José Luis López Bonilla

National Polytechnic Institute | IPN

We construct the sequences of Fibonacci and Lucas in any quadratic field $\mathbb{Q}(\sqrt{d}\,)$ with $d>0$ square free, noting that the general properties remain valid as those given by the classical sequences of Fibonacci and Lucas for the case $d = 5$, under the respective variants. For this construction, we use the fundamental unit of $\mathbb...

In this paper, we show that a specific extension of the Schrödinger ansatz for two free particles, under the requirement that this function (ansatz) be analytical, is compatible with the expected physical result for a quantum description at the quantum-classical boundary ; that is, its total erasure in the transition to a description compatible wit...

In this paper, we constructed a solution of the Schrödinger equation for the ammonia molecule, modeled as a particle with a permanent electric dipole moment, which can access only two quantum states, as part of a beam of these molecules entering a region where an electrostatic field with a weak gradient act. In this solution, the contribution of th...

In this paper, we consider a matrix approach to the simple pendulum system. Specifically, we define the following inverse problem: Construct a “mass matrix” whose eigenvalues correspond to the value of the mass of the pendulum particle. In subsequent development, we find solutions with degeneracy in the eigenvalues of the mass matrix.

In this paper, we give some interesting congruences involving harmonic numbers and sequences in the form $W_n = xa^n+yb^n$, related to Fermat quotients, such as $\sum_{n=1}^m W_n H_n \equiv f (mod p)$, for the cases $m = p−1$ and $m = \frac{(p−1)}{2}$, where $q_p(a) = \frac{a^{p−1}−1}{p}$, $a,b \in Z−pZ$.

We employ the Z-transform to obtain a recurrence relation to determine the coefficients \(g_n\) in: \[ \left(\sum_{k=0}^\infty f_k x^k \right)^p = \sum_{n=0}^\infty g_n x^n, \] in terms of the quantities \(f_k\), where \(p\) is an arbitrary real or complex number.

In this paper, we establish the role of Euler's operator in the Sun's binomial inversion formulae and arrive at useful results.

In this article, we present mathematical simulations of non-separable functions (those that would "correspond" to two entangled quantum particles) that lose this character only as a result of approaching the quantum-classical frontier. No mathematical representation of the action of deteriorating agents of quantum entanglement was included in the s...

We show that the Hermite polynomials and the number of self-conjugate permutations of {1, 2, …, n} can be written in terms of Bell polynomials.

We know that the resolvent of a matrix is the Laplace transform of exp(), which allow us to give an elementary deduction of Balakrishnan's formula for the fractional power of a matrix.

After introducing the famous Nörlund-Rice integral formula, we apply it to Laguerre polynomials, Melzak’s relation, and Stirling numbers of the second kind to obtain nice expressions.

Abstract. In this paper, considering the usual Schrödinger equation, in the one-dimensional case, and assuming a certain formal expression for the potential, which is written in terms of wave functions, we identify what would be the complementary equation to be verified by said functions, which would lead the potential to be a solution of the Korte...

In this article, which celebrates 100 years of the Stern-Gerlach experiment, we identify and discuss some limitations of the mathematical field we intend to represent as the Stern-Gerlach magnetic field. We extend some recent theoretical results concerning Stern-Gerlach eigenenergies, where what we call "the gradient effect" manifests itself: the c...

In this short communication we will explore Bell's exponential function in order to obtain new identities for a series involving the function pk(n), the number of partitions of n into k colors. Keywords: k-colored partitions, Convergent series, Bell polynomials.

We employ the Z-transform to study a result of Merca involving the partition function $p(n)$ and Euler's totient. Besides, we obtain an identity valid for pentagonal numbers and an arbitrary prime number. Keywords: Partition function, Dirichlet character, Euler totient function, Z-transform, Möbius function.

We exhibit the participation of Euler's operator in the Sun's binomial inversion formulae and we use the Z-transform to give elementary proofs of such formulae.

The knowledge of how the human motions is performed helps to understand how the human body works. This paper presents a method to estimate the human limbs angles through a kinematic model depicted by Roll-Pitch-Yaw rotationmatrix and the mimic of those angles on a humanoid robot. The advantage of this model is the detailed representation of each jo...

We show determinantal identities verified by (),the number of ways that a positive integer n can be written as sum of k squares, which are implied by the polynomial structure of ().

We exhibit that the 3-rotations and Lorentz mappings generate Möbius transformations in the complex plane. Resumen. En este trabajo se muestra que las 3-rotaciones y los mapeos de Lorentz generan transformaciones de Möbius en el plano complejo. Möbius transformation and the Riemann sphere.

In this article, we present a simple proof of a mathematical result that is presented in some books without proof. This is the electrostatic equivalence between two distributions of electrical charge: (1) On the one hand, the one resulting from the superposition of two dielectric spheres, with volumetric densities of electrical charge equal to "+ρ....

In this article, we emphasize some aspects related to the difficulty encountered when identifying physical entanglement in quantum systems. We offer some comments that can alert us when it comes to distinguishing the purely mathematical aspects of entanglement from the physical aspects in real-world quantum systems. We show the case of an unusual t...

Abstract. In this article, we present a simple proof of a mathematical result that is presented in some books without proof. This is the electrostatic equivalence between two distributions of electrical charge: (1) On the one hand, the one resulting from the superposition of two dielectric spheres, with volumetric densities of electrical charge equ...

We show an identity involving Stirling numbers which can be considered the companion relation of the Akiyama-Tanigawa's formula.

Mestrovic [1] proved the following congruence: ∑ () ()() (1) For any prime where () is the Fermat quotient of p to base 2. In this brief Note, we use a result of Sun [2] to obtain an extensión of (1) involving Bernoulli numbers [3].

In this article, we show two independent proofs, which use of different properties of the Lanczos-Dirac Delta, of which, the Delta cannot be separable. Furthermore, the conditions that should be imposed on an ordinary function are identified so that it presents the translation property of the Lanczos-Dirac Delta.

In this article, we will answer a question posed in the book Classical Mechanics by H. Goldstein: "Is the Hamilton-Jacobi equation the short wavelength limit of the Schrödinger equation?" But, before that, we will identify an essential element that will take us from the Hamilton-Jacobi equation to the dynamic equation of non-relativistic quantum me...

It is known a recurrence relation for the Ramanujan’s tau-function involving the sum of divisors function𝝈(𝒏), whose solution gives a closed formula for 𝝉(𝒏) in terms of complete Bell polynomials, and a determinantal expression for 𝝈(𝒎) where participate the values 𝝉(𝒌).

Diophantine Equations named after ancient Greek mathematician Diophantus, plays a vital role not only in number theory but also in several branches of science. In this paper, we will solve one of the quadratic Diophantine equations where the right hand side are odd positive integral powers of 17 and provide its complete solutions. The method adopte...

We study certain type of convolution sums involving an arbitrary arithmetic function f, which it is applied to Ramanujan’s tau function when f coincides with the sum of divisors function.

An arithmetic function q is specially multiplicative if it is the Dirichlet product of two completely multiplicative functions, thus for a fixed prime number p, we know that the sequence {

Diophantine Equations has been of special interest to many mathematicians for several centuries. Equations whose solutions are expected to be in integers are usually termed as Diophantine equations. Thousands of equations of Diophantine type has been solved by several mathematicians. Some equations have created a great legacy like Fermat's Last The...

We derive some congruences for r k (n) and t k (n) using their generating functions, where r k (n) and t k (n) denote the number of representations of n as a sum of k squares and number of representations of n as a sum of k triangular numbers, respectively. The sum of squares function, denoted by r k (n), gives the number of representations of n as...

In this article, we represent a recurrence relation of the arithmetic function connected with an ascending factorial function, Lah and Stirling numbers. We then obtain a relation of harmonic numbers and again extend the coefficients of these arithmetic functions involving Bell polynomials through introducing the sequence of Hankel type integrals. O...

We show that certain type of generating functions allow to construct Cauchy convolutions for the Apostol-Euler and Apostol-Bernoulli polynomials and their connection with partial Bell Polynomials.

In this article, we consider the polynomials consisting of the coefficients of complete and the partial Bell polynomials and the polynomial expressions for the arithmetic functions and , the number of representations of n as a sum of k triangular numbers and k squares, respectively, and also the color partitions Then making an appeal to the Fourier...

In this article, we consider the polynomials consisting of the coefficients of complete and the partial Bell polynomials and the polynomial expressions for the arithmetic functions and , the number of representations of n as a sum of k triangular numbers and k squares, respectively, and also the color partitions Then making an appeal to the Fourier...

The main results of our paper are relations involving sums of Multiple Zeta Riemann functions (MZVs). Using the techniques described here, we found a simple relation for z({a}n) = z(a, a, ..., a^n) for a positive integer n and a complex a where Re{a} > 1.

In this paper, a mathematical simulation for the simplest version of the Zeeman Effect of a magnetic field is presented. In this simulation, a matrix treatment of an inverse problem is adopted. The problem that we will address focuses on determining a matrix from a set of numbers, which, by imposition, will be the eigenvalues of this matrix. Starti...

In the paper, the authors simply review recent results of inequalities, monotonicity, signs of determinants, determinantal formulas, closed-form expressions, and identities of the Bernoulli numbers and polynomials, establish an identity involving the differences between the Bernoulli polyno-mials and the Bernoulli numbers, present two identities am...

Resumo. Neste artigo, consideramos a equação de Von Neumann com o objetivo de redefini-la de tal forma que ela passe de um contexto matricial (com matrizes de dimensão finita de ordem $N×N$) para um contexto vetorial-matricial, em termos de um vetor coluna (de ordem $N^2 ×1$) e uma matriz maior (de ordem $N^2$ × $N^2$). Com esse procedimento, passa...

En este artículo se presenta una simulación matemática de la versión más simple del efecto Zeeman de un campo magnético. En esta simulación, se adopta un tratamiento matricial de un problema inverso. El problema consiste en determinar una matriz a partir de un conjunto de números que, por imposición, serán los valores propios de dicha matriz. Parti...

In this paper, we exhibit a connection between the sum of divisors function and the number of representations of a positive integer as a sum of squares.

In this article, we show two independent proofs, which use of different properties of the Lanczos-Dirac Delta, of which, the Delta cannot be separable. Furthermore, the conditions that should be imposed on an ordinary function are identified so that it presents the translation property of the Lanczos-Dirac Delta.

We show that the Kuznetsov's polynomials can be written in terms of the complete Bell polynomials, and we deduce a direct relationship between the and the Bernoulli numbers.

We use the Z-transform to motivate the Baldoni et al algorithm to solve homogeneous linear recurrence relations, with applications to Fibonacci numbers and Chebyshev polynomials.

We show an alternative proof of an identity given by Persson-Strang for the well known Legendre polynomials.

We obtain a recurrence relation for the Ramanujan’s tau-function involving the sum of divisors function, and the solution of this recurrence gives a closed formula for 𝝉(𝒏) in terms of the complete Bell polynomials

If 𝒓𝒌(𝒏) is the number of representations of a positive integer n as the sum of k squares, then 𝒓𝒌(𝒏) is a polynomial in k of degree n; here we exhibit expressions for certain coefficients through this polynomial.

In this paper, considering a 1-dimensional physical system and Weierstrass’s polynomial approximation theorem, a polynomial version of Hamilton’s principle is constructed, in which, instead of considering numerous continuous curves that connect two fixed points, polynomials are considered representative of these curves. As a result of the mathemati...

This is an Open Access Journal / article distributed under the terms of the Creative Commons Attribution License (CC BY-NC-ND 3.0) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. All rights reserved. Apostol's theorem is applied to Euler totient and Möbius functions.

For arbitrary geometries with Petrov types O, N, III, and D (empty), we construct the Andersson-Edgar’s generator for the Lanczos spinor.

In this paper we perform the exact decoupling of a coupled system of two 1-dimensional stationary Schroedinger equations that were originally considered by P.L. Christiansen, J.J. Rasmussen and M.P. Sorensen. Based on the characteristics of the uncoupled equations found, we argue that the coupled system of equations is spurious for the problem addr...

In this paper we analyze a Klein-Gordon equation, which arises in the context of the physics of antiferromagnetic magnons, in order to determine whether a beam of linearly polarized spin wave could be separated into two secondary beams, in each of which one can find both a left and a right circularly polarized mode, which can be called (similar to...

We show that the recurrence relation deduced by Robbins and Osler et al for the sum of divisors function can be solved in terms of the complete Bell polynomials. Besides, the connection between and the number of representations of n as the sum of four triangular numbers allows obtain arecurrence relation where only participate the values of with m...

For several orthogonal polynomials, Cohen proved that their roots are the eigenvalues of symmetric tridiagonal matrices. In this paper, we give examples of this Cohen’s result for the Legendre, Laguerre, and Hermite polynomials, which are useful in applications to quantum mechanics and numerical analysis.

We employ a method of factorization to obtain the general solution of the second order linear differential equation, which is an alternative procedure to the usual Variation of Parameters method of Lagrange. We consider that our approach can be adapted to linear differential equations of the third and fourth order.

The concept of Fourier Series is widely used in several Engineering problems like Wave Equations, Heat Equations, Laplace Equations, Signal Processing and much more. The concept of Discrete Fourier Transforms is the equivalent of continuous Fourier Transforms (DFT) for signals transmitted at finite number of points. The interpolation process allows...

We use the Z-transform to solve a type of recurrence relation satisfied by the number of representations of an integer n as the sum of squares and as the sum of triangular numbers, and also by the color partitions of n; the corresponding solution is in terms of the complete Bell polynomials. Keywords: sum of divisors function, Z-transform, sums of...

We use the Apostol-Robbins theorem to determine for several arithmetic functions of interest in number theory

By considering a triangular array of numbers for first six rows, we introduce a combinatorial game, whose solution depends on a number triangle. The conclusion brings us with a surprising consequence in deciding the result of the game. This paper analyzes the game and present the solution in detail using the number triangle which resembles the famo...

We consider two recurrence relations for the number of representations of an integer as sums of squares, and we show that the corresponding coefficients into such recurrences can be written in terms of theta functions.

The study of numbers has fascinated humans for more than two millennia and it continues to do so forever. Over the period of time, several mathematicians have contributed enormously to the growth of number theory as we know today. Arithmetic functions play a significant role in understanding of numbers. Divisors function, Sum of divisors function,...

We exhibit a relationship between q-shifted factorial, (q; q)n, and the incomplete exponential Bell polynomials and also evaluate several q-hypergeometric series using the q-version of Petkovsek-WilfZeilberger’s algorithm. Finally, we write the partition function p(n) in terms of Qm(k), the number of partitions of m using (possibly repeated) parts...

We study the evolution of the Lorentz mapping, and of its associated unimodular complex matrix, between two Frenet-Serret's tetrads on an arbitrary time-like world line.

We exhibit polynomial expressions for the arithmetic functions () and (), the number of representations of n as a sum of k squares and k triangular numbers, respectively, and also for the color partitions ().

Here we exhibit alternative proofs of the identities given by Persson-Strang and (Huat-Chan)-Wan-Zudilin for the Legendre polynomials. Besides, we show the connection between the Lanczos derivative and these polynomials via the Rangarajan-Purushothaman’s formula.

We exhibit polynomial expressions for the arithmetic functions and , the number of representations of n as a sum of k squares and k triangular numbers, respectively, and also for the color partitions 2020 Mathematics Subject Classification:-11B73, 11P70, 11P99.

In the present paper, we show that the partial Bell polynomials allow for obtaining identities involving the generalized Bernoulli numbers. Then, on applying these identities we derive different generating and bilateral generating functions. 2020 Mathematical Sciences Classification: 05A15, 05A19, 05A30.

In this article, we represent a recurrence relation of the arithmetic function connected with an ascending factorial function, Lah and Stirling numbers. We then obtain a relation of harmonic numbers and again extend the coefficients of these arithmetic functions involving Bell polynomials through introducing the sequence of Hankel type integrals. O...

We exhibit that the coefficients of the characteristic polynomial of any matrix Anxn can be written in terms of the complete Bell polynomials, and this result is applied to Chebyshev matrices which generates the concept of Associated Polynomials of Chebyshev.

We exhibit that the coefficients of the characteristic polynomial of any matrix Anxn can be written in terms of the complete Bell polynomials, and this result is applied to Chebyshev matrices which generates the concept of Associated Polynomials of Chebyshev.

In this article, we represent a recurrence relation of the arithmetic function connected with an ascending factorial function, Lah and Stirling numbers. We then obtain a relation of harmonic numbers and again extend the coecients of these arithmetic functions involving Bell polynomials through introducing the sequence of Hankel type integrals. On t...

We exhibit a recurrence relation for $r_k (n)$, that is, for the number of representations of a positive integer as a sum of squares, so it is possible to determine $r_k (n)$ if we know $r_k (m) , m=0, 1, 2,\ldots, n-1$.

We show that, in Minkowski spacetime, any Faraday tensor verifying the Maxwell equations in vacuum allows construct a tensor of 4th order with the same algebraic symmetries than the Riemann tensor, which implies the presence of a third-order tensor with the algebraic and differential properties of the Lanczos generator, and finally this indicates t...

We exhibit a relationship between $q-$shifted factorial, $(q; q)_n$, and the incomplete exponential Bell polynomials and also evaluate several $q-$hypergeometric series using the $q-$version of Petkovsek-Wilf-Zeilberger's algorithm. Finally, we write the partition function $p(n)$ in terms of $Q_m(k)$, the number of partitions of $m$ using (possibly...

In this paper, we adopt a matrix treatment to solve the variational problem that consists of determining the physical path traveled by light between two points in a medium whose refractive index depends on a spatial coordinate. The considered treatment begins with the trivial repetition of the expression of the value of the considered functional, r...

In this paper, we adopt a matrix treatment to solve the variational problem that consists of determining the physical path traveled by light between two points in a medium whose refractive index depends on a spatial coordinate. The considered treatment begins with the trivial repetition of the expression of the value of the considered functional, r...

We establish the generalized $n^{th}$-Order Opial's integral inequality via (p,q)-calculus with some extensions. The other analytical tools used to establish the results were $(p,q)$-Cauchy repeated integration formula and $(p,q)$-Cauchy-Schwarz's integral inequality.

In this paper, we exhibit short deductions of the Jha and Malenfant expressions for the partition function, and we show several connections between the Euler, divisor and partition functions via the partial Bell polynomials and Hessenberg determinants. Besides, we show that the Gandhi's recurrence relation for colour partitions implies the Osler-Ha...

We exhibit representations of the coefficients of the characteristic polynomial of any matrix An×n, especially in terms of the (exponential) complete Bell polynomials. Besides, we use the Faddeev-Sominsky method to obtain the Lanczos formula for the resolvent of a matrix. We indicate that the Newton's recurrence formula can be solved via the invers...

We apply the binomial coefficient polynomial to write the harmonic numbers in terms of Stirling numbers of the first kind, and we employ the Melzak's identity to exhibit representations of. We also deduce identities involving the harmonic numbers via their connection with the derivatives of binomial coefficients.

En este artículo adoptamos un tratamiento matricial para resolver el problema variacional que consiste en determinar el camino físico tomado por la luz entre dos puntos en un medio cuyoíndice de refracción depende de una coordenada espacial. El tratamiento considerado comienza con la repetición trivial de la expresión del valor del funcional consid...

In this paper, the factorization of 3-rotation matrices in terms of the Euler-Olinde Rodrigues parameters from the generation of Lorentz transformations through quaternions is obtained.

We study combinatorial identities associated with the normalized binomial mid-coefficients and the self-conjugate permutations.

The article has to motive to derive new class of differential equations and associated integral equations for some hybrid family of truncated-exponential-based Appell polynomials. We derive the recurrence relation, differential equations, integral equations and integro-differential equations of the truncated exponential polynomials by using the fac...

We show an alternative deduction for the Rathie-Paris formula satisfied by the Gauss hypergeometric function.

The exactly solvable Position Dependent Mass Schrödinger Equation (PDMSE) for Mie-type potentials is presented. To that, by means of a point canonical transformation the exactly solvable constant mass Schrödinger equation is transformed into a PDMSE. The mapping between both Schrödinger equations lets obtain the energy spectra and wave functions fo...

In recent years SCMA has been proposed as a strong candidate to perform as access technique for fifth-generation mobile communications systems; usually for the uplink, bit pairs are mapped directly to code words and MPA is used in the receiver. In this work, we proposed to encode three bits to increase the capacity of data transmission in the uplin...