ARTICLE | doi:10.20944/preprints201803.0008.v1
Subject: Mathematics & Computer Science, Analysis Keywords: Extended beta function, hypergeometric functions, Hurwitz-Lerch Zeta functions, Mellin transform, transformation formula and summation formula.
Online: 1 March 2018 (14:46:40 CET)
In this paper, we define a (p,v)-extension of Hurwitz-Lerch Zeta function by considering an extension of beta function defined by Parmar et al. [J. Classical Anal. 11 (2017) 81–106]. We obtain its basic properties which include integral representations, Mellin transformation, derivative formulas and certain generating relations. Also, we establish the special cases of the main results.
ARTICLE | doi:10.20944/preprints202105.0192.v1
Subject: Keywords: entries in Gradshteyn and Rhyzik, Lerch function, Logarithm function, Contour Integral, Cauchy, Infinite Integral
Online: 10 May 2021 (13:51:04 CEST)
We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage of using special functions is their analytic continuation which widens the range of the parameters of the definite integral over which the formula is valid. We give as examples definite integrals of logarithmic functions times a trigonometric function. In various cases these generalizations evaluate to known mathematical constants such as Catalan’s constant and π
ARTICLE | doi:10.20944/preprints202204.0176.v1
Subject: Mathematics & Computer Science, Algebra & Number Theory Keywords: ceiling function; floor function; Fibonacci Number; Generalised Dirichlet series; Lerch - Zeta Function; Hurwitz - Zeta function; Polylogarithm; Riemann-Zeta function
Online: 19 April 2022 (06:03:59 CEST)
In this part of the series of two papers, we extend the theorems discussed in part I for infinite series. We then use these theorems to develop distinct novel results involving the Hurwitz zeta function, Riemann zeta function, Polylogarithm and Fibonacci numbers. In continuation, we obtain some zeros of the newly developed zeta functions and explain their behaviour using plots in complex plane. Furthermore, we provide particular cases for the theorems and corollaries which show that our results generalise the currently available functions and series such as the Riemann zeta function and the geometric series. Finally, we provide four miscellaneous examples to showcase the vast scope of the developed theorems.
ARTICLE | doi:10.20944/preprints201704.0026.v1
Subject: Mathematics & Computer Science, Analysis Keywords: simple form; explicit form; differential equation; Lerch transcendent; logarithmic function
Online: 4 April 2017 (16:53:18 CEST)
In the note, the authors find several simple and explicit forms for a family of inhomogeneous linear ordinary differential equations studied in "D. Lim, Differential equations for Daehee polynomials and their applications, J. Nonlinear Sci. Appl. 10 (2017), no. 4, 1303-1315; Available online at http://dx.doi.org/10.22436/jnsa.010.04.02".
ARTICLE | doi:10.20944/preprints202002.0245.v1
Subject: Mathematics & Computer Science, Analysis Keywords: limit values; modular relation; Lerch zeta-function; Hurwitz zetafunction; Laurent coefficients
Online: 17 February 2020 (15:10:08 CET)
Boundary behavior of a given important function or its limit values are essential in the whole spectrum of mathematics and science. We consider some tractable cases of limit values in which either a difference of two ingredients or a difference equation is used coupled with the relevant functional equations to give rise to unexpected results. This involves the expression for the Laurent coefficients including the residue, the Kronecker limit formulas and higher order coefficients as well as the difference formed to cancel the inaccessible part, typically the Clausen functions. We also state Abelian results which yield asymptotic formulas for weighted summatory function from that for the original summatory function
ARTICLE | doi:10.20944/preprints201611.0146.v1
Subject: Mathematics & Computer Science, Analysis Keywords: explicit form; inhomogeneous linear ordinary differential equation; derivative; Lerch transcendent; absolute monotonicity; complete monotonicity; Bernstein function; inequality; diagonal recurrence relation; Stirling numbers of the first kind; logarithmic function
Online: 29 November 2016 (08:00:53 CET)
In the paper, the authors present an explicit form for a family of inhomogeneous linear ordinary differential equations, find a more significant expression for all derivatives of a function related to the solution to the family of inhomogeneous linear ordinary differential equations in terms of the Lerch transcendent, establish an explicit formula for computing all derivatives of the solution to the family of inhomogeneous linear ordinary differential equations, acquire the absolute monotonicity, complete monotonicity, the Bernstein function property of several functions related to the solution to the family of inhomogeneous linear ordinary differential equations, discover a diagonal recurrence relation of the Stirling numbers of the first kind, and derive an inequality for bounding the logarithmic function.