The Lerch transcendent from the point of view of Fourier analysis

Navas, Luis M.; Ruiz, Francisco J.; Malumbres, Juan

doi:10.1016/j.jmaa.2015.05.048

Cited by 5 publications

(4 citation statements)

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“…In Section 3, we give a new proof of the functional equations (3.1) and (3.3) by using the integral representations (2.4) and (2.9), and modifying the fifth method of the proof of the functional equation for the Riemann zeta function (see [16,Section 2.8]). It should be noted that in the 21st century, Knopp and Robins [7], and Navas, Ruiz and Varona [9] gave new proofs of the functional equation for ζ(s, a) by using Poisson summation of the Lipschitz summation formula (see [7, p. 1916]) and the uniqueness of Fourier coefficients (see [9, p. 190]), respectively. 1.2.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Real zeros of Hurwitz–Lerch zeta functions in the interval (−1,0)

Nakamura

2016

Journal of Mathematical Analysis and Applications

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Abstract. For 0 < a ≤ 1, s, z ∈ C and 0 < |z| ≤ 1, the Hurwitz-Lerch zeta function is defined by Φ(s, a, z) := ∞ n=0 z n (n + a) −s when σ := ℜ(s) > 1. In this paper, we show that Φ(σ, a, z) = 0 when σ ∈ (−1, 0) if and only if [I] z = 1 and (z ∈ R and 0 < a ≤ 1. In addition, we give a new proof of the functional equation of Φ(s, a, z).1. Introduction and statement of main results 1.1. Main results. The Hurwitz-Lerch zeta function is defined by as follows.We can easily see that the Riemann zeta function ζ(s) and the Hurwitz zeta function ζ(s, a) are expressed as Φ(s, 1, 1) and Φ(s, a, 1), respectively. The Dirichlet series of Φ(s, a, z) converges absolutely in the half-plane σ > 1 and uniformly in each compact subset of this half-plane. When z = 1, the function Φ(s, a, z) is analytically continuable to the whole complex plane. However, the Hurwitz zeta function ζ(s, a) is a meromorphic function with a simple pole at s = 1. In this paper, we show the following. Note that b (−1, 0). In Section 3, we give a new proof of the functional equations (3.1) and (3.3) by using the integral representations (2.4) and (2.9), and modifying the fifth method of the proof of the functional equation for the Riemann zeta function (see [16, Section 2.8]). It should be noted that in the 21st century, Knopp and Robins [7], and Navas, Ruiz and Varona [9] gave new proofs of the functional equation for ζ(s, a) by using Poisson summation of the Lipschitz summation formula (see [7, p. 1916]) and the uniqueness of Fourier coefficients (see [9, p. 190]), respectively.2010 Mathematics Subject Classification. Primary 11M35.

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Real zeros of Hurwitz–Lerch zeta functions in the interval (−1,0)

Nakamura

2016

Journal of Mathematical Analysis and Applications

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“…The following Fourier expansion of the Hurwitz zeta function ζ(s, x) is well-known. We will briefly discuss the proof given in [11], but without entering into too much detail, with a view towards using similar ideas in the case of the Lerch function.…”

Section: Fourier Series For Hurwitz and Lerch Zeta Functionsmentioning

confidence: 99%

“…Note that the difference between the theorems is due to the fact that the generating function of the Apostol-Bernoulli polynomials has a zero at t = 0 while that of the Bernoulli polynomials does not. That is why in (11) we only need to remove one term from the Taylor series, while in (8) we need two. It is often convenient to assume A(0) = 0 in order to normalize results.…”

Section: Fourier Series For Hurwitz and Lerch Zeta Functionsmentioning

confidence: 99%

“…Multiple proofs of these formulas are found in the literature (see for instance the classical references [4, §1.10 and §1.11] or [8, §1.6]) as well as extensive studies of their many other properties. In [11] we give a direct simple argument, based on the Mellin transform, for finding the Fourier coefficients in the case of the Hurwitz zeta function. Here we show how that reasoning applies also to the Lerch function.…”

Section: Introductionmentioning

confidence: 99%

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A note on Appell sequences, Mellin transforms and Fourier series

Navas

Ruiz

Malumbres

2019

Journal of Mathematical Analysis and Applications

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A large class of Appell polynomial sequences {p n (x)} ∞ n=0 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 various 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.

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Multi-class deep learning architecture for classifying lung diseases from chest X-Ray and CT images

Al-Sheikh,

Al Dandan,

Al-Shamayleh

et al. 2023

Sci Rep

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Medical imaging is considered a suitable alternative testing method for the detection of lung diseases. Many researchers have been working to develop various detection methods that have aided in the prevention of lung diseases. To better understand the condition of the lung disease infection, chest X-Ray and CT scans are utilized to check the disease’s spread throughout the lungs. This study proposes an automated system for the detection multi lung diseases in X-Ray and CT scans. A customized convolutional neural network (CNN) and two pre-trained deep learning models with a new image enhancement model are proposed for image classification. The proposed lung disease detection comprises two main steps: pre-processing, and deep learning classification. The new image enhancement algorithm is developed in the pre-processing step using k-symbol Lerch transcendent functions model which enhancement images based on image pixel probability. While, in the classification step, the customized CNN architecture and two pre-trained CNN models Alex Net, and VGG16Net are developed. The proposed approach was tested on publicly available image datasets (CT, and X-Ray image dataset), and the results showed classification accuracy, sensitivity, and specificity of 98.60%, 98.40%, and 98.50% for the X-Ray image dataset, respectively, and 98.80%, 98.50%, 98.40% for the CT scans dataset, respectively. Overall, the obtained results highlight the advantages of the image enhancement model as a first step in processing.

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The Lerch transcendent from the point of view of Fourier analysis

Cited by 5 publications

References 10 publications

Real zeros of Hurwitz–Lerch zeta functions in the interval (−1,0)

Real zeros of Hurwitz–Lerch zeta functions in the interval (−1,0)

A note on Appell sequences, Mellin transforms and Fourier series

Multi-class deep learning architecture for classifying lung diseases from chest X-Ray and CT images

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