Cohen, Lewin and Zagier found four ladders that entail the polylogarithms Lin( k 1 ) := P r>0 kr 1 =r n at order n = 16, with indices k 360, and 1 being the smallest known Salem number, i.e. the larger real root of Lehmer's celebrated polynomial 10 + 9 7 6 5 4 3 + + 1, with the smallest known non-trivial Mahler measure. By adjoining the index k = 630, we generate a fth ladder at order 16 and a ...

We present rapidly converging series for the Khintchine constant and for general “Khintchine means” of continued fractions. We show that each of these constants can be cast in terms of an efficient free-parameter series, each series involving values of the Riemann zeta function, rationals, and logarithms of rationals. We provide an alternative, polylogarithm series for the Khintchine constant a...

The Rayleigh equation 3/2R+RR+pρ(-1)=0 with initial conditions R(0)=R(0), R(0)=0 models the collapse of an empty spherical bubble of radius R(T) in an ideal, infinite liquid with far-field pressure p and density ρ. The solution for r≡R/R(0) as a function of time t≡T/T(c), where R(T(c))≡0, is independent of R(0), p, and ρ. While no closed-form expression for r(t) is known, we find that r(0)(t)=(...

in this paper, we consider a new class of analytic functions in the unit disk using polynomials of order alpha. we give some sufficient conditions for functions belonging to this class.