Asymptotic behaviour of the sums

We will use the following results concerning the asymptotic behaviour for N — 00 of the following sums where n is an integer [18]  

Similar behaviours are obtained when j is replaced by j — 1/2. Moreover we will use the inequalities  

In a similar fashion one can proceed to the second order. However, the algebra becomes rather cumbersome and we performed the symbolic computations with the program Mathematica [19]. 

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Appendix C. Asymptotic behaviour of the sums Appendix D. Derivation of equations (48) and (49)... 

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