Non-standard orthogonality for Meixner polynomials (Q1572607)

Meixner polynomials \(M_n^{(\gamma,\mu)}\) are defined as a terminating \({}_2F_1\) hypergeometric series with two parameters \(\mu\) and \(\gamma\) and degree \(n\). For \(0 < \mu < 1\) and \(\gamma > 0\) these polynomials are orthogonal on the integers \(\{0,1,2,\ldots\}\) with respect to the negative binomial distribution (Pascal distribution). In the present paper the authors allow \(\gamma\) to take any real value and they show that for \(\gamma\) real the Meixner polynomials are orthogonal with respect to an inner product involving differences (a discrete Sobolev inner product). They also define a difference operator which is symmetric for this inner product and show that the Meixner polynomials are eigenfunctions for this operator. The case when \(\gamma = -N\), with \(N\) a positive integer, is treated in more detail. For \(n \leq N\) one then has Krawchuk polynomials, and for \(n > N\) the Meixner polynomial is the product of a Pochhammer polynomial \((x-N)_{N+1}\) and a Meixner polynomial with \(\gamma = N+2\). Some parts in the paper may confuse the reader: on page 2 lines 6-7 the authors state that for \(\gamma > 0\) the Meixner polynomials are orthogonal on \([0,\infty)\), but in reality they are orthogonal on \(\{0,1,2,\ldots\}\); on page 6, Equation (3.2), the inner product is defined as a sum ranging over \(x \in \{0,1,2,\ldots\}\), but on the same line they state that \(x \in [0,\infty)\); a similar thing happens on page 7, line 3.