Generalized Bell numbers and peirce matrix via Pascal matrix (Q1729008)

Summary: With the Stirling matrix \(S\) and the Pascal matrix \(T\), we show that \(T^k S(k \geq 0)\) satisfies a type of generalized Stirling recurrence. Then, by expressing the sum of components of each row of \(T^k S\) as \(k\)-Bell number, we investigate properties of \(k\)-Bell numbers as well as \(k\)-Peirce matrix.