On the distributional orthogonality of the general Pollaczek polynomials (Q1914761)

Pollaczek polynomials depend on three parameters \(\lambda\), \(a\), \(b\) and satisfy a three-term recurrence relation \[ xP_n(x)= P_{n+ 1}(x)+ B_nP_n(x)+ C_nP_{n- 1}(x), \] with coefficients given by \(B_n =-b/(\lambda+ a+ n)\) and \(C_{n+ 1}= (n+1)(n+ 2\lambda)/[4(\lambda+ a+n)(\lambda+ a+n+ 1)]\). If \(C_{n+ 1}>0\) for all \(n\geq 0\) then these polynomials are orthogonal with respect to a positive measure on the real line, which essentially lives on the interval \([-1,1]\), but there are cases for which there are mass points outside this interval (and they are relevant for a study of the hydrogen atom). In this paper, the authors study the case where \(C_{n+ 1}\neq 0\) are not all positive. In this case, the polynomials will be orthogonal with respect to some moment functional \({\mathcal L}_\lambda\) acting on polynomials. It is shown that \({\mathcal L}_{\lambda+ 1}= q_\lambda(x){\mathcal L}_\lambda\), where \(q_\lambda\) is a polynomial, so that the moment functional can be obtained recursively from the moment functional \({\mathcal L}_{\lambda+ m}\), which will be positive definite when \(m\) is large enough. The authors obtain the moment functional \({\mathcal L}_\lambda\) in terms of Dirac distributions (and their derivatives) and the measure corresponding to \({\mathcal L}_{\lambda+ m}\), whenever \(\lambda\), \(a\), \(b\) are real, \(2\lambda\) and \(\lambda\pm a\) are not nonnegative integers and \((\lambda+ j)^2+ b^2\geq a^2\) for \(j= 0,1,\dots,m-1\), where \(m\) is a positive integer such that \(0<\lambda+ a+ m<1\) and \(-1/2<\lambda+ m<0\) or \(m> 0\) and \(\lambda+ a+ m>0\).