New connections between functions from additive and multiplicative number theory
Let \(f\) and \(g\) be two functions that are connected by the relation \[g(n) = \sum_{d\mid n} f(d).\] Moreover, let \(p(n)\) be the number of partitions of \(n\), and let \(S_{n,k}^{(r)}\) be the total number of \(k\)'s in all partitions of \(n\) whose smallest part is at least \(r\). In this paper, the following general identity is proven: \[\sum_{k=r}^{n+r} f(k) S_{n+r,k}^{(r)} = \sum_{j=0}^n a_{r,j} p(n-j),\] where the coefficients \(a_{r,j}\) are given by the recurrence relations \[a_{r+1,j} = a_{r,j+1} - f(r) b_{r,j+1} - \begin{cases} 0 & j+1 < r, \\ a_{r,j+1-r} & j+1 \geq r,\end{cases}\] and \[b_{r+1,j} = b_{r,j} - \begin{cases} 0 & j < r, \\ b_{r,j-r} & j \geq r,\end{cases}\] with initial conditions \(a_{1,j} = g(j+1)\) and \(b_{1,j} = \delta_{0,j}\). The coefficients \(a_{r,k}\) can also be expressed in terms of the function \[g_r(n) = \sum_{d \mid n, d \geq r} f(d),\] namely: \[a_{r,n} = \sum_{k=0}^n b_{r,k} g_r(n+r-k)\] for \(n \geq 0\) and \(r \geq 1\). The identity can also be restated in terms of another partition function: let \(p_r(n)\) denote the number of partitions of \(n\) whose smallest part is at least \(r\) (with the convention that \(p_r(0) = 1\) and \(p_r(n) = 0\) for \(0 < n < r\)). Then \[\sum_{k=r}^{n+r} f(k) S_{n+r,k}^{(r)} = \sum_{k=r}^{n+r} g_r(k) p_r(n+r-k).\] Upon specialisation, one obtains many different identities involving classical arithmetic functions such as Euler's totient function, the divisor function, and others. A similar family of identities is derived based on Euler's pentagonal number theorem.
scientific article; zbMATH DE number 6879876
scientific article