Spectral transformations and generalized Pollaczek polynomials (Q1582334)

The authors study orthogonal polynomials belonging to the so called \textit{generalized Pollaczek class}. The coefficients from the recurrence relation for the monic orthogonal polynomials \[ P_{n+1}(x)+u_nP_n(x)+b_nP_{n-1}(x)=xP_n(x) \] are rational functions in \(n\), whereas the weight function \(w(x)\) is known explicitly (the authors do not impose asymptotic conditions on the coefficients). Using iterated spectral transformations of the form \[ {\tilde P}_n(x)={P_{n+1}(x)-A_nP_n(x)\over x-\alpha} \] where \(A_n=P_{n+1}(\alpha)/P_n(\alpha)\), with \(\alpha\) satisfying \(P_n(\alpha)\not= 0\) and Christoffel transformations preserving the positivity properties of the weight functions (where the spectrum of the starting sequence of polynomials is assumed to be discrete), the authors derive a general scheme to construct a.o. the Racah polynomials and the modified Charlier polynomials on defected lattices. Moreover they study duality properties, generating functions and give a connection with factorization chains. The final section gives an interesting discussion of the method and formulates the not yet studied problem of finding all orthogonal polynomials generated by \[ F_1(x,t)\exp{F_2(x,t)}=\sum_{n=0}^{\infty} P_n(x)t^n, \] where \(F_1, F_2\) both are polynomials in \(x\) with coefficients depending on \(t\).