Fourier Series of the Periodic Bernoulli and Euler Functions

We give some properties of the periodic Bernoulli functions and study the Fourier series of the periodic Euler functions which are derived periodic functions from the Euler polynomials. And we derive the relations between the periodic Bernoulli functions and those from Euler polynomials by using the Fourier series

Appell and Sheffer sequences: on their characterizations through functionals and examples

The aim of this paper is to present a new simple recurrence for Appell and Sheffer sequences in terms of the linear functional that defines them, and to explain how this is equivalent to several well-known characterizations appearing in the literature. We also give several examples, including integral representations of the inverse operators associated to Bernoulli and Euler polynomials, and a new integral representation of the re-scaled Hermite $d$-orthogonal polynomials generalizing the Weierstrass operator related to the Hermite polynomials

Revisiting the Formula for the Ramanujan Constant of a Series

The main contribution of this paper is to propose a closed expression for the Ramanujan constant of alternating series, based on the Euler–Boole summation formula. Such an expression is not present in the literature. We also highlight the only choice for the parameter a in the formula proposed by Hardy for a series of positive terms, so the value obtained as the Ramanujan constant agrees with other summation methods for divergent series. Additionally, we derive the closed-formula for the Ramanujan constant of a series with the parameter chosen, under a natural interpretation of the integral term in the Euler–Maclaurin summation formula. Finally, we present several examples of the Ramanujan constant of divergent series.info:eu-repo/semantics/publishedVersio

An Alternative Form of the Functional Equation for Riemann's Zeta Function, II

This paper treats about one of the most remarkable achievements by Riemann, that is the symmetric form of the functional equation for {\zeta}(s). We present here, after showing the first proof of Riemann, a new, simple and direct proof of the symmetric form of the functional equation for both the Eulerian Zeta function and the alternating Zeta function, connected with odd numbers. A proof that Euler himself could have arranged with a little step at the end of his paper "Remarques sur un beau rapport entre les s\'eries des puissances tant direct que r\'eciproches". This more general functional equation gives origin to a special function, here named {\cyr \E}(s), which we prove that it can be continued analytically to an entire function over the whole complex plane using techniques similar to those of the second proof of Riemann. Moreover we are able to obtain a connection between Jacobi's imaginary transformation and an infinite series identity of Ramanujan. Finally, after studying the analytical properties of the function {\cyr \E}(s), we complete and extend the proof of a Fundamental Theorem, both on the zeros of Riemann Zeta function and on the zeros of Dirichlet Beta function, using also the Euler-Boole summation formula.Comment: 26 pages, 2 figure

De las sumas de potencias a las sucesiones de Appell y su caracterización a través de funcionales.

El estudio de las sumas de potencias de enteros positivos ha sido un tema de interés desde la antigüedad [8] que se mantiene vigente hasta nuestros días, ver por ejemplo [29, 24]. Actualmente estas fórmulas son de conocimiento común, sobre todo para potencias bajas, y las encontramos como ejemplos sencillos de inducción matemática y de introducción a la integral de Riemann. Históricamente, existen evidencias de su desarrollo desde la escuela pitagórica, pero fue hasta el siglo XVII que Jacob Bernoulli [9, 10], motivado por investigaciones en probabilidad, quien alcanzó el triunfo de descifrar una fórmula general de tipo polinomial para cualquier potencia. Su método llevó además al descubrimiento de la sucesión de números que hoy llevan su nombre, a saber, los números de Bernoulli. Estos aparecen naturalmente en múltiples fórmulas del análisis matemático, por ejemplo como coeficientes en la expansión de Taylor de funciones trigonométricas y en el cálculo de sumas de series [32]. En efecto, ellos permiten el cálculo efectivo de (2k), donde k es un entero no nulo y denota la famosa función zeta de Riemann.Magister en Matemáticas AplicadasMaestrí