Real-Analyticity of Generalized Sine Functions with Two Parameters

We identify the maximal real interval on which $\sin_{p,n}$ is real-analytic for any real number $p>1$ and any integer $n>1$. We achieve this by first proving that $\sin_{p,n}$ is analytic at $(1/2){\pi}_{p,n}$ iff $p=m/(m-1)$ for some integer $m>1$, in which case we determine the radius of convergence of the Taylor series at $(1/2){\pi}_{p,n}$.