Related papers: Variations on a Theme by James Stirling
We present details of logically simplest integral sufficient for deducing the Stirling asymptotic formula for n!. It is the Newton integral, defined as the difference of values of any primitive at the endpoints of the integration interval.…
This paper presents invariants under gamma correction and similarity transformations. The invariants are local features based on differentials which are implemented using derivatives of the Gaussian. The use of the proposed invariant…
We discuss an earlier application of the Ritz variational method for strongly nonlinear problems. We clearly prove that the results derived for several extremely simple problems of supposedly physical and mathematical interest do not…
We describe proofs of the standard generating formulas for unsigned and signed Stirling numbers of the first kind that follow from a natural combinatorial interpretation based on cycle-colored permutations.
The Thomas - Fermi equation describing the screening of the Coulomb potential inside heavy neutral atoms is reconsidered. An accurate representation for its numerical solution was obtained by means of the variational principle. The proposed…
In this paper we present a method to derive Pi(x) and other Arithemtical functions that can be generated by a Dirichlet series by variational principles,we use a variational method to determine the solution for a Fredholm integral equation…
Our main aim is to apply the theory of regularly varying functions to the asymptotical analysis at infinity of solutions of Friedmann cosmological equations. A new constant $\Gamma$ is introduced related to the Friedmann cosmological…
A method is given for obtaining equivalence subgroups of a family of differential equations from the equivalence group of simpler equations of a similar form, but in which the arbitrary functions specifying the family element depend on…
Recently, the degenerate Stirling numbers of the first kind were introduced. In this paper, we give some formulas for the degenerate Stirling numbers of the first kind in the terms of the complete Bell polynomials with higher-order harmonic…
Dynamics of the delay rational difference equation $\displaystyle{z_{n+1}=\frac{\alpha+\beta z_{n-k}}{\gamma - z_{n}}}$ with complex parameters $\alpha$, $\beta$, $\gamma$ and arbitrary complex initial conditions is investigated. Existence…
We give short elementary expositions of combinatorial proofs of some variants of Euler's partitition problem that were first addressed analytically by George Andrews, and later combinatorially by others. Our methods, based on ideas from a…
Some extensions of an inequality from IMO'2001 are proven by means of the Lagrange multiplier criterion.
The discrete distribution of the length of longest increasing subsequences in random permutations of $n$ integers is deeply related to random matrix theory. In a seminal work, Baik, Deift and Johansson provided an asymptotics in terms of…
We study some asymptotic variants of the club principle. Along the way, we construct some forcings and use them to separate several of these principles
We study a sequence of differences related to the problem of finding the smallest factorial $n!$ greater than or equal to $a^n$, where $a > 1$, using the gamma function.
In this letter, we prove an inequality involving alternating binomial logarithmic sums by exploiting the variance of the logarithm of the maximum of independent and identically distributed exponential random variables. This inequality was…
In this work we look at the original fractional calculus of variations problem in a somewhat different way. As a simple consequence, we show that a fractional generalization of a classical problem has a solution without any restrictions on…
In this paper we obtain some sophisticated combinatorial congruences involving binomial coefficients and confirm two conjectures of the author and Davis. They are closely related to our investigation of the periodicity of the sequence…
We study a general type of series and relate special cases of it to Stirling series, infinite series discussed by Choi and Hoffman, and also to special values of the Arakawa-Kaneko zeta function, complementing and generalizing earlier…
The method of monotonization of difference schemes is being considered in the paper. The method was earlier proposed by the author for stationary problems. It is investigated in the paper more profoundly. The idea of the method is to build…