Related papers: Some new properties of the beta function and Raman…
We consider convexity and monotonicity properties for some functions related to the $q$-gamma function. As applications, we give a variety of inequalities for the $q$-gamma function, the $q$-digamma function $\psi_q(x)$, and the $q$-series.…
This paper proposes a reformulation of the Riemann Xi function in order to investigate its properties. The reformulated function, which depicts the Xi function as the weighted sum of incomplete gamma functions, is validated, and a number of…
In this expository and survey paper, along one of main lines of bounding the ratio of two gamma functions, we look back and analyse some inequalities, the complete monotonicity of several functions involving ratios of two gamma or $q$-gamma…
We consider integral and series transformations, which are associated with Ramanujan's identities, involving various arithmetic functions and a ratio of products of Riemann's zeta functions of different arguments. Reciprocal inversion…
The Gauss hypergeometric function ${}_2F_1(a,b,c;z)$ can be computed by using the power series in powers of $z, z/(z-1), 1-z, 1/z, 1/(1-z),(z-1)/z$. With these expansions ${}_2F_1(a,b,c;z)$ is not completely computable for all complex…
In the first part we establish a connection between the Euler-Maclaurin summation formula and the Rota-Baxter functional equation. In the second part we give a simple proof of a formula, due to Ramanujan, on the summation of certain…
Motivated by open questions in the papers " Refinements and sharpenings of some double inequalities for bounding the gamma function" and "Complete monotonicity and monotonicity of two functions defined by two derivatives of a function…
The variable change w=exp(u) is applied to establish novel integral representations of the incomplete gamma-function, hypergeometric F-function,confluent hypergeometric /Phi-function and beta-function, and to analyze these functionsas as…
Integral representations of hypergeometric functions proved to be a very useful tool for studying their properties. The purpose of this paper is twofold. First, we extend the known representations to arbitrary values of the parameters and…
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…
Recently, Garthwaite-Penniston have shown that the coefficients of Ramanujan's mock theta function $\omega$ satisfy infinitely many congruences of Ramanujan-type. In this work we give the first explicit examples of congruences for…
We improve the upper bounds of the following inequalities proved in [H. Alzer and N. Batir, Monotonicity properties of the gamma function, Appl. Math. Letters, 20(2007), 778-781]. \begin{equation*}…
We provide new representations for the finite parts at the poles and the derivative at zero of the Barnes zeta function in any dimension in the general case. These representations are in the forms of series and limits. We also give an…
In this paper, we present a new method for finding identities for hypergeoemtric series, such as the (Gauss) hypergeometric series, the generalized hypergeometric series and the Appell-Lauricella hypergeometric series. Furthermore, using…
In this paper our aim is to establish new integral representations for the Fox--Wright function ${}_p\Psi_q[^{(\alpha_p,A_p)}_{(\beta_q,B_q)}|z]$ when $$\mu=\sum_{j=1}^q\beta_j-\sum_{k=1}^p\alpha_k+\frac{p-q}{2}=-m,\;\;m\in\mathbb{N}_0.$$…
In the paper, we present a monotonicity result of a function involving the gamma function and the logarithmic function, refine a double inequality for the gamma function, and improve some known results for bounding the gamma function.
In this paper, we obtain analytical solutions of some definite integrals of Srinivasa Ramanujan [Mess. Math., XLIV, 75-86, 1915] in terms of Meijer's $G$-function by using Laplace transforms of $ \sin(\beta x^{2}),\cos(\beta x^{2}),…
In this paper we derive rapidly converging series for Catalan's constant and for Ap\'ery's constant. The method may be easily generalised to produce new series representations for other values of the Riemann zeta function and the Dirichlet…
In this paper, additional properties of the lower gamma functions and the error functions are introduced and proven. In particular, we prove interesting relations between the error functions and Laplace transform.
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 $G_{2,3}$.…