In this note, we show that Radford's formula for the fourth power of the antipode can be proven for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This of course not only includes the case of a finite-dimensional Hopf algebra but also the case of any Hopf algebra with integrals (co-Frobenius Hopf algebras). The proof follows in a few lines from well-known formula...

this paper presents a proof of stirling's formula using haar wavelets and some properties of hilbert space, such as parseval's identity. the present paper shows a connection between haar wavelets and certain sequences.

The idea is that the equality of the Euler product and Riemann-Hadamard product for zeta allows extraction of an exact formula for a suitably-weighted counting of primes, a sum over zeros of zeta, via a contour integration of the logarithmic derivatives. As observed by [Guinand 1947] and [Weil 1952], [Weil 1972], the classical formulas are equalities of values of a certain distribution, in the ...

How do the two most important fundamental constants of mathematics, e and π, find their way into an asymptotic formula for the product of integers? We give two very different arguments (one will not show the full formula) that, between them, illustrate a good number of basic asymptotic methods. The formal language of Asymptopia, such as o(n) and O(n), is deferred to Chapter 2. Two further argum...