Related papers: An affirmative answer to an open problem on Ramanu…
In this paper, the power series and hypergeometric series representations of the beta and Ramanujan functions \begin{equation*} \mathcal{B}\left( x\right) =\frac{\Gamma \left( x\right)^{2}}{\Gamma \left( 2x\right) }\text{ and…
In this paper, a generalization of Ramanujan's cubic transformation, in the form of an inequality, is proved for zero-balanced Gaussian hypergeometric function $F(a,b;a+b;x)$, $a,b>0$.
In the paper, necessary and sufficient conditions are presented for a function involving a ratio of gamma functions to be logarithmically completely monotonic. This extends and generalizes the main result in [\emph{Inequalities and…
In this paper we investigate the monotonicity properties related to the ratio of gamma functions, from which some related asymptotics and inequalities are established. Some special cases also confirm the conjectures of C.-P. Chen…
We will use a discrete analogue of the classical \emph{Laplace} method to show that for infinitely many positive integers $n$, the main term of the asymptotic expansions of scaled confluent basic hypergeometric functions, including the…
We extend the validity range of a Ramanujan's hypergeometric transformation formula proved by Berndt, Bhargava and Garvan, Trans. Amer. Math. Soc. 347, 4163 (1995) and study its implications. Relations to special values of complete elliptic…
In this paper, we obtain analytical solution of an unsolved integral $\textbf{R}_{C}(m,n)$ of Srinivasa Ramanujan [$\textit{Mess. Math}$., XLIV, 75-86, 1915], using hypergeometric approach, Mellin transforms, Infinite Fourier cosine…
This paper systematically investigates the absolute monotonicity of two function families associated with the Gaussian hypergeometric function $F(a, b; c; x)$ (where $a,b,c\in\mathbb{R}_+$): $\mathcal{F}_p(x)=(1-x)^pF(a,b;c;x)$ and…
In this article, we study the asymptotics of harmonic functions. A typical method is by proving monotonicity formulas of a version of rescaled Dirichlet energy, and use it to study the renormalized solution -- the Almgren's blowup. However,…
In this paper, we establish the irrationality of some open problems in mathematics based on using a recursive formula that generate the complete sequence of numbers. see [1] But before getting into that we begin with some Ramanujan notable…
We introduce completely monotonic functions of order $r>0$ and show that the remainders in asymptotic expansions of the logarithm of Barnes double gamma function and Euler's gamma function give rise to completely monotonic functions of any…
For $m,n\in \mathbb{N}$, let $0 < \alpha_i,\beta_j,\lambda_{ij} \leq 1$ be such that $\sum_{j=1}^n \lambda_{ij} = \alpha_i$, $\sum_{i=1}^m \lambda_{ij} = \beta_j$, and $\sum_{i=1}^m \alpha_i = \sum_{j=1}^n \beta_j \leq 1$. We prove that the…
The Ramanujan sequence $ \{\theta_{n}\}_{n \geq 0}$, defined as $$ \theta_{0}= \frac{1}{2} \ , \ \ \ \theta_{n} = \left(\ \ \frac{e^{n}}{2} - \sum_{k=0}^{n-1} \frac{n^{k}}{k !} \ \ \right) \cdot \frac{n !}{n^{n}} \ , \ \ n \geq 1 \ ,$$ has…
In this paper, the authors present sharp approximations in terms of sine function and polynomials for the so-called Ramanujan constant (or the Ramanujan $R$-function) $R(a)$, by showing some monotonicity, concavity and convexity properties…
We construct a monotonicity formula for a class of free boundary problems associated with the stationary points of the functional \[ J(u)=\int_\Omega F(|\nabla u|^2)+\mbox{meas}(\{u>0\}\cap \Omega), \] where $F$ is a density function…
In the paper, the authors concisely survey and review some functions involving the gamma function and its various ratios, simply state their logarithmically complete monotonicity and related results, and find necessary and sufficient…
We investigate conditions for logarithmic complete monotonicity of a quotient of two products of gamma functions, where the argument of each gamma function has different scaling factor. We give necessary and sufficient conditions in terms…
In 1993 one of the authors formulated some conjectures on monotonicity of ratios for exponential series sections. They lead to more general conjecture on monotonicity of ratios of Kummer hypergeometric functions and was not proved from…
Some necessary and sufficient conditions for the existence of Cohen-Ramanujan expansions for arithmetical functions were provided by these authors in [\textit{arXive preprint arXive:2205.08466}, 2022]. Given two arithmetical functions $f$…
In [Temme N.M., Special functions. An introduction to the classical functions of mathematical physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996, Section 11.3.3.1] a uniform asymptotic expansion for the…