Some Multiple Power Series with Zero-One Coefficients

Abstract

The paper is concerned with sums of the type \[S_{n,j} = \sum {x_1^{a_1 } x_2^{a_2 } \cdots x_n^{a_n } } \quad (n > 1),\] where the summation is over either \[( * )\qquad ja_i \leqq a_1 + a_2 + \cdots + a_n \quad (1 \leqq j \leqq n;1 \leqq i \leqq n)\] or \[( * * )\qquad a_1 + a_2 + \cdots + a_n = ja_i + (n - j)b_i \quad (0 \leqq j \leqq n;1 \leqq i \leqq n);\] the $a_i $ and $b_i $ are nonnegative integers. It is proved, for example, that for the first type with $j = 2$, the sum is a rational function with denominator equal to $\prod _{1 \leqq i < k \leqq n} (1 - x_i x_k )$. Several combinatorial applications are obtained by specializing the $x_i $. For example it is proved that the number of nonnegative solutions of the system \[a_1 + a_2 + \cdots + a_n = N,\quad (n - 1)a_i \leqq N,\quad (1 \leqq i \leqq n)\] is equal to the binomial coefficient \[\left( {\begin{array}{*{20}c} {k + n - s - 1} \\ {n - 1} \\ \end{array} } \right)\quad (N = k(n - 1) + s,0 \leqq s < n - 1).\] The final section of the paper is concerned with multiple Dirichlet series \[{\bf \Phi} _{n,j} = \sum {m_1^{ - s_1 } m_2^{ - s_2 } \cdots m_n^{ - s_n } } ,\] where the smmation is over all positive integers $m_i $ such that \[m_i^j \mid m_1 m_2 \cdots m_n \quad (1 \leqq j \leqq n;1 \leqq i \leqq n).\] The ${\bf \Phi} _{n,j} $ are expressed as products involving series satisfying (*); in particular \[{\bf \Phi} _{n,n - 1} = \frac{{\zeta (\alpha )\zeta (\sigma - s_1 ) \cdots \zeta (\alpha - s_n )}}{{\zeta ((n - 1)\sigma )}},\] where $\sigma = s_1 + \cdots + s_n $ and $\zeta (s)$ is the Riemann zeta-function.