Related papers: Wallis's Formula and the Probability Integral

We show that it is consistent that the Borel Conjecture and the dual Borel Conjecture hold simultaneously.

This note generalizes the notion of conditional probability to Riesz spaces using the order-theoretic approach. With the aid of this concept, we establish the law of total probability and Bayes' theorem in Riesz spaces; we also prove an…

New integral formulas involving the Meijer $G$-function are derived using recent results concerning distributional characterisations and distributional transformations in probability theory.

We present a proof of Moessner's theorem by double induction, using only basic rules of arithmetic. No prerequisite knowledge is assumed. Familiarity with summation is advised.

In this note we give two proofs of Brooks' Theorem. The first is obtained by modifying an earlier proof and the second by combining two earlier proofs. We believe these proofs are easier to teach in Computer Science courses.

In this short note I restate and simplify the proof of the impossibility of probabilistic induction from Popper (1992). Other proofs are possible (cf. Popper (1985)).

By changing variables in a suitable way and using dominated convergence methods, this note gives a short proof of Stirling's formula and its refinement.

The arithmetic function of two variables is defined. Some properties of the function are given along with the formula that is an analog of the so-called Mobius' inversion formula. A heuristic statement is suggested.

The purpose of this note is to give an affirmative answer to a conjecture appearing in [Integral Transforms Spec. Funct. 26 (2015) 90-95].

This note formally defines the concept of coinductive validity of judgements, and contrasts it with inductive validity. For both notions it shows how a judgement is valid iff it has a formal proof. Finally, it defines and illustrates the…

We provide elementary proof of several congruences involving single sum and multisums of binomial coefficients.

In this note we give a detailed proof of a theorem of Aubin.

We prove a recent conjecture of Hadjicostas concerning a double integral formula involving the zeta and the gamma functions.

This short note present a "proof" of $P\neq NP$. The "proof" with double quotation marks is to indicate that we do not know whether the proof is correct or not (We're confused because we do know in which we make the mistakes).

This paper is devoted to establishing several new formulas relating Bernoulli and Stirling numbers of both kinds.

A very simple but useful almost sure convergence theorem of probability is given.

We formulate several polynomial identities. One side of these identities has a nice simple form. Whereas the other has a form of a polynomial whose coefficients contain binomial coefficients double factorials or (and) rising factorials. The…

We will give a simple proof of the ambiguous class number formula.

We present an astonishingly simple and elegant proof of the celebrated Basel problem.

We give positive answer to two conjectures posed by M. E. H Ismail in his monograph [Classical and quantum orthogonal polynomials in one variable, Cambridge University Press, 2005].