Balmer's formula and lagurre polynomials
When solving Schrodinger equation different potentials are included depending on the problem at hand, for example, particle in a box, harmonic oscillator, H atom etc. Then boundary conditions are applied; for a particle in a box, for example, the wavefunction has to be zero at either end of the box and this ensures that the solutions are quantised, i.e the wavefunction depends both on position x and an integer n and the energy also depends on an integer.

The different solutions to different types of differential equations were worked out many many years ago, and often by French mathematicians consequently the polynomial solutions obtained often carry their names, Laguerre, Hermite, Legendre and Chebychev (although he was Russian). These polynomials all depend on position x and one or more integers. The Hermite polynomials, for example, which form solutions to the harmonic oscillator, are often labelled as $H_n(x)$ where different polynomials are found with  $n=0,1,2 ...$, e.g. $H_0(x)=1, H_1(x)=1, H_2(x)=4x^2-2 \cdots$. 

In solving the H atom the radial and angular solutions are separated and each solved independently. The associated Laguerre polynomials ($L_n(x)$) are used in finding the radial solutions, and the angular solutions  depend on the spherical harmonic polynomials which themselves depend on the associated Legendre polynomials $P_l^m(\cos(\theta)$. 

(Notes. The polynomials are often described by recursion formulae meaning that once the first couple are known the rest are easily found. The associated polynomials are found by repeated differentiation of the polynomial, via Rodrigues' formulae. The polynomials are orthogonal.)