On sieved orthogonal polynomials. IV. Generating functions
The most elementary sieved orthogonal polynomials satisfy the three term recurrence relation \[ 2xa_ nP_ n(x)=b_ nP_{n+1}(x)+c_ nP_ n(x)+d_ nP_{n-1}(x) \] with \(a_ n=b_ n=d_ n=1\), \(c_ n=0\) unless k-1 divides n, and in this case the coefficients are either all polynomials in n or else all polynomials in \(q^ n\). Here k is an integer at least 2 and q is real. Generating functions of the form \(\sum^{\infty}_{n=0}P_{kn+\ell}(x)r^ n\), \(\ell =0,1,...,k-1\), are considered and are shown to be expressible in terms of the case \(\ell =0\). For some polynomials this generating function for \(\ell =0\) can be found, and from this the asymptotics of \(P_ n(x)\) can be found. From this the orthogonality can sometimes be found. One interesting example treated in this paper is \[ a_ n=d_ n=1,\quad c_ n=0,\quad b_{mk- 1}=1-q^ m,\quad b_{mk+\ell}=1,\quad \ell =0,1,...,m-2. \] When \(k=1\) these are the continuous q-Hermite polynomials of Rogers. The orthogonality for \(k=2,3,..\). is given for the first time in this paper.
scientific article; zbMATH DE number 4052019
scientific article