Related papers: Bounds for the logarithm of the Euler gamma functi…
The asymptotic expansion of the Touchard polynomials $T_n(z)$ (also known as the exponential polynomials) for large $n$ and complex values of the variable $z$, where $|z|$ may be finite or allowed to be large like $O(n)$, has been recently…
We establish an asymptotic formula for the logarithmic mean value of a 1-bounded multiplicative function that is sharp in many cases of interest. We derive from it a variety of applications, making progress on several old problems. As a…
In a closed manifold of positive dimension $n$, we estimate the expected volume and Euler characteristic for random submanifolds of codimension $r\in \{1,...,n\}$ in two different settings. On one hand, we consider a closed Riemannian…
We consider the uniform asymptotic expansion for the Gauss hypergeometric function \[{}_2F_1(a+\epsilon\lambda,b;c+\lambda;x),\qquad 0<x<1\] as $\lambda\to+\infty$ in the neigbourhood of $\epsilon x=1$ when the parameter $\epsilon>1$ and…
We show in this paper how to construct Symanzik polynomials and the Schwinger parametric representation of Feynman amplitudes for gauge theories in an unspecified covariant gauge. The complete Mellin representation of such amplitudes is…
Hafner and Stopple proved a conjecture of Zagier, that the inverse Mellin transform of the symmetric square $L$-function associated to the Ramanujan tau function has an asymptotic expansion in terms of the non-trivial zeros of the Riemann…
We introduce approximation functions of $li(x)$ for all $x\ge e$: (1) $\displaystyle li_{\underline{\omega},\alpha}(x) = \frac{x}{\log(x)}\left( \alpha\frac{\underline{m}!}{\log^{\underline{m}}(x)} +…
We study the isentropic Euler equations with time-dependent damping, given by $\frac{\mu}{(1+t)^\lambda}\rho u$. Here, $\lambda,\mu$ are two non-negative constants to describe the decay rate of damping with respect to time. We will…
We develop a unified analytical and dynamical framework for the qualitative study of the one-parameter family of generalized Dirichlet eta functions $\eta_{a}(t)=\sum_{m\ge0}(-1)^{m}(am+1)^{-t}$, $a>0$, $t>0$, which includes the classical…
Several asymptotic expansions of parabolic cylinder functions are discussed and error bounds for remainders in the expansions are presented. In particular Poincar{\'e}-type expansions for large values of the argument $z$ and uniform…
The electrostatic interpretation of zeros of Jacobi polynomials, due to Stieltjes and Schur, enables us to obtain the complete asymptotic expansion as $n \to \infty$ of the minimal logarithmic potential energy of $n$ point charges…
A pair of linearly independent asymptotic solutions are constructed for the second-order linear difference equation {equation*} P_{n+1}(x)-(A_{n}x+B_{n})P_{n}(x)+P_{n-1}(x)=0, {equation*} where $A_n$ and $B_n$ have asymptotic expansions of…
Asymptotic expansions of series $\sum_{k=0}^\infty \epsilon^k(k+a)^\gamma e^{-(k+a)^\alpha x}$ and $\sum_{k=0}^\infty \epsilon^k(k+a)^\gamma / (x(k+a)^\alpha+1)^\mu}$ in powers of $x$ as $x\to+0$ are found, where $\epsilon=1$ or…
In this paper we derive new representations for the incomplete gamma function, exploiting the reformulation of the method of steepest descents by C. J. Howls (Howls, Proc. R. Soc. Lond. A 439 (1992) 373--396). Using these representations,…
In this paper some Tur\'an type inequalities for the general Bessel function, monotonicity and bounds for its logarithmic derivative are derived. Moreover we find the series representation and the relative extrema of the Tur\'anian of…
In this paper, we investigate a nonlocal equation involving the logarithmic Laplacian with indefinite nonlinearities: \begin{equation*} \left\{ \begin{array}{ll} L_\Delta u(x)=a(x_n)f(u), & x\in\Omega, \\ u(x)=0,& x\in…
The transformations of the sum identities for generalized harmonic and oscillatory numbers, obtained earlier in our recent report [1], enable us to derive the new identities expressed in terms of the corresponding square roots of x. At…
In this paper, the logarithmically complete monotonicity property for a functions involving $q$-gamma function is investigated for $q\in(0,1).$ As applications of this results, some new inequalities for the $q$-gamma function are…
We prove characterizations of Appell polynomials by means of symmetric property. For these polynomials, we establish a simple linear expression in terms of Bernoulli and Euler polynomials. As applications, we give interesting examples. In…
The real and complex zeros of the parabolic cylinder function $U(a,z)$ are studied. Asymptotic expansions for the zeros are derived, involving the zeros of Airy functions, and these are valid for $a$ positive or negative and large in…