  This paper presents two new non-classical Lagrange basis functions which are
based on the new Jacobi-M\"untz functions presented by the authors recently.
These basis functions are, in fact, generalizations form of the newly generated
Jacobi based functions. With respect to these non-classical Lagrange basis
functions, two non-classical interpolants are introduced and their error bounds
are proved in detail. The pseudo-spectral differentiation (and integration)
matrices have been extracted in two different manners. Some numerical
experiments are provided to show the efficiency and capability of these newly
generated non-classical Lagrange basis functions.
