Can anybody help me to do 7a iii? I have done the rest.?

The term (m+r-L) will become 0 when m = L-r and so c_m will be 0 for all m >= L-r by the recurrence relation. Consequently the series effectively stops and is a polynomial.

For r = 1, L=3, the polynomial is
c_0 (x - (5/3)x^3)