On Sylvester sums of compound sequence semigroup complements (Q2401185)

A compound sequence is a generalization of a geometric sequence. Let \(A = (a_1,\ldots, a_k)\) and \(B = (b_1,\ldots, b_k)\) where \(\gcd(a_i, b_j) = 1\) for all \(i \geq j\). Let \(g_i = b_1\cdots b_i a_{i+1} \cdots a_k\). We say that \((g_i)_{i=0}^k\) is a compound sequence and denote it by \(G(A,B)\). The numerical semigroup \(R(A,B)\) generated by \(G(A,B)\) is the set of all integers representable as linear combinations of elements of \(G(A,B)\) with non-negative coefficients. Let \(NR(A,B)\) be the set of positive integers in the complement of \(R(A,B)\). The \(m\)th Sylvester sum is defined by \[ S_m(A,B) = \sum_{n \in NR(A,B)} n^m. \] The authors prove generalizations of theorems of \textit{H. J. H. Tuenter} [J. Number Theory 117, No. 2, 376--386 (2006; Zbl 1097.11010)] and \textit{Ö. J. Rödseth} [Fibonacci Q. 32, No. 5, 407--408 (1994; Zbl 0840.11009)], improving on results for numerical semigroups generated by two elements. They give explicit formulas for \(S_m(A,B)\) for small values of \(m\) and give a general formula involving Bernoulli numbers. The final part of the paper is about Weierstrass points on algebraic curves. For \(1 \leq i \leq k\) let \(f_i(x) \in \mathbb{C}[x]\) be a separable polynomial of degree \(b_i\). Consider the affine curve \[ A_k = \{ (x_0,\ldots, x_k) \in \mathbb{C}^{k+1} : x_i^{a_i} = f_i(x_{i-1}) \text{ for } 1 \leq i \leq k\}. \] A plane curve defined by a single equation \(x_i^{a_i} = f_i(x_{i-1})\) with \(a_i \geq 2\) is called a superelliptic curve. Assume that \(A_k\) is non-singular and let \(C_k\) be its nonsingular projective model. The authors show that the genus of \(C_k\) is \(S_0(A,B)\), give a basis for the vector space of holomorphic \(q\)-differentials on \(C_k\), and compute the \(q\)-Weierstrass weight of the point at infinity on \(C_k\) in terms of \(S_0(A,B)\).