Related papers: An Elementary Proof of Stirling's Formula
We describe a simple method that produces automatically closed forms for the coefficients of continued fractions expansions of a large number of special functions. The function is specified by a non-linear differential equation and initial…
Recall that a Stirling permutation is a permutation on the multiset $\{1,1,2,2,\ldots,n,n\}$ such that any numbers appearing between repeated values of $i$ must be greater than $i$. We call a Stirling permutation ``flattened'' if the…
In 1655, John Wallis whilst at the University of Oxford discovered the famous and beautiful formula for pi, now known as Wallis' Product. Since then, several analogous formulae have been discovered generalising the original. One more modern…
We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional…
We show that an elementary proof of Fermat's Last Theorem (FLT) exists. Our paper also extends the scope of FLT from integers to all rational numbers.
Extensions of the $Stirling$ numbers of the second kind and $Dobinski$ -like formulas are proposed in a series of exercises for graduates. Some of these new formulas recently discovered by me are to be found in the source paper $ [1]$.…
We give a new proof of Brooks' theorem that immediately implies a strengthening of Brooks' theorem, known as Catlin's theorem.
An astonishingly simple analytical frequency approximation formula for a class of strongly nonlinear oscillators is derived and applied to various example systems yielding useful quick estimates.
We will prove the Brannan conjecture for particular values of the parameter. The basic tool of the study is an integral representation published in a recent work [3].
We give elementary proofs of several Stirling's precise bounds. We first improve all the precise bounds from the literature and give new precise bounds. In particular, we show that for all $n\ge 8$ $$\sqrt{2\pi n}\left(\frac{n}{e}\right)^n…
Possibilities are considered to simplify the computation of several statistical functions used to test statistical hypotheses when processing observations: the inverse normal distribution, the Student's t-distribution, and the criterion for…
We introduce a new approach to the the asymptotic iteration method (AIM) by means of which we establish the standard AIM connection with the continued fractions technique and we develop a novel termination condition in terms of the…
Leibniz's rule for the $n$-th derivative of a product is a very well known and extremely useful formula. In this article, we introduce an analogous explicit formula for the $n$-th derivative of a quotient of two functions. Later, we use…
In this paper, we obtain some formulae for harmonic sums, alternating harmonic sums and Stirling number sums by using the method of integral representations of series. As applications of these formulae, we give explicit formula of several…
In this paper we present a short and elementary proof for the error in Simpson's rule.
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.
In this note we investigate mixed partitions with extra condition on the sizes of the blocks. We give a general formula and the generating function. We consider in more details a special case, determining the generating functions, some…
Factorization -- a simple form of standardization -- is concerned with reduction strategies, i.e. how a result is computed. We present a new technique for proving factorization theorems for compound rewriting systems in a modular way, which…
An analytic proof is proposed of Wiener's theorem on factorization of positive definite matrix-functions.
The superiority of stochastic symplectic methods over non-symplectic counterparts has been verified by plenty of numerical experiments, especially in capturing the asymptotic behaviour of the underlying solution process. How can one…