  We propose an extended framework for continuous-stage Runge-Kutta methods
which enables us to treat more complicated cases especially for the case
weighting on infinite intervals. By doing this, various types of weighted
orthogonal polynomials (e.g., Jacobi polynomials, Laguerre polynomials, Hermite
polynomials etc.) can be used in the construction of Runge-Kutta-type methods.
Particularly, families of Runge-Kutta-type methods with geometric properties
can be constructed in this new framework. As examples, some new symplectic
integrators by using Legendre polynomials, Laguerre polynomials and Hermite
polynomials are constructed.
