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Gamma : exploring Euler's constant
\"Among the many constants that appear in mathematics, [pi], e, and i are the most familiar. Following closely behind is [gamma] or gamma, a constant that arises in many mathematical areas yet remains profoundly mysterious. Introduced by the Swiss mathematician Leonhard Euler (1707-1783), who figures prominently in this book, gamma is defined as the limit of the sum of 1 + 1/2 + 1/3 + ... up to 1/n , minus the natural logarithm of n -- and the numerical value is 0.5772156 ... But unlike its more celebrated colleagues [pi] and e, the exact nature of gamma remains a mystery. In fact, we don't even know if gamma is a fraction. In this tantalizing blend of history and mathematics, Julian Havil takes readers on a journey through logarithms and the harmonic series, the two defining elements of gamma, toward the first account of gamma's place in mathematics. Sure to be popular with not only students and instructors but all math aficionados, Gamma takes us through countries, centuries, lives, and works, unfolding along the way the stories of some remarkable mathematics from some remarkable mathematicians.\"--Back cover.
Book
On the Definition of Higher Gamma Functions
We extent our definition of Euler Gamma function to higher Gamma functions, and we give a unified characterization of Barnes higher Gamma functions, Mellin Gamma functions, Barnes multiple Gamma functions, Jackson
q
-Gamma function, and Nishizawa higher
q
-Gamma functions in the space of finite order meromorphic functions. The method extends to more general functional equations and unveils the multiplicative group structure of solutions that appears as a cocycle equation. We also generalize Barnes hierarchy of higher Gamma function and multiple Gamma functions. With the new definition, Barnes–Hurwitz zeta functions are no longer necessary in the definition of Barnes multiple Gamma functions. This simplifies the classical definition, without the analytic preliminaries about the meromorphic extension of Barnes–Hurwitz zeta functions, and defines a larger class of Gamma functions. For some algebraic independence conditions on the parameters, we prove uniqueness of the solutions. Hence, this implies the identification of classical Barnes multiple Gamma functions as a subclass of our multiple Gamma functions.
Journal Article
Extended k-Gamma and k-Beta Functions of Matrix Arguments
Various k-special functions such as k-gamma function, k-beta function and k-hypergeometric functions have been introduced and investigated. Recently, the k-gamma function of a matrix argument and k-beta function of matrix arguments have been presented and studied. In this paper, we aim to introduce an extended k-gamma function of a matrix argument and an extended k-beta function of matrix arguments and investigate some of their properties such as functional relations, inequality, integral formula, and integral representations. Also an application of the extended k-beta function of matrix arguments to statistics is considered.
Journal Article
Special Functions and their Application
This short text gives clear descriptions and explanations of the Gamma function, the Probability Integral and its related functions, Spherical Harmonics Theory, The Bessel function, Hermite polynomials and Laguerre polynomials.
eBook
A Study of Szász–Durremeyer-Type Operators Involving Adjoint Bernoulli Polynomials
This research work introduces a connection of adjoint Bernoulli’s polynomials and a gamma function as a sequence of linear positive operators. Further, the convergence properties of these sequences of operators are investigated in various functional spaces with the aid of the Korovkin theorem, Voronovskaja-type theorem, first order of modulus of continuity, second order of modulus of continuity, Peetre’s K-functional, Lipschitz condition, etc. In the last section, we extend our research to a bivariate case of these sequences of operators, and their uniform rate of approximation and order of approximation are investigated in different functional spaces. Moreover, we construct a numerical example to demonstrate the applicability of our results.
Journal Article
On the Asymptotic Expansions of the (p,k)-Analogues of the Gamma Function and Associated Functions
General asymptotic expansion of the (p,k)-gamma function is obtained and various approaches to this expansion are studied. The numerical precision of the derived asymptotic formulas is shown and compared. Results are applied to the analogues of digamma and polygamma functions, and asymptotic expansion of the quotient of two (p,k)-gamma functions is also derived and analyzed. Various examples and application to the k-Pochhammer symbol are presented.
Journal Article
Efficient and Accurate Algorithms for the Computation and Inversion of the Incomplete Gamma Function Ratios
Algorithms for the numerical evaluation of the incomplete gamma function ratios $P(a,x)=\\gamma(a,x)/\\Gamma(a)$ and $Q(a,x)=\\Gamma(a,x)/\\Gamma(a)$ are described for positive values of $a$ and $x$. Also, inversion methods are given for solving the equations $P(a,x)=p$, $Q(a,x)=q$, with $0
Journal Article
Inhomogeneous Whittaker Equation with Initial and Boundary Conditions
In this study, a semi-analytical solution to the inhomogeneous Whittaker equation is developed for both initial and boundary value problems. A new class of special integral functions Ziκ,μf(x), along with their derivatives, is introduced to facilitate the construction of the solution. The analytical properties of Ziκ,μf(x) are rigorously investigated, and explicit closed-form expressions for Ziκ,μf(x) and its derivatives are derived in terms of Whittaker functions Mκ,μ(z) and Wκ,μ(z), confluent hypergeometric functions, and other special functions including Bessel functions, modified Bessel functions, and the incomplete gamma functions, along with their respective derivatives. These expressions are obtained for specific parameter values using symbolic computation in Maple. The results contribute to the broader analytical framework for solving inhomogeneous linear differential equations with applications in engineering, mathematical physics, and biological modeling.
Journal Article
Analytical Properties of k ‐Generalized Digamma and Polygamma Functions Derived From Dilcher‐Type Gamma 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.
Journal Article
On the integral operators pertaining to a family of incomplete I-functions
This paper introduces a new incomplete I-functions. The incomplete I-function is an extension of the I-function given by Saxena [1] which is a extension of a familiar Fox's H-function. Next, we find the several interesting classical integral transform of these functions and also find the some basic properties of incomplete I-function. Further, numerous special cases are obtained from our main results among which some are explicitly indicated. Incomplete special functions thus obtained consisting of probability theory has many potential applications which are also presented. Finally, we find the solution of non-homogeneous heat conduction equation in terms of Incomplete I-function.
Journal Article