On the linearization problem involving Pochhammer symbols and their \(q\)-analogues
Given three sequences \((P_n(x))\), \((Q_n(x))\), \((R_n(x))\) of polynomials, the \(n\)th polynomial in each sequence being of degree \(n\), the linearization problem is the problem of finding the coefficients \(L_{mjn}\) in the expansion \(Q_m(x)R_j(x)=\sum _{n=0} ^{m+j}L_{mjn}P_n(x)\). Under the assumption that the sequences \((Q_n(x))\) and \((R_n(x))\) satisfy second-order \(q\)-difference equations and that the polynomials \((P_n(x))\) are orthogonal and satisfy a fourth-order \(q\)-difference equation, an algorithm is presented which finds a recurrence of the form \(\sum _{k} ^{}c_k(i,j,n)L_{m,j,n+k}=0\) for the coefficients that have to be found. It is shown, in some examples in which the polynomials are shifted \(q\)-factorials and \(q\)-Charlier polynomials, how this recurrence can be used to find closed forms for the coefficients.
scientific article; zbMATH DE number 1326530
scientific article