Tomorrow is my mathematical method exam where we have studied different kind of special functions named Legendre, Bessel's, Hermite and Laguerre functions. I solve their associated differential equation. Interestingly, I noticed to solve a specific differential equation associated with a specific function, they choose different series (ascending or descending) order in the Frobenius method. Like for Legendre and Hermite function, they use descending order $\left(\sum_{r=0}^\infty a_r x^{k-r}\right)$.
Is there any advantage of this order choosing?
Another thing is,
choosing values for $a_0($for ascending$)$ or $a_n($for descending$)$ in the series solution. I have got some intuition but is there any tips to guess them?
I can find the series solution, but to prove some recurrence relation I need to use closed format of the special functions. I was wondering,
Is there any way to remember the closed format of those functions?
I will appreciate any kind of help or solution. Thanks for your time and consideration.
