scholarly journals Some Difference Equations for Srivastava’s λ-Generalized Hurwitz–Lerch Zeta Functions with Applications

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 311 ◽  
Author(s):  
Asifa Tassaddiq
Keyword(s):  
Integral Representation ◽  
Difference Equations ◽  
Zeta Functions ◽  
Numerical Computations ◽  
Direct Evaluation ◽  
Mellin Transforms ◽  
The Family

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.

scholarly journals Computational Analysis of Generalized Zeta Functions by Using Difference Equations

2020 ◽  
Author(s):  
Asifa Tassaddiq
Keyword(s):  
Difference Equations ◽  
Taylor Series ◽  
Computational Analysis ◽  
Zeta Functions ◽  
Series Expansions ◽  
Graphical Representations ◽  
Computational Power ◽  
Numerical Computations ◽  
Taylor Series Expansions

In this article, author performs computational analysis for the generalized zeta functions by using computational software Mathematica. To achieve the purpose recently obtained difference equations are used. These difference equations have a computational power to compute these functions accurately while they can not be computed by using their known integral represenations. Several authors investigated such functions and their analytic properties, but no work has been reported to study the graphical representations and zeors of these functions. Author performs numerical computations to evaluate these functions for different values of the involved parameters. Taylor series expansions are also presented in this research.

scholarly journals Martingale decomposition of an L2 space with nonlinear stochastic integrals

2019 ◽  
Vol 56 (4) ◽  
pp. 1231-1243
Author(s):  
Clarence Simard
Keyword(s):  
Integral Representation ◽  
General Result ◽  
Optimization Problem ◽  
Stochastic Integrals ◽  
Polynomial Functions ◽  
Spatial Parameter ◽  
The Family

AbstractThis paper generalizes the Kunita–Watanabe decomposition of an $L^2$ space. The generalization comes from using nonlinear stochastic integrals where the integrator is a family of continuous martingales bounded in $L^2$ . This result is also the solution of an optimization problem in $L^2$ . First, martingales are assumed to be stochastic integrals. Then, to get the general result, it is shown that the regularity of the family of martingales with respect to its spatial parameter is inherited by the integrands in the integral representation of the martingales. Finally, an example showing how the results of this paper, with the Clark–Ocone formula, can be applied to polynomial functions of Brownian integrals.

scholarly journals Analysis and combinatorics of partition zeta functions

2020 ◽  
pp. 1-10
Author(s):  
Robert Schneider ◽  
Andrew V. Sills
Keyword(s):  
Zeta Function ◽  
Explicit Formula ◽  
Zeta Functions ◽  
Complete Information ◽  
Combinatorial Proof ◽  
Fixed Length ◽  
The Family ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Partial Fraction Decomposition

We examine “partition zeta functions” analogous to the Riemann zeta function but summed over subsets of integer partitions. We prove an explicit formula for a family of partition zeta functions already shown to have nice properties — those summed over partitions of fixed length — which yields complete information about analytic continuation, poles and trivial roots of the zeta functions in the family. Then we present a combinatorial proof of the explicit formula, which shows it to be a zeta function analog of MacMahon’s partial fraction decomposition of the generating function for partitions of fixed length.

Experimental and computational studies of mixing in complex Stokes flows: the vortex mixing flow and multicellular cavity flows

1994 ◽  
Vol 269 ◽  
pp. 199-246 ◽  
Author(s):  
Sadhan C. Jana ◽  
Guy Metcalfe ◽  
J. M. Ottino
Keyword(s):  
Broad Class ◽  
Merit Function ◽  
Global Optimum ◽  
Boundary Integral ◽  
Computational Studies ◽  
Complex Flow ◽  
Numerical Computations ◽  
Stokes Flows ◽  
The Family

A complex Stokes flow has several cells, is subject to bifurcation, and its velocity field is, with rare exceptions, only available from numerical computations. We present experimental and computational studies of two new complex Stokes flows: a vortex mixing flow and multicell flows in slender cavities. We develop topological relations between the geometry of the flow domain and the family of physically realizable flows; we study bifurcations and symmetries, in particular to reveal how the forcing protocol's phase hides or reveals symmetries. Using a variety of dynamical tools, comparisons of boundary integral equation numerical computations to dye advection experiments are made throughout. Several findings challenge commonly accepted wisdom. For example, we show that higher-order periodic points can be more important than period-one points in establishing the advection template and extended regions of large stretching. We demonstrate also that a broad class of forcing functions produces the same qualitative mixing patterns. We experimentally verify the existence of potential mixing zones for adiabatic forcing and investigate the crossover from adiabatic to non-adiabatic behaviour. Finally, we use the entire array of tools to address an optimization problem for a complex flow. We conclude that none of the dynamical tools alone can successfully fulfil the role of a merit function; however, the collection of tools can be applied successively as a dynamical sieve to uncover a global optimum.

scholarly journals On Global Attractivity of a Class of Nonautonomous Difference Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
Wanping Liu ◽  
Xiaofan Yang ◽  
Jianqiu Cao
Keyword(s):  
Positive Solution ◽  
Difference Equations ◽  
Global Attractivity ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Positive Equilibrium ◽  
Global Behavior ◽  
Open Problems ◽  
The Family ◽  
Recursive Equations

We mainly investigate the global behavior to the family of higher-order nonautonomous recursive equations given byyn=(p+ryn-s)/(q+ϕn(yn-1,yn-2,…,yn-m)+yn-s),n∈ℕ0, withp≥0,r,q>0,s,m∈ℕand positive initial values, and present some sufficient conditions for the parameters and mapsϕn:(ℝ+)m→ℝ+,n∈ℕ0, under which every positive solution to the equation converges to zero or a unique positive equilibrium. Our main result in the paper extends some related results from the work of Gibbons et al. (2000), Iričanin (2007), and Stević (vol. 33, no. 12, pages 1767–1774, 2002; vol. 6, no. 3, pages 405–414, 2002; vol. 9, no. 4, pages 583–593, 2005). Besides, several examples and open problems are presented in the end.

scholarly journals Periodic Problems of Difference Equations and Ergodic Theory

2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
B. A. Biletskyi ◽  
A. A. Boichuk ◽  
A. A. Pokutnyi
Keyword(s):  
Difference Equations ◽  
Periodic Boundary Condition ◽  
Generalized Inverse ◽  
Bounded Operator ◽  
Inverse Operator ◽  
Sufficient Conditions ◽  
Linear Bounded Operator ◽  
Periodic Problems ◽  
The Family ◽  
Necessary And Sufficient

The necessary and sufficient conditions for solvability of the family of difference equations with periodic boundary condition were obtained using the notion of relative spectrum of linear bounded operator in the Banach space and the ergodic theorem. It is shown that when the condition of existence is satisfied, then such periodic solutions are built using the formula for the generalized inverse operator to the linear limited one.

scholarly journals HYPERGEOMETRIC ZETA FUNCTIONS

2010 ◽  
Vol 06 (01) ◽  
pp. 99-126 ◽  
Author(s):  
ABDUL HASSEN ◽  
HIEU D. NGUYEN
Keyword(s):  
Integral Representation ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Special Functions ◽  
Zeta Functions ◽  
Functional Inequality ◽  
Bernoulli Numbers ◽  
Classical Counterpart ◽  
New Family ◽  
Riemann Zeta

This paper investigates a new family of special functions referred to as hypergeometric zeta functions. Derived from the integral representation of the classical Riemann zeta function, hypergeometric zeta functions exhibit many properties analogous to their classical counterpart, including the intimate connection to Bernoulli numbers. These new properties are treated in detail and are used to demonstrate a functional inequality satisfied by second-order hypergeometric zeta functions.

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