Related papers: An accurate approximation formula for gamma functi…

In this article, we obtain a super-exponential rate of convergence in total variation between the traces of the first $m$ powers of an $n\times n$ random unitary matrices and a $2m$-dimensional Gaussian random variable. This generalizes…

For $f$ analytic and close to convex in $D=\{z: |z|< 1\}$, we give sharp estimates for the logarithmic coefficients $\gamma_{n}$ of $f$ defined by $\log \dfrac{f(z)}{z}=2\sum_{n=1}^{\infty} \gamma_{n}z^{n}$ when $n=1, 2,3$.

A recently published result states inequalities of the harmonic mean of the digamma function. In this work, we prove among others results that for all positive real numbers $x\neq 1$, $$-\gamma<-\gamma…

We prove that the functions Phi(x)=[Gamma(x+1)]^{1/x}(1+1/x)^x/x and log Phi(x) are Stieltjes transforms.

In this paper, we prove Raabe-type integral formulas for gamma function via left and right sided Riemann-Liouville fractional integrals. As corollaries, we give the left and right sided repeated integration formulas for the log-gamma and…

In this work we construct Gaussian beam approximations to solutions of the high frequency Helmholtz equation with a localized source. Under the assumption of non-trapping rays we show error estimates between the exact outgoing solution and…

An exact solution is obtained for the gravitational bending of light in static, spherically symmetric metrics which includes the Schwarzschild-de Sitter spacetime and also the Mannheim-Kazanas metric of conformal Weyl gravity. From the…

Simple upper and lower bounds are obtained for the integral $\int_0^x\mathrm{e}^{-\gamma t}t^\nu I_\nu(t)\,\mathrm{d}t$, $x>0$, $\nu>-\frac{1}{2}$, $0<\gamma<1$. Most of our bounds for this integral are tight as $x\rightarrow\infty$. We…

Many popular learning algorithms (E.g. Regression, Fourier-Transform based algorithms, Kernel SVM and Kernel ridge regression) operate by reducing the problem to a convex optimization problem over a vector space of functions. These methods…

Simple bounds are obtained for the integral $\int_0^x\mathrm{e}^{-\gamma t}t^\nu I_\nu(t)\,\mathrm{d}t$, $x>0$, $\nu>-1/2$, $0\leq\gamma<1$, together with a natural generalisation of this integral. In particular, we obtain an upper bound…

This paper presents a novel systematic methodology to obtain new simple and tight approximations, lower bounds, and upper bounds for the Gaussian Q-function, and functions thereof, in the form of a weighted sum of exponential functions.…

Accurate fitting formulae to the synchrotron function, $F(x)$, and its complementary function, $G(x)$, are performed and presented. The corresponding relative errors are less than $0.26\%$ and $0.035\%$ for $F(x) $ and $G(x)$, respectively.…

This paper is inspired by the work of J. S\'{a}ndor in 2006. In the paper, the authors establish some double inequalities involving the ratio $ \frac{\Gamma_{q}(x+1)}{ \Gamma_{q} \left( x+\frac{1}{2}\right)}$, where $\Gamma_{q}(x)$ is the…

The classical Lerch's formula states the following normalized product: $$\prod_{n=0}^\infty(x+n)=\frac{\sqrt{2\pi}}{\Gamma(x)},\quad \textrm{Re}(x)>0, $$ where $\Gamma(x)$ is the Euler gamma function. In this note, by using Barnes' multiple…

In our previous publications we have introduced the cosine product-to-sum identity [17] $$ \prod\limits_{m = 1}^M {\cos \left( {\frac{t}{{{2^m}}}} \right)} = \frac{1}{{{2^{M - 1}}}}\sum\limits_{m = 1}^{{2^{M - 1}}} {\cos \left( {\frac{{2m -…

We consider a family of non-local and non-convex functionals, and we prove that their Gamma-liminf is bounded from below by a positive multiple of the Sobolev norm or the total variation. As a by-product, we answer some open questions…

An approximation, in the sense of $\Gamma$-convergence and in any dimension $d\geq1$, of Griffith-type functionals, with $p-$growth ($p>1$) in the symmetrized gradient, is provided by means of a sequence of non-local integral functionals…

The topic of gamma type functions and related functional equation $f(x+1)=g(x)f(x)$ has been seriously studied from first half of the twentieth century till now. Regarding unique solutions of the equation the asymptotic condition…

The gamma density function is usually defined in interval between zero and infinity. This paper introduces an upper and a lower boundary to this distribution. The parameters which characterize the truncated gamma distribution are evaluated.…

We exhibit a class of "relatively curved" $\vec{\gamma}(t) := (\gamma_1(t),\dots,\gamma_n(t))$, so that the pertaining multi-linear maximal function satisfies the sharp range of H\"{o}lder exponents, \[ \left\| \sup_{r > 0} \ \frac{1}{r}…