Recurrence coefficients of a new generalization of the Meixner polynomials (Q431332)

The recurrence coefficients of orthogonal polynomials are often related to the Painlevé equations (see overview in [\textit{L. Boelen, G. Filipuk} and \textit{W. Van Assche}, J. Phys. A, Math. Theor. 44, No. 3, Article ID 035202, 19 p. (2011; Zbl 1217.34135)]). In the paper under review a new example of such a relation is given.NEWLINENEWLINEThe generalized Meixner orthogonal polynomials \(p_n(k)\) are orthogonal polynomials with respect to the scalar product NEWLINE\[NEWLINE\sum_{k=0}^{\infty}p_n(k)p_m(k)w(k)=\delta_{n,m},NEWLINE\]NEWLINE where NEWLINE\[NEWLINEw(x)=\frac{\Gamma(\beta)\Gamma(\gamma+x)e^x}{\Gamma(\gamma)\Gamma(\beta+x)\Gamma(x+1)}, \text{ and hence } w(k)=\frac{(\gamma)_ke^k}{(\beta)_k k!}.NEWLINE\]NEWLINE This weight was introduced in [C. Smet and W. Van Assche, Orthogonal polynomial on a bi-lattice, Constr Approx, to appear, \url{arxiv:1101.1817}], see also [\textit{T. S. Chihara}, Mathematics and its Applications. Vol. 13. New York - London -Paris: Gordon and Breach, Science Publishers. (1978; Zbl 0389.33008)] for the information about classical Meixner weight.NEWLINENEWLINEBoth classical and generalized Meixner polynomials satisfy the recurrence relation NEWLINE\[NEWLINExp_n(x)=a_{n+1}p_{n+1}(x)+b_np_n(x)+a_np_{n-1}(x),NEWLINE\]NEWLINE where \(a_k, b_k\) must be obtained from some other recurrence relations. In [\textit{C. Smet} and \textit{W. Van Assche}, Constr. Approx. 36, No. 2, 215--242 (2012; Zbl 1262.34103)] these recurrence relation were explicitly written in the case of generalized Meixner polynomials.NEWLINENEWLINEIn the case of classical Meixner polynomials (the case \(w(k)=\frac{(\beta)_k c^k}{k!}\), \(\beta>0\), \(0<c<1\)) the solutions of the recurrence relations for \(a_n\), \(b_n\) are known explicitly. In another particular case of generalized Meixner polynomials corresponding to \(w(k)=\frac{(\beta)_k c^k}{(k!)^2}\), \(\beta>0\), \(c>0\) the solution of these recurrence relations are related to solutions of the fifth Painlevé equation L. Boelen, G. Filipuk and W. Van Assche [loc. cit.].NEWLINENEWLINEThe main result of the present paper is the solution of this recurrence relations for all generalized Meixner polynomials. It is proved that the solutions can be expressed through the solutions of the fifth Painlevé equation. Several forms of this expression are presented, the different forms are transformed one into another using Bäcklund transformations. The parameters of the fifth Painlevé equation equation are expressed explicitely through the parameters of the generalized Meixner weight.