  We study a family of the Laurent biorthogonal polynomials arising from the
Hermite continued fraction for a ratio of two complete elliptic integrals.
Recurrence coefficients, explicit expression and the weight function for these
polynomials are obtained. We construct also a new explicit example of the
Szeg\"o polynomials orthogonal on the unit circle. Relations with associated
Legendre polynomials are considered.
