The Wallis products for Fermat curves (Q6632153)

The purpose of this article is to study hypergoniometric functions, generalizations of trigonometric functions, which satisfy certain remarkable ``Pythagorean formulas'':\N\[\N\text{sn}_p^p(t)+\text{cn}_p^p(t)=1,\qquad p>1.\N\]\NFirst, a general theorem about the power series expansions of these functions is proved. Then, integral expressions for the inverses of these functions are used to find their power series expansions. Similar formulas hold for the corresponding hyperbolic functions. Finally, the \(p\)-analogues of \(\pi\) are computed together with corresponding \(p\)-analogues of the circle and the ellipse.