Acceleration of generalized hypergeometric functions through precise remainder asymptotics

We express the asymptotics of the remainders of the partial sums {s_n} of the generalized hypergeometric function q+1_F_q through an inverse power series z^n n^l \sum_k c_k/n^k, where the exponent l and the asymptotic coefficients {c_k} may be recursively computed to any desired order from the hypergeometric parameters and argument. From this we derive a new series acceleration technique that can be applied to any such function, even with complex parameters and at the branch point z=1. For moderate parameters (up to approximately ten) a C implementation at fixed precision is very effective at computing these functions; for larger parameters an implementation in higher than machine precision would be needed. Even for larger parameters, however, our C implementation is able to correctly determine whether or not it has converged; and when it converges, its estimate of its error is accurate.Comment: 36 pages, 6 figures, LaTeX2e. Fixed sign error in Eq. (2.28), added several references, added comparison to other methods, and added discussion of recursion stabilit

International prevalence, recognition, and treatment of cardiovascular risk factors in outpatients with atherothrombosis

CONTEXT: Atherothrombosis is the leading cause of cardiovascular morbidity and mortality around the globe. To date, no single international database has characterized the atherosclerosis risk factor profile or treatment intensity of individuals with atherothrombosis. OBJECTIVE: To determine whether atherosclerosis risk factor prevalence and treatment would demonstrate comparable patterns in many countries around the world. DESIGN, SETTING, AND PARTICIPANTS: The Reduction of Atherothrombosis for Continued Health (REACH) Registry collected data on atherosclerosis risk factors and treatment. A total of 67,888 patients aged 45 years or older from 5473 physician practices in 44 countries had either established arterial disease (coronary artery disease [CAD], n = 40,258; cerebrovascular disease, n = 18,843; peripheral arterial disease, n = 8273) or 3 or more risk factors for atherothrombosis (n = 12,389) between 2003 and 2004. MAIN OUTCOME MEASURES: Baseline prevalence of atherosclerosis risk factors, medication use, and degree of risk factor control. RESULTS: Atherothrombotic patients throughout the world had similar risk factor profiles: a high proportion with hypertension (81.8%), hypercholesterolemia (72.4%), and diabetes (44.3%). The prevalence of overweight (39.8%), obesity (26.6%), and morbid obesity (3.6%) were similar in most geographic locales, but was highest in North America (overweight: 37.1%, obese: 36.5%, and morbidly obese: 5.8%; P or =3 risk factors to 85.6% for CAD), and other evidence-based risk reduction therapies. Current tobacco use in patients with established vascular disease was substantial (14.4%). Undertreated hypertension (50.0% with elevated blood pressure at baseline), undiagnosed hyperglycemia (4.9%), and impaired fasting glucose (36.5% in those not known to be diabetic) were common. Among those with symptomatic atherothrombosis, 15.9% had symptomatic polyvascular disease. CONCLUSION: This large, international, contemporary database shows that classic cardiovascular risk factors are consistent and common but are largely undertreated and undercontrolled in many regions of the world