Historical Note

From the information supplied by Prof. Muldoon it seems that, since the time of Euler, aspects of the subject of these sums has been discovered and rediscovered, possibly more than just once. The explicit expressions of the sums as functions of $\nu$ were given, for the first $12$ cases, by Lehmer [8]. The relation of the sums to the diffusion equation was first established by Kapitsa [9,10], who also worked on the calculation of the sums. The recursive general solution presented here seems to be equivalent to a known recursion formula, first derived by Meiman [11] and later rediscovered by Kishore [12].