Writing
    ◆    Explanation w.r.t. Debye Series

Programming
    ◆    Implementation of recursive algorithm in Mathematica
    ◆    Resurrecting the idea of saving data for Legendre Polynome
    ◆    Improving IDL code

Saving the data of Legendre Polynome and later reusing it for the Mathematica code doesn’t save necessarily lotta time. Just 36% of time has been saved, i.e. in case of 100 mulitiplicity where it takes 20613 seconds, I can complete this with 2168.3865 min. i.e. 36 Hrs. instead of 57 Hrs.
The next step would be to try recursive algorithm in Mathematica.