Integral polynomial sequences arising from matrix powers of order 2
For the matrix \( A= \left( \begin{matrix} 1 & 1 \\ x & -x \\ \end{matrix} \right) \), let \[ u_n(x)=\sum_{k=0}^{n-1}\alpha(n,k)x^k \quad\text{and}\quad v_n(x)=\sum_{k=0}^{n-1}\beta(n,k)x^k \] be the polynomials of degree \(n-1\) arising from the \(n\)-th power matrix \( A^n= \left( \begin{matrix} u_n(x) & v_n(x) \\ \ast & \ast \\ \end{matrix} \right) \). Both sequences \(\{u_n(x)\}_{n=1}^\infty\) and \(\{v_n(x)\}_{n=1}^\infty\) satisfy the recurrence relation \[ \chi_{n+2}(x)+(x-1)\chi_{n+1}(x)-2x\chi_{n}(x)=0\, . \] The authors analyze the double integral sequences \(\{\alpha(n,k)\}\) and \(\{\beta(n,k)\}\) and show that \(\alpha(n,k)=\beta(n+1,k)+\beta(n,k-1)\). Several interesting relations with, for example, a weighted Delannoy number, Vandermonde matrix, mirrored \(\Gamma\)-matrix, Riordan array, Jacobi polynomial are established.
scientific article; zbMATH DE number 6127341