The two solutions of the Legendre's Differential Equation obtained by series solution method are :
and
Now according to my textbook, for the useful polynomial for n equal to a positive integer, the constant $a_{0}$ in the first case is chosen as $a_{0}={\frac{1.3.5...(2n-1)}{n!}}$ and the solution is then called Legendre's polyinomial or coffecient or Zonal Harmonics of the first kind.
For n being the negative integer the constant $a_{0}$ in the second case is chosen as $a_{0}={\frac{n!}{1.3.5...(2n+1)}}$ and the corresponding polynomial is defined as Legendre's function of the second kind.
Now the choice of $a_{0}$ seems rather arbitrary and exactly how is such a choice useful?
P.S.: While solving the Hermite Equation we choose $a_{0}$ in a similar (and apparently) arbitrary manner. Is the reason behind the choice somehow related?
The complete solution of Legendre's equation using series solution method can be found here