Identities arising from binomial-like formulas involving divisors of numbers (Q6054723)

The main focus of the paper under review is to present a large number of identities involving \(\omega(n)\), the number of distinct prime divisors of \(n\). The identities included involve, the Fibonacci, Lucas, Stirling and Lah numbers, as well as the Pochhammer symbols. Often the identities are simpler when \(n\) is squarefree. A small selection of the many identities are included here. Let \(L_k\) be the \(k\)th Lucas number, and let \(F_k\) be the \(k\)th Fibonacci number. If \(n\) is squarefree then: \[\sum_{d|n} \frac{L_{\omega(d)+1}}{\omega(d)+1} = \frac{L_{2\omega(n)+2}-1}{\omega(n)+1}, \] \[\sum_{d|n} \frac{F_{\omega(d)+1}}{\omega(d)+1} = \frac{F_{2\omega(n)+2}}{\omega(n)+1}, \] and \[\sum_{d|n,d >1} \frac{L_{\omega(d)}}{\omega(d)} = \sum_{k=1}^{\omega(n)} \frac{L_{2k}-1}{k}. \] These identities arise as special cases of more general identities. A major role is played by the following theorem. Assume that \(n\) has as distinct prime factors \(p_1\), \(p_2, \dots p_{\omega(n)}\) and \(n=p_1^{a_1} \cdots p_k^{a_{\omega(n)}}\). Then \[\prod_{i=1}^{\omega(n)} (x+a_i y) = \sum_{d|n} x^{\omega(n)-\omega(d)}y^{\omega(d)}.\]