ON MINAKSHISUNDARAM-PLEIJEL ZETA-FUNCTIONS OF SPHERES

The aim of this paper is to show that the Minakshisundaram-Pleijel zet a function Z(k)(s) of k-dimensional sphere S-k, k greater than or equa l to 2 (defined in Re(s) > k/2 by GRAPHICS with (k - 1)!P-k(n) = R(n 1, k - 2)(2n + k - 1) where the ''rising factorial'' R(x, n) = x(x 1)...(x + n - 1) is defined for real number x and n nonnegative intege r) can be put in the form GRAPHICS where B-k,B-j are explicitly comput ed. The above formula allows us to find explicitly the residue of Z(k) (s) at the pole s = k/2 - n, n is an element of N, GRAPHICS In passing , we also obtain apparently new relations among the Stirling numbers.