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Search Results (4)

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Keywords = λ-generalized Hurwitz–Lerch zeta functions

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23 pages, 539 KB  
Article
On Convoluted Forms of Multivariate Legendre-Hermite Polynomials with Algebraic Matrix Based Approach
by Mumtaz Riyasat, Amal S. Alali, Shahid Ahmad Wani and Subuhi Khan
Mathematics 2024, 12(17), 2662; https://doi.org/10.3390/math12172662 - 27 Aug 2024
Viewed by 988
Abstract
The main purpose of this article is to construct a new class of multivariate Legendre-Hermite-Apostol type Frobenius-Euler polynomials. A number of significant analytical characterizations of these polynomials using various generating function techniques are provided in a methodical manner. These enactments involve explicit relations [...] Read more.
The main purpose of this article is to construct a new class of multivariate Legendre-Hermite-Apostol type Frobenius-Euler polynomials. A number of significant analytical characterizations of these polynomials using various generating function techniques are provided in a methodical manner. These enactments involve explicit relations comprising Hurwitz-Lerch zeta functions and λ-Stirling numbers of the second kind, recurrence relations, and summation formulae. The symmetry identities for these polynomials are established by connecting generalized integer power sums, double power sums and Hurwitz-Lerch zeta functions. In the end, these polynomials are also characterized Svia an algebraic matrix based approach. Full article
(This article belongs to the Section E: Applied Mathematics)
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6 pages, 254 KB  
Article
A Quadruple Integral Containing the Gegenbauer Polynomial Cn(λ)(x): Derivation and Evaluation
by Robert Reynolds and Allan Stauffer
Symmetry 2022, 14(2), 205; https://doi.org/10.3390/sym14020205 - 21 Jan 2022
Viewed by 2355
Abstract
A four-dimensional integral containing g(x,y,z,t)Cn(λ)(x) is derived. Cn(λ)(x) is the Gegenbauer polynomial, [...] Read more.
A four-dimensional integral containing g(x,y,z,t)Cn(λ)(x) is derived. Cn(λ)(x) is the Gegenbauer polynomial, g(x,y,z,t) is a product of the generalized logarithm quotient functions and the integral is taken over the region 0≤x≤1,0≤y≤1,0≤z≤1,0≤t≤1. The integral is difficult to compute in general. Special cases are given and invariant index forms are derived. The zero distribution of almost all Hurwitz–Lerch zeta functions is asymmetrical. All the results in this work are new. Full article
(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)
16 pages, 347 KB  
Article
Some Difference Equations for Srivastava’s λ-Generalized Hurwitz–Lerch Zeta Functions with Applications
by Asifa Tassaddiq
Symmetry 2019, 11(3), 311; https://doi.org/10.3390/sym11030311 - 1 Mar 2019
Cited by 6 | Viewed by 2652
Abstract
In this article, we establish some new difference equations for the family of λ-generalized Hurwitz–Lerch zeta functions. These difference equations proved worthwhile to study these newly defined functions in terms of simpler functions. Several authors investigated such functions and their analytic properties, but [...] Read more.
In this article, we establish some new difference equations for the family of λ-generalized Hurwitz–Lerch zeta functions. These difference equations proved worthwhile to study these newly defined functions in terms of simpler functions. Several authors investigated such functions and their analytic properties, but no work has been reported for an estimation of their values. We perform some numerical computations to evaluate these functions for different values of the involved parameters. It is shown that the direct evaluation of involved integrals is not possible for the large values of parameter s ; nevertheless, using our new difference equations, we can evaluate these functions for the large values of s . It is worth mentioning that for the small values of this parameter, our results are 100% accurate with the directly computed results using their integral representation. Difference equations so obtained are also useful for the computation of some new integrals of products of λ-generalized Hurwitz–Lerch zeta functions and verified to be consistent with the existing results. A derivative property of Mellin transforms proved fundamental to present this investigation. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
20 pages, 362 KB  
Article
A New Representation for Srivastava’s λ-Generalized Hurwitz-Lerch Zeta Functions
by Asifa Tassaddiq
Symmetry 2018, 10(12), 733; https://doi.org/10.3390/sym10120733 - 8 Dec 2018
Cited by 12 | Viewed by 2558
Abstract
Taking inspiration principally from some of the latest research, we develop a new series representation for the λ-generalized Hurwitz-Lerch zeta functions. This representation led to important new results. The Fourier transform played a foundational role in this work. The duality property of [...] Read more.
Taking inspiration principally from some of the latest research, we develop a new series representation for the λ-generalized Hurwitz-Lerch zeta functions. This representation led to important new results. The Fourier transform played a foundational role in this work. The duality property of the Fourier transform became significant for checking the consistency of the results. Some known data has been verified as special cases of the results obtained in this investigation. Full article
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)
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