Treffer: Ladder operators for \(q^{-1}\)-Hermite polynomials

The author computes raising and lowering operators and derives formulas of Rodrigues' type for the \(q^{-1}\)-Hermite polynomials and shows that these polynomials are solutions of \(q\)-Sturm-Liouville problems. He also gives two different Rodrigues' formulas and exhibits two distinct \(q\)- Sturm-Liouville equations satisfied by the \(q^{-1}\)-Hermite polynomials. The \(q^{-1}\)-Hermite polynomials \(\{h_ n(x\mid q)\}_ 0^ \infty\) are generated by \(h_ 0 (x\mid q)=1\), \(h_ 1 (x\mid q)= 2x\) and \[ h_{n+1} (x\mid q)= 2x h_ n(x\mid q)- q^{-n} (1-q^ n) h_{n-1} (x\mid q), \qquad n>0 \] erzeugt.