Gradimir V. Milovanovic, Vijay Gupta, Neha Malik
In the present paper, we consider (p,q)-analogue of the Beta operators and using it, we propose the integral modification of the generalized Bernstein polynomials. We estimate some direct results on local and global approximation. Also, we illustrate some graphs for the convergence of (p,q)-Bernstein-Durrmeyer operators for different values of the parameters p and q using Mathematica package.
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In the present paper, we consider $(p,q)$-analogue of the Baskakov-Beta operators and using it, we estimate some direct results on approximation. Also, we represent the convergence of these operators graphically using MATLAB.
The present paper deals with the Stancu type generalization of the Kantorovich discrete $q$-Beta operators. We establish some direct results, which include the asymptotic formula and error estimation in terms of the modulus of continuity and weighted approximation.
The goal in the paper is to advertise Dunkl extension of Szasz beta type operators. We initiate approximation features via acknowledged Korovkin and weighted Korovkin theorem and obtain the convergence rate from the point of modulus of continuity, second order modulus of continuity, the Lipschitz class functions, Peetre's K-functional and modulus of weighted continuity by Dunkl generalization of Szasz beta type operators.
This paper deals with the modified q-Stancu-Beta operators and we have investigated the statistical approximation theorems for these operators with the help of the Korovkin type approximation theorem. We have also established the rates of statistical convergence by means of the modulus of continuity and the Lipschitz type maximal function. Our results show that the rates of convergence of our operators are at least as fast as the classical Stancu-Beta operators.
On the generalized Dirichlet beta and Riemann zeta functions and Ramanujan-type formulae for beta and zeta values
本文利用双曲正割与余割函数的幂积分定义广义Dirichlet beta和Riemann zeta函数，建立其函数方程，并由奇整数zeta值的Ramanujan恒等式导出新的Ramanujan型公式。
The main object of this paper is to present generalizations of gamma, beta and hypergeometric functions. Some recurrence relations, transformation formulas, operation formulas and integral representations are obtained for these new generalizations.
The beta distribution is a two-parameter family of probability distributions whose distribution function is the (regularised) incomplete beta function. In this paper, the inverse incomplete beta function is studied analytically as univariate function of the first parameter. Monotonicity, limit results and convexity properties are provided. In particular, logarithmic concavity of the inverse incomplete beta function is established. In addition, we provide monotonicity results on inverses of a larger class of parametrised distributions that may be of independent interest.
The classical beta function B(x; y) is one of the most fundamental special functions, due to its important role in various fields in the mathematical, physical, engineering and statistical sciences. Useful extensions of the classical Beta function has been considered by many authors. In the present paper, our main objective is to study the convergence of extensions of classical beta function and introduce modified extension of classical beta function. It is interpreted numerically and geometrically in the view of convergence, further properties and integral presentations are established.
In this paper, we introduce and investigate a new extension of the beta function by means of an integral operator involving a product of Bessel-Struve kernel functions. We also define a new extension of the well-known beta distribution, the Gauss hypergeometric function and the confluent hypergeometric function in terms of our extended beta function. In addition, some useful properties of these extended functions are also indicated in a systematic way.
All beta-type functions, which are p-homogeneous, are determined. Applying this result, we show that a beta-type function is a homogeneous mean iff it is the harmonic one. A reformulation of a result due to Heuvers in terms of a Cauchy difference and the harmonic mean is given.
We aim to introduce a new extension of beta function and to study its important properties. Using this definition, we introduce and investigate new extended hypergeometric and confluent hypergeometric functions. Further, some hybrid representations of this extended beta function are derived which include some well known special functions and polynomials.
The main object of this paper is to present a new generalized beta function which defined by three parametres Mittag-Leffler function. We also introduce new generalizations of hypergeometric and confluent hypergeometric functions with the help of new generalized beta function. Furthermore, we obtained various properties of these functions such as integral representations, Mellin transforms, differentiation formulas, transformation formulas, recurrence relations and summation formula.
The logarithmic convexity of restrictions of the Beta functions to rays parallel to the main diagonal and the functional equation \[ ϕ\left( x+1\right) =\frac{x\left( x+k\right) }{\left( 2x+k+1\right) \left( 2x+k\right) }ϕ\left( x\right) ,\ \ \ \ \ \ x>0, \] for $k>0$ allow to get a characterizations of the Beta function. This fact and a notion of the beta-type function lead to a new characterization of the Gamma function.
We derive new reduction formulas for the incomplete beta function and the Lerch transcendent in terms of elementary functions. As an application, we calculate some new integrals. Also, we use these reduction formulas to test the performance of the algorithms devoted to the numerical evaluation of the incomplete beta function.
In this present paper, we establish the log-convexity and Turán type inequalities of extended $(p,q)$-beta functions. Also, we present the log-convexity, the monotonicity and Turán type inequalities for extended $(p,q)$-confluent hypergeometric function by using the inequalities of extended $(p,q)$-beta functions.
We present the evaluation of definite integrals in the classical table by I. S. Gradshteyn and I. M. Ryzhik that can be reduced to the beta function.
The table of Gradshteyn and Rhyzik contains some trigonometric integrals that can be expressed in terms of the beta function. We describe the evaluation of some of them.
In this paper we derive rapidly converging series for Catalan's constant and for Apéry's constant. The method may be easily generalised to produce new series representations for other values of the Riemann zeta function and the Dirichlet beta function.
Some inequalities for the beta function and its ratios
本文研究了贝塔函数差值和比值的不等式，推导了相关数学性质。
This paper studies the log-convexity of the extended beta functions. As a consequence, Turán-type inequalities are established.The monotonicity, log-convexity, log-concavity of extended hypergeometric functions are deduced by using the inequalities on extended beta functions,. The particular cases of those results also gives the Turán-type inequalities for extended confluent and extended Gaussian hypergeometric functions. Some reverse of Turán-type inequalities also derived.
The table of Gradshteyn and Ryzhik contains some integrals that can be expressed in terms of the incomplete beta function. We describe some elementary properties of this function and use them to check some of the formulas in the mentioned table.
Motivated mainly by certain interesting recent extensions of the Gamma, Beta and hypergeometric functions, we introduce here new extensions of the Beta function, hypergeometric and confluent hypergeometric functions. We systematically investigate several properties of each of these extended functions, namely their various integral representations, Mellin transforms, derivatives, transformations, summation formulas and asymptotics. Relevant connections of certain special cases of the main results presented here are also pointed out.
In this paper, we investigate inhomogeneous and simultaneous Diophantine approximation in beta dynamical systems. For $β>1$ let $T_β$ be the $β$-transformation on $[0,1]$. We determine the Lebesgue measure and Hausdorff dimension of the set \[\left\{(x,y)\in [0,1]^2: |T_β^nx-f(x,y)|<φ(n)\text{ for infinitely many }n\in\mathbb{N}\right\},\] where $f:[0,1]^2\to [0,1]$ is a Lipschitz function and $φ$ is a positive function on $\mathbb{N}$. Let $β_2\geq β_1>1$, $f_1,f_2:[0,1]\to [0,1]$ be two Lipschitz functions, $τ_1,τ_2$ be two positive continuous functions on $[0,1]$. We also determine the Haus…
A new definition for the Dirichlet beta function for positive integer arguments is discovered and presented for the first time. This redefinition of the Dirichlet beta function, based on the polygamma function for some special values, provides a general method for obtaining all special constants associated with Dirichlet beta function. We also show various new and fundamental relations between the polygamma function, Riemann zeta, the even-indexed euler numbers, the Dirichlet beta functions in a way never seen or imagined before.
We introduce new generalizations of the Gamma and the Beta functions. Their properties are investigated and known results are obtained as particular cases.
Israel M. Gelfand gave a geometric interpretation for general hypergeometric functions as sections of the tautological bundle over a complex Grassmannian $G_{k,n}$. In particular, the beta function can be understood in terms of $G_{2,3}$. In this manuscript, we construct one of the simplest generalizations of the Euler beta function by adding arbitrary-many odd variables to the classical setting. We also relate the beta super-function to the gamma and the hypergeometric function.
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