http://hdl.handle.net/10366/4147 Sat, 02 May 2026 04:00:39 GMT 2026-05-02T04:00:39Z http://hdl.handle.net/10366/168737 [EN]One can find in the mathematical literature many recent papers studying the generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, defined by means of generating functions. In this article we clarify the range of parameters in which these definitions are valid and when they provide essentially different families of polynomials. In particular, we show that, up to multiplicative constants, it is enough to take as the ``main family'' those given by $$\Bigl(\frac{2}{\lambda e^{t}+1}\Bigr)^{\alpha} e^{xt} = \sum_{n=0}^{\infty} \mathcal{E}^{(\alpha)}_{n}(x;\lambda) \frac{t^n}{n!}, \qquad \lambda \in \mathbb{C}\setminus \lbrace -1\rbrace,$$ and as an ``exceptional family'' $$\Bigl(\frac{t}{e^t-1} \Bigr)^\alpha e^{xt} = \sum_{n=0}^{\infty} \mathcal{B}^{(\alpha)}_{n}(x) \frac{t^n}{n!},$$ both of these for $\alpha \in \mathbb{C}$. Tue, 01 Jan 2019 00:00:00 GMT http://hdl.handle.net/10366/168737 2019-01-01T00:00:00Z http://hdl.handle.net/10366/168519 [EN]We consider the series $\sum_{n=1}^{\infty} z^{n} (a_{n} + x)^{-s}$ where $\{a_{n}\}$ satisfies a linear recurrence of arbitrary degree with integer coefficients. Under appropriate conditions, we prove that it can be continued to a meromorphic function on the complex $s$-plane. Thus we may associate a Lerch-type zeta function $\varphi(z,s,x)$ to a general recurrence. This subsumes all previous results which dealt only with the ordinary zeta and Hurwitz cases and degrees $2$ and $3$. Our method generalizes a formula of Ramanujan for the classical Hurwitz-Riemann zeta functions. We determine the poles and residues of $\varphi$, which turn out to be polynomials in $x$. In addition we study the dependence of $\varphi(z,s,x)$ on $x$ and $z$, and its properties as a function of three complex variables. Wed, 27 Aug 2025 00:00:00 GMT http://hdl.handle.net/10366/168519 2025-08-27T00:00:00Z http://hdl.handle.net/10366/168506 [EN]The Dirichlet series associated to the Fibonacci sequence $\{F_{n}\}$, $$\sum_{n=1}^{\infty} F_{n}^{-s},$$ converges for $s\in \mathbb{C}$ with $\Re s > 0$. The analytic function $\varphi(s)$ it defines on the right half-plane is known as the Fibonacci zeta function. Here we consider its logarithmic derivative $\varphi'(s)/\varphi(s)$, which formally corresponds to the Dirichlet series $$-\sum_{l=1}^{\infty} \Lambda_{\mathcal{F}}(l) l^{-s},$$ where the arithmetical function $\Lambda_{\mathcal{F}}(l)$ can be considered analogous to the classical von Mangoldt function $\Lambda(s)$, which is defined by $\zeta'(s)/\zeta(s) = -\sum_{n=1}^{\infty} \Lambda(n) n^{-s}$ where $\zeta(s)$ is the Riemann zeta function. This paper studies some properties of the function $\Lambda_{\mathcal{F}}(l)$ along with the domain of convergence of this Dirichlet series. Wed, 01 Jan 2025 00:00:00 GMT http://hdl.handle.net/10366/168506 2025-01-01T00:00:00Z http://hdl.handle.net/10366/168501 [EN]We consider a Dirichlet series $\sum_{n=1}^{\infty} a_{n}^{-s}$ where $a_{n}$ satisfies a linear recurrence of arbitrary degree with integer coefficients. Under suitable hypotheses, we prove that it has a meromorphic continuation to the complex plane, giving explicit formulas for its pole set and residues, as well as for its finite values at negative integers, which are shown to be rational numbers. To illustrate the results, we focus on some concrete examples which have also been studied previously by other authors. Tue, 23 May 2023 00:00:00 GMT http://hdl.handle.net/10366/168501 2023-05-23T00:00:00Z http://hdl.handle.net/10366/168492 [EN]A large class of Appell polynomial sequences $\{p_{n}(x)\}_{n=0}^{\infty}$ are special values at the negative integers of an entire function $F(s,x)$, given by the Mellin transform of the generating function for the sequence. For the Bernoulli and Apostol-Bernoulli polynomials, these are basically the Hurwitz zeta function and the Lerch transcendent. Each of these have well-known Fourier series which are proved in the literature using varied techniques. Here we find the latter Fourier series by directly calculating the coefficients in a straightforward manner. We then show that, within the context of Appell sequences, these are the only cases for which the polynomials have uniformly convergent Fourier series. In the more general context of Sheffer sequences, we find that there are other polynomials with uniformly convergent Fourier series. Finally, applying the same ideas to the Fourier transform, considered as the continuous analog of the Fourier series, the Hermite polynomials play a role analogous to that of the Bernoulli polynomials. Thu, 15 Aug 2019 00:00:00 GMT http://hdl.handle.net/10366/168492 2019-08-15T00:00:00Z http://hdl.handle.net/10366/168476 [EN]The aim of this work is to offer a method for studying the consis tence and computing the set of solutions of some infinite systems of differential equations from the exponential map of a finite potent endomorphism. This method is related with the Drazin inverse of a finite potent operator on an infinite-dimensional vector space that was introduced by the author in 2019. Moreover, explicit examples of the exponential of a finite potent endomorphism and the set of solutions of an infinite system of differential equations are provided. Thu, 04 Sep 2025 00:00:00 GMT http://hdl.handle.net/10366/168476 2025-09-04T00:00:00Z http://hdl.handle.net/10366/168475 [EN]The aim of this work is to study new properties of the Drazin-Star and the Star-Drazin inverses of a bounded finite potent operator on a Hilbert space. Given a bounded finite potent operator $\varphi \in \ed_k (\mathcal H)$, we prove that the pseudo-characteristic polynomials of $\varphi^{D,*}$ and $\varphi^{*,D}$ coincide. Accordingly, we obtain that $\sigma (\varphi^{D,*}) = \sigma (\varphi^{*,D})$, $\tr_{\mathcal H} (\varphi^{D,*}) = \tr_{\mathcal H} (\varphi^{*,D})$ and $\Det_{\mathcal H} (\text{Id} + \varphi^{D,*}) = \Det_{\mathcal H} (\text{Id} + \varphi^{*,D})$. In particular, these results hold for a finite square complex matrix $A$. Moreover, we offer the explicit characterization of the AST-decompositions of $\mathcal H$ induced by the Group-Star and the Star-Group inverses of a bounded linear operator $\psi$ on $\mathcal H$ with $i(\psi)\leq 1$. Wed, 03 Dec 2025 00:00:00 GMT http://hdl.handle.net/10366/168475 2025-12-03T00:00:00Z http://hdl.handle.net/10366/166722 [EN] This article is devoted to the generalization of the Drazin pre-order and the G-Drazin partial order to Core-Nilpotent endomorphisms over arbitrary k-vector spaces, namely, infinite dimensional ones. The main properties of this orders are described, such as their respective characterizations and the relations between these orders and other existing ones, generalizing the existing theory for finite matrices. In order to do so, G-Drazin inverses are also studied in this framework. Also, it includes a generalization of the space pre-order to linear operators over arbitrary k-vector spaces. Mon, 01 Jan 2024 00:00:00 GMT http://hdl.handle.net/10366/166722 2024-01-01T00:00:00Z http://hdl.handle.net/10366/164313 [EN] The aim of thiswork is to offer the definition and themain properties of a pseudo-characteristic polynomial of a finite potent endomorphism. From this polynomial we can characterize the spectrum of a bounded finite potent linear operator on a Hilbert space andwe study arithmetic properties of complete algebraic curves. In particular, we provide a new algebraic proof of the Residue Theorem. Wed, 01 Jan 2025 00:00:00 GMT http://hdl.handle.net/10366/164313 2025-01-01T00:00:00Z http://hdl.handle.net/10366/164142 [EN] The aim of this paper is to give an abstract formulation of the classical reciprocity laws for function fields that could be generalized to the case of arbitrary (non-commutative) reductive groups as a first step to finding explicit non-commutative reciprocity laws. The main tool in this paper is the theory of determinant bundles over adelic Sato Grassmannians and the existence of a Krichever map for rank n vector bundles. Tue, 01 Jan 2008 00:00:00 GMT http://hdl.handle.net/10366/164142 2008-01-01T00:00:00Z http://hdl.handle.net/10366/164141 [EN]The aim of this note is to offer a new explicit expression of the Contou-Carrère symbol that depends only on a product of a finite number of terms. As an application, we obtain an explicit formula for a Witt Residue. Mon, 01 Jan 2007 00:00:00 GMT http://hdl.handle.net/10366/164141 2007-01-01T00:00:00Z http://hdl.handle.net/10366/164140 [EN] The aim of this note is to offer a method for the explicit computation of all 1-inverses of an arbitrary square matrix. The main tool to obtain this algorithm is the characterization of the set of 1-inverses of a finite potent endomorphism. Sun, 01 Jan 2023 00:00:00 GMT http://hdl.handle.net/10366/164140 2023-01-01T00:00:00Z http://hdl.handle.net/10366/164137 [EN] The aim of this work is to show the consistency of all systems of nonhomogeneous linear difference equations of the form ϕ(x_n+1) = x_n + v_0, where ϕ ∈ End_k(V) is a finite potent endomorphism of an arbitrary vector space V and v_0 ∈ V. An algorithm to compute the set of solutions of these systems is given. In particular, the method offered is valid for computing the explicit solutions of the system of non-homogeneous difference equations A(x_n+1) = x_n + b, with A being a finite square matrix. Sat, 01 Jan 2022 00:00:00 GMT http://hdl.handle.net/10366/164137 2022-01-01T00:00:00Z http://hdl.handle.net/10366/164136 [EN] The aim of this work is to offer necessary and sufficient conditions for the coincidence of the Drazin inverse and the DMP inverses of finite square complex matrices. This characterization is deduced from statements valid for finite potent endomorphisms and, in particular, several properties of matrices and generalized inverses are given. Sat, 01 Jan 2022 00:00:00 GMT http://hdl.handle.net/10366/164136 2022-01-01T00:00:00Z http://hdl.handle.net/10366/164135 [EN] The aim of this work is to extend to finite potent endomorphisms the notion of G-Drazin inverse of a finite square matrix. Accordingly, we determine the structure and the properties of a G-Drazin inverse of a finite potent endomorphism and, as an application, we offer an algorithm to compute the explicit expression of all G-Drazin inverses of a finite square matrix. Sat, 01 Jan 2022 00:00:00 GMT http://hdl.handle.net/10366/164135 2022-01-01T00:00:00Z http://hdl.handle.net/10366/164134 [EN] The aim of this work is to extend to finite potent endomorphisms the notion of the Drazin-Moore-Penrose inverse of a finite matrix. Several properties of this generalized inverse are proved that are also valid for some infinite matrices. Moreover, Drazin-Reflexive inverses associated with arbitrary reflexive generalized inverses of a finite potent endomorphism are introduced. Fri, 01 Jan 2021 00:00:00 GMT http://hdl.handle.net/10366/164134 2021-01-01T00:00:00Z