  This paper presents two new classes of M\"untz functions which are called
Jacobi-M\"untz functions of the first and second types. These newly generated
functions satisfy in two self-adjoint fractional Sturm-Liouville problems and
thus they have some spectral properties such as: orthogonality, completeness,
three-term recurrence relations and so on. With respect to these functions two
new orthogonal projections and their error bounds are derived. Also, two new
M\"untz type quadrature rules are introduced. As two applications of these
basis functions some fractional ordinary and partial differential equations are
considered and numerical results are given.
