On repdigits which are sums or differences of two $k$-Pell numbers

Let $k\geq 2$. A generalization of the well-known Pell sequence is the $k$-Pell sequence whose first $k$ terms are $0,\ldots,0,1$  and each term afterwards is given by the linear recurrence  $$P_n^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)}.$$

The goal of this talk is to show that $11, 33, 55, 88$ and $99$ are only repdigits expressible as sum or difference of two $k$-Pell numbers. The proof of this main theorem uses lower bounds for linear forms in logarithms of algebraic numbers (Baker's method) and  a modified version of Baker-Davenport reduction method (due to Dujella and Pethö). Our result extends that of  Bravo and Herrera.