λ-Generalized Gamma Functions

This article presents a systematic investigation of various integrals and computational representations for some families of generalized Hurwitz–Lerch Zeta functions which are introduced here. We first establish their relationship with the -function, which enables us to derive the Mellin–Barnes type integral representations for nearly all of the generalized and specialized Hurwitz–Lerch Zeta functions. The integral expressions studied in this paper provide extensions of the corresponding results given by many authors, including (for example) Garg et al. [A further study of general Hurwitz-Lerch zeta function, Algebras Groups Geom. 25 (2008), pp. 311–319] and Lin and Srivastava [Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations, Appl. Math. Comput. 154 (2004), pp. 725–733]. We also derive a further analytic continuation formula which provides an elegant extension of the well-known analytic continuation formula for the Gauss hypergeometric function. Fractional derivatives associated with the generalized Hurwitz–Lerch Zeta functions are obtained. The relationship between the generalized Hurwitz–Lerch Zeta function and the -function, which was given by Garg et al., is seen to be erroneous and we give its corrected version here. Finally, a unification and extension of the Hurwitz–Lerch Zeta function, introduced in this article, is presented and two of its interesting special cases associated with the Mittag–Leffler type functions due to Barnes [The asymptotic expansion of integral functions defined by Taylor series, Philos. Trans. Roy. Soc. London Ser. A 206 (1906), pp. 249–297] and the generalized M-series considered recently by Sharma and Jain [A note on a generalzed M-series as a special function of fractional calculus, Fract. Calc. Appl. Anal. 12 (2009), pp. 449–452.] are deduced.

Numerous special functions, including the beta function, hypergeometric functions, and other orthogonal polynomials, are closely connected to the gamma function. Recently, gamma function has been enhanced by adding a new parameter. As a consequence, this gamma is called the parametric gamma function or b-gamma function. By utilizing the enhanced gamma function (or the parametric gamma function), we have present a generalization for the special function Rabotnov function. Consequently, new fractal-fractional operators (derivative and integral) involving the generalized Rabotnov function are defined. Analysis is introduced to discover the main properties of the suggested operators. Since the main challenge in the calculus of fractal-fractional operators is to design examples, we illustrate a set of examples including power series. We explore the boundedness of the recommended operators. There are many gains for checking the boundedness of fractal operators in general and fractal-fractional operators in particular. Moreover, as an application, we establish the existence and uniqueness solution of abstract fractal-fractional equation. Examples are presented at the end of the effort. Graphics and computations are observed using MATHEMATICA 13.3 software.•By using the parametric gamma function, the Rabotnov special function is generalized. New fractal-fractional operators are presented using the generalized Rabotnov function with examples;•Boundedness of these operators is investigated, where bounded operators give a strong foundation for understanding linear transformations across normed spaces, with many applications in science and mathematics.•Conditions of the existence and uniqueness of solutions of fractal-fractional differential abstract equation are established.

Integral representations play a prominent role in the analysis of entire functions. The representations of generalized Mittag-Leffler type functions and their asymptotics have been (and still are) investigated by plenty of authors in various conditions and cases. The present paper explores the integral representations of a special function extending to two variables the two-parametric Mittag-Leffler type function. Integral representations of this functions within different variation ranges of its arguments for certain values of the parameters are thus obtained. Asymptotic expansion formulas and asymptotic properties of this function are also established for large values of the variables. This yields corresponding theorems providing integral representations as well as expansion formulas.

On some series representations of the Hurwitz zeta function

It is impossible to overplay this function's pervasive role in mathematics. From its humble origin as a complex-valued generalization of the shifted factorial function to its more sophisticated guise as the Mellin transform of e-x, the gamma function surfaces in the study of special functions everywhere. On the other hand it can be argued, following Weierstrass, that the reciprocal of the gamma function is a more natural player, one equally deserving of a central part. In this article we focus on Laplace's basic result and its numerous ramifications. Our first goal is to provide two simple proofs of his identity. We avoid popular techniques: messy contour integration (the kind that deals with branch points) and the theory of Fourier transforms, which invariably invokes the Fourier inversion formula. The only facts about the gamma function that we use follow from Euler's definition. These are its recurrence relation

In this paper, some convexity properties and some inequalities forthe (p; k)-analogue of the Gamma function,a (p; k)-analogue of the celebrated Bohr-Mollerup theorem is given. Furthermore, a (p; k)-analogue of the Riemann zeta function,p;k(x)is introduced andsome associated inequalities are derived. The established results provide the(p; k)-generalizations of some known results concerning the classical Gammafunction

Zeta and q-Zeta Functions and Associated Series and Integrals is a thoroughly revised, enlarged and updated version of Series Associated with the Zeta and Related Functions. Many of the chapters and sections of the book have been significantly modified or rewritten, and a new chapter on the theory and applications of the basic (or q-) extensions of various special functions is included. This book will be invaluable because it covers not only detailed and systematic presentations of the theory and applications of the various methods and techniques used in dealing with many different classes of series and integrals associated with the Zeta and related functions, but stimulating historical accounts of a large number of problems and well-classified tables of series and integrals. Detailed and systematic presentations of the theory and applications of the various methods and techniques used in dealing with many different classes of series and integrals associated with the Zeta and related functions

The use of differintegral operators $D^\nu $ with $D = {d / {dz}}$ and $\nu \in \mathbb{C}$ and hyperdifferential operators, together with some operational calculus manipulations leads to new formulae of representation of usual special functions. These formulae are useful for obtaining generating functions and differential recurrence relations of orthogonal polynomials.

Summation and recurrence formula involving the central factorial numbers and zeta function

Let $$\mathbb {R}_+$$ ℝ + denote the open half-line (0, ∞) and let Δ denote the forward difference operator on the space of functions from $$\mathbb {R}_+$$ ℝ + to $$\mathbb {R}$$ ℝ . In this book, we are interested in the classical difference equation Δf = g on $$\mathbb {R}_+$$ ℝ + , which can be written explicitly as $$\displaystyle f(x+1)-f(x) ~=~ g(x),\qquad x>0, $$ f ( x + 1 ) − f ( x ) = g ( x ) , x > 0 , where $$g\colon \mathbb {R}_+\to \mathbb {R}$$ g : ℝ + → ℝ is a given function. This equation appears naturally in the theory of the Euler gamma function, with $$f(x) = \ln \Gamma (x)$$ f ( x ) = ln Γ ( x ) and $$g(x) = \ln x$$ g ( x ) = ln x , but also in the study of many other special functions such as the Barnes G-function and the Hurwitz zeta function (see Examples 1.6 and 1.7 below).

Considering the equations for some functions involving the first or the second derivatives of the biconfluent Heun function, we construct two expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta functions. The first series applies single Beta functions as expansion functions, while the second one involves a combination of two Beta functions. The coefficients of expansions obey four- and five-term recurrence relations, respectively. It is shown that the proposed technique is potent to produce series solutions in terms of other special functions. Two examples of such expansions in terms of the incomplete Gamma functions are presented.

Fast spherical Bessel transform via fast Fourier transform and recurrence formula

JRAF: A Julia package for computation of relativistic molecular auxiliary functions

This study presents a detailed investigation into the analytical properties of the generalized Gamma function introduced by Dilcher. We establish a fundamental recurrence relation and derive novel reflection formula for this generalized function, extending the classical identities known for the Euler Gamma function. Further, we analyze the generalized digamma function, the logarithmic derivative of the generalized Gamma function, and obtain a significant difference equation that characterizes its behavior. Explicit expressions for the evaluation of this generalized digamma function at half‐integer arguments are provided. The study also introduces the natural extension to generalized polygamma functions and reports on several special values for both classical and generalized cases. To complement the theoretical analysis, a graphical representation of the generalized digamma function is included, alongside supporting numerical computations that illustrate its properties and validate the derived results. Our findings provide a comprehensive framework that enhances the understanding of these generalized special functions and underscores their potential for broader application in mathematical physics and number theory.

We study a series of issues related with integral representations of Gamma functions and its quotients.The base of our study is two classical results in the theory of functions.One of them is a well-known first Binet formula, the other is a less known Malmsten formula.These special formulae express the values of the Gamma function in an open right half-plane via corresponding improper integrals.In this work we show that both results can be extended to the imaginary axis except for the point = 0.Under such extension we apply various methods of real and complex analysis.In particular, we obtain integral representations for the argument of the complex quantity being the value of the Gamma function in a pure imaginary point.On the base of the mentioned Malmsten formula at the points ̸ = 0 in the closed right half-plane, we provide a detailed derivation of the integral representation for a special quotient expressed via the Gamma function: () ≡ Γ( + 1 2 )/Γ( + 1).This fact on the positive semi-axis was mentioned without the proof in a small note by Dušan Slavić in 1975.In the same work he provided two-sided estimates for the quantity () as > 0 and at the natural points () coincided with the normalized central binomial coefficient.These estimates mean that () is enveloped on the positive semi-axis by its asymptotic series.In the present paper we briefly discuss the issue on the presence of this property on the asymptotic series () in a closed angle | arg | /4 with a punctured vertex.By the new formula representing () on the imaginary axis we obtain explicit expressions for the quantity |()| 2 and for the set Arg () as > 0. We indicate a way of proving the second Binet formula employing the technique of simple fractions.