On p-adic valuations of certain m colored p-ary partition functions

Let k is an element of N->= 2 and for given m is an element of Z\{0} consider the sequence (S-k,S-m(n))(n is an element of N) defined by the power series expansion 1/(1-x)(m) (infinity)(i=0 Pi)1/(1-x(ki))(m(k-1)) = (n=0)Sigma(infinity)Sk(,m)(n)x(n). The number S-k,S-m(n) for m is an element of N+ has a natural combinatorial interpretation: it counts the number of representations ofnas sums of powers of k, where the part equal to 1 takes one among mk colors and each part>1 takes m (k-1) colors. We concentrate on the case when k=p is a prime. Our main result is the computation of the exact value of the p-adic valuation of S-p,S-m(n). In particular, in each case the set of values of nu(p)(S-p,S-m(n)) is finite and the maximum value is bounded by max{nu(p)(m)+1,nu(p)(m+1)+1}. Our results can be seen as a generalization of earlier work of Churchhouse and recent work of Gawron, Miska and Ulas, and the present authors.