Bipartite Chebyshev polynomials and elliptic integrals expressible by elementary functions

The article is concerned with polynomials $g(x)$ whose graphs are “partially packed” between two horizontal tangent lines. We assume that most of the local maximum points of $g(x)$ are on the first horizontal line, and most of the local minimum points on the second horizontal line, except several “exceptional” maximum or minimum points, that locate above or under two lines, respectively. In addition, the degree of $g(x)$ is exactly the number of all extremum points $+1$. Then we call $g(x)$ a multipartite Chebyshev polynomial associated with the two lines. Under a certain condition, we show that $g(x)$ is expressed as a composition of the Chebyshev polynomial and a polynomial defined by the $x$-component data of the exceptional extremum points of $g(x)$ and the intersection points of $g(x)$ and the two lines. Especially, we study in detail bipartite Chebyshev polynomials, which has only one exceptional point, and treat a connection between such polynomials and elliptic integrals.