DIFFERENTIABILITY PROPERTIES OF p-TRIGONOMETRIC FUNCTIONS

p-trigonometric functions are generalizations of the trigonometric functions. They appear in context of nonlinear differential equations and also in analytical geometry of the p-circle in the plain. The most important ptrigonometric function is sinp(x). For p > 1, this function is defined as the unique solution of the initial-value problem (|u′(x)|p−2u′(x))′ = (p− 1)|u(x)|p−2u(x), u(0) = 0, u′(0) = 1 , for any x ∈ R. We prove that the n-th derivative of sinp(x) can be expressed in the form 2n−2−1 X k=0 ak,n sin qk,n p (x) cos 1−qk,n p (x) , on (0, πp/2), where πp = R 1 0 (1− s p)−1/pds, and cosp(x) = sinp(x). Using this formula, we proved the order of differentiability of the function sinp(x). The most surprising (least expected) result is that sinp(x) ∈ C∞(−πp/2, πp/2) if p is an even integer. This result was essentially used in the proof of theorem, which says that the Maclaurin series of sinp(x) converges on (−πp/2, πp/2) if p is an even integer. This completes previous results that were known e.g. by Lindqvist and Peetre where this convergence was conjectured.