Effective Laguerre asymptotics (2007)
Borwein, David, Borwein, Jonathan, Crandall, Richard
It is known that the generalized Laguerre polynomials can enjoy sub-exponential growth for large primary index. Specifically, for certain fixed parameter pairs $(a,z)$ one has the large-$n$...
Effective Laguerre asymptotics (2007)
Borwein, David, Borwein, Jonathan, Crandall, Richard
It is known that the generalized Laguerre polynomials can enjoy sub-exponential growth for large primary index. Specifically, for certain fixed parameter pairs $(a,z)$ one has the large-$n$...
Hypergeometric forms for Ising-class integrals (2006)
Borwein, Jonathan M., Crandall, Richard, Bailey, David H.
We apply experimental-mathematical principles to analyze integrals Cn. These are generalizations of a previous integral Cn := Cn,1 relevant to the Ising theory of solid-state physics [8]. We find...
Integrals of the Ising class (2006)
Borwein, Jonathan M., Crandall, Richard, Bailey, David H.
From an experimental-mathematical perspective we analyze “Isingclass” integrals. These are structurally related n-dimensional integrals we call Cn, Dn, En, where Dn is a magnetic susceptibility...
Borwein, Jonathan M., Bailey, David H., Crandall, Richard
By a ``box integral'' we mean here an expectation $\langle |\vec r - \vec q|^s \rangle$ where $\vec r$ runs over the unit $n$-cube, with $\vec q$ and $s$ fixed, explicitly: \begin{eqnarray*}...
Prime numbers: a computational perspective. Second Edition (2005)
Richard Crandall, Carl Pomerance, Richard Crandall, Carl Pomerance
Cover illustration: The cover shows a magnified view—through a watchmaker’s loupe—of a very small portion of an actual poster giving the 7.2 million decimal digits of the prime 2 24036583-1....
On the dynamics of certain recurrence relations (2004)
Borwein, Jonathan M., Crandall, Richard, Borwein, David, Mayer, Raymond
In previous analyses \cite{borcra1, borcra2} the remarkable AGM continued fraction of Ramanujan---denoted ${\cal R}_1(a,b)$---was proven to converge for almost all complex parameter pairs $(a,b)$. It...
On the Ramanujan AGM fraction. Part I: the Real-parameter Case (2003)
Borwein, Jonathan M., Crandall, Richard, Fee, Greg
The Ramanujan AGM continued fraction is a construct enjoying attractive algebraic properties such as a striking arithmetic-geometric mean (AGM) relation and elegant connections with elliptic-function...
On the Ramanujan AGM fraction. Part II: the Complex-parameter Case (2003)
Borwein, Jonathan M., Crandall, Richard
The Ramanujan continued fraction is interesting in many ways; e.g. for certaiun complex parameters (eta, a, b) one has an attractive AGM relation Reta(a,b) + Reta(b,a) = 2Reta((a+b)/2, sqrt{ab})....
On the Ramanujan AGM fraction (2003)
Borwein, Jonathan M., Crandall, Richard, Fee, Greg
The Ramanujan AGM fraction is a construct $${\cal R}_\eta(a,b) =\,\frac{a}{\displaystyle \eta+\frac{b^2}{\displaystyle \eta +\frac{4a^2}{\displaystyle \eta+\frac{9b^2}{\displaystyle...
On the binary expansions of algebraic numbers (2003)
Borwein, Jonathan M., Bailey, David H., Crandall, Richard, Pomerance, Carl
Employing concepts from additive number theory, together with results on binary evaluations and partial series, we establish bounds on the density of 1's in the binary expansions of real algebraic...
Algorithms for localized dart-throwing (2000)
Abstract. How does one “throw localized darts, ” that is, place random points in a subregion? It is inefficient just to find points in a larger region, only to check painstakingly every generated...
Computational strategies for the Riemann zeta function (1999)
Borwein, Jonathan M., Bradley, David M., Crandall, Richard
We provide a compendium of evaluation methods for the Riemann zeta function, presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new....
A Search for Wieferich and Wilson Primes (1997)
Richard Crandall, Karl Dilcher, Carl Pomerance
Abstract. An odd prime p is called a Wieferich prime if 2 p−1 ≡ 1 (mod p 2); alternatively, a Wilson prime if (p − 1)! ≡−1 (mod p 2). To date, the only known Wieferich primes are p = 1093...
On the Khintchine Constant (1995)
Bailey, David H., Borwein, Jonathan M., Crandall, Richard
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...
On the Evaluation of Euler Sums (1995)
Richard Crandall, Richard E. Cr, Joe P. Buhler
this paper can be applied to these alternative sums, to yield corresponding converging series for each. These methods will perhaps be applicable in the future to multiple zeta sums