Rational divide-and-conquer relations (Q541804)

Summary: A rational divide-and-conquer relation, which is a natural generalization of the classical divide-and-conquer relation, is a recursive equation of the form \(f(bn) = R (f(n), f(n), \dots, f(b- 1)n) + g(n)\), where \(b\) is a positive integer \(\geq 2\); \(R\) a rational function in \(b - 1\) variables and \(g\) a given function. Closed-form solutions of certain rational divide-and-conquer relations which can be used to characterize the trigonometric cotangent-tangent and the hyperbolic cotangent-tangent function solutions are derived and their global behaviors are investigated.