Related papers: A Probabilistic Proof of a Wallis-type Formula for…
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…
Using mostly elementary results and functions from probability, we prove Wallis's formula for pi: pi/2 = prod_n (2n * 2n) / ((2n-1) * (2n+1)). The proof involves normalization constants and the Gamma function, Standard normal, and the…
In this note we show that any proof of Wallis's formula or of the probability integral formula proves both assertions.
We generalize the derivation of the Wallis formula for $\pi$ from a variational computation of the spectrum of the Hydrogen atom. We obtain infinite product formulas for certain combinations of gamma functions, which include irrational…
Stirling's formula is a powerful asymptotic approximation of the factorial function. Many well-known proofs of this formula are grounded in integral calculus. In this paper, we present an alternative proof of Stirling's formula using only…
We give a new proof for a product formula of Jacobi which turns out to be equivalent to a $q$-trigonometric product which was stated without proof by Gosper. We apply this formula to derive a $q$-analogue for the Gauss multiplication…
In this Note, we start off with the primary representation of e and from there present an elementary short proof for the Wallis formula for $\pi$.
We consider products of $q$-gamma functions with rational arguments, and prove several $q$-generalizations of recent works concerning products of gamma functions. In particular, we consider products indexed by Dirichlet characters, and…
We generalize Wallis's 1655 infinite product for $\pi/2$ to one for $(\pi/K)\csc(\pi/K)$, as well as give new Wallis-type products for $\pi/4, 2, \sqrt{2+\sqrt2}, 2\pi/3\sqrt3,$ and other constants. The proofs use a classical infinite…
We comment on the Friedmann and Hagen's quantum mechanical derivation of the Wallis formula for $\pi$. In particular, we demonstrate that not only the Gaussian trial function, used by Friedmann and Hagen, but also the Lorentz trial function…
In this article, we derive, using Fourier series and multiple derivative of the function $\pi/\sin(\pi x)$, series representations for positive powers of $\pi$. We also show that the Euler-Wallis product can be easily obtained from the same…
Based on the $\kappa$-deformed functions ($\kappa$-exponential and $\kappa$-logarithm) and associated multiplication operation ($\kappa$-product) introduced by Kaniadakis (Phys. Rev. E \textbf{66} (2002) 056125), we present another…
We show how the formulas in paper Variae considerationes circa series hypergeometricas written by Euler imply the duplication formula for the Gamma-function. This paper can be seen as an Addendum to a previous paper by the author.
In this paper we will give a proof of a certain summation formula for Gamma functions utilizing Gegenbauer polynomials.
We introduce a gamma function $\Ga(x,z)$ in two complex variables which extends the classical gamma function $\Ga(z)$ in the sense that $\lim_{x\to 1}\Ga(x,z)=\Ga(z)$. We will show that many properties which $\Ga(z)$ enjoys extend in a…
Two kinds of infinite product representations for Vign\'eras multiple gamma function are presented. As an application of these formulas, a multiplication formula for the function is derived.
The Stirling approximation formula for $n!$ dates from 1730. Here we give new and instructive proofs of this and related approximation formulae via tools of probability and statistics. There are connections to the Central Limit Theorem and…
We give an asymptotic expansion (the higher Stirling formula) and an infinite product representation (the Weierstrass product formula) of the Vign\'{e}ras multiple gamma function by considering the classical limit of the multiple q-gamma…
An asymptotic expansion of a ratio of products of gamma functions is derived. It generalizes a formula which was stated by Dingle, first proved by Paris, and recently reconsidered by Olver.
New proofs of the duplication formulae for the gamma and the Barnes double gamma functions are derived using the Hurwitz zeta function. Concise derivations of Gauss's multiplication theorem for the gamma function and a corresponding one for…