Models to be expressed as proofs in another Induction Models. The notion of reduction allows us to capture equivalence among Induction Models.

scholarly journals Matrix Manipulations for Properties of Pell p-Numbers and their Generalizations

2020 ◽  
Vol 28 (3) ◽  
pp. 89-102
Author(s):  
Özgür Erdağ ◽  
Ömür Deveci ◽  
Anthony G. Shannon
Keyword(s):  
Binet Formula ◽  
Generating Matrix ◽  
Combinatorial Representation

AbstractIn this paper, we define the Pell-Pell p-sequence and then we discuss the connection of the Pell-Pell p-sequence with Pell and Pell p-sequences. Also, we provide a new Binet formula and a new combinatorial representation of the Pell-Pell p-numbers by the aid of the nth power of the generating matrix the Pell-Pell p-sequence. Furthermore, we obtain an exponential representation of the Pell-Pell p-numbers and we develop relationships between the Pell-Pell p-numbers and their permanent, determinant and sums of certain matrices.

Sign in / Sign up

Export Citation Format

Share Document