These orbitals are the solutions of schrodinger equations. In quantum mechanics the basis functions is used to express/represent electronic state...

Because the quantum nomber n must be larger than angular momentum quantum nomber

Thanks. . .

But be careful with potentials other than the Coulomb potential.

For example, for the spherically symmetric oscillator the states 1p, 1d, 2d exist...

The solution of Schrodinger equation vanishes if such restrictions are not there..I mean if the value of angular quantum number (l) is allowed to go beyond the values defined by principal quantum number, then the solution of the SE for such values of (l) does not exist.

Remind how one normally introduces the quantum number N for arbitrary spherical potential. The radial Schodinger equation for a spherical potential has fixed form for a given \ell. Then we find square integrable radial functions (eigenfunctions) and corresponding eigenvalues (energies) satisfying this equation and numerate the solutions for the given \ell: N=1, 2, 3... starting from the lowest eigenvalue. (see e.g. the example by Gert of the spherical well). The lowest eigenvalue corresponds to the nodeless eigenfunction, then comes function with one node. etc. In such a way functions 1s,2s,3s,...;1p,2p,3p,...; 1d,2d,3d,...;... appear for the arbitrary spherical potential. In case of the Coulomb potential, some eigenvalues of equations with different \ell coincide: second (N=2) eigenvalue for \ell=0 coincides with first eigenvalue (N=1) for \ell=1; third (N=3) eigenvalue for \ell=0 coincides with second (N=2) for \ell=1 and first (N=1) for \ell=2, and so on (there are deep group-theoretical grounds for this, according to V. Fock, but this will take us too far). Then for the Coulomb field we introduce for convenience the new 'principal quantum number' n = N + \ell, which alone determines the energy by the well-known formula. This n also marks then the eigenfunctions together with \ell. So, we stay with n and \ell, where n is always greater than \ell. Therefore the inequality n > \ell for the Coulomb field is in a sense "artificial", i.e. according to the definition.