On the Konhauser sets of biorthogonal polynomials suggested by the Laguerre polynomials (Q1393647)

Recently, \textit{J. D. E. Konhauser} [J. Math. Anal. Appl. 11, 242--260 (1965; Zbl 0125.31501); Pac. J. Math. 21, 303--314 (1967; Zbl 0156.07401)] discussed two polynomial sets \(\{Y_n^\alpha(x;k)\}\) and \(\{Z_n^\alpha(x;k)\}\), which are biorthogonal with respect to the weight function \(x^\alpha e^{-x}\) over the interval \((0,\infty)\), where \(\alpha > -1\) and \(k\) is a positive integer. For the polynomials \(\{Y_n^\alpha(x;k)\}\), the following bilateral generating function is derived in this paper: \[ \sum_{n=0}^\infty Y_n^\alpha(x;k) \zeta_n(y) t^n = (1 - t)^{-(\alpha +1)/k} \exp\{x [1- (1 - t)^{-1/k}]\} G[x(1 - t)^{-1/k}, yt/(1-t)], \tag{1} \] where \[ G[x,t] = \sum_{n=0}^\infty \lambda_n Y_n^\alpha(x;k) t^n, \tag{2} \] the \(\lambda_n \ne 0\) are arbitrary constants, and \(\zeta_n(y)\) is a polynomial of degree \(n\) in \(y\) given by \[ \zeta_n(y) = \sum_{n=0}^r \binom{n}{r} \lambda_r y^r. \tag{3} \] It is also shown that the polynomials \(\{Z_n^\alpha(x;k)\}\) can be expressed as a finite sum of \(\{Z_n^\alpha(x;k)\}\) in the form \[ Z_n^\alpha(x;k) = \left(\frac{x}{y}\right)^{kn} \sum_{n=0}^r \binom{\alpha+kn}{kr}\frac{(kr)!}{r!} [(y/x)^k - 1]^r Z_{n-r}^\alpha(y;k), \tag{4} \] which can readily be translated in terms of a multiplication formula for \(Z_n^\alpha(x;k)\). For \(k=2\), formulas (1) and (4) yield the corresponding properties of the polynomials introduced earlier by \textit{S. Preiser} [J. Math. Anal. Appl. 4, 38--64 (1962; Zbl 0109.29502)]. Moreover, when \(k=1\), both (1) and (4) would reduce to similar results involving the generalized Laguerre polynomials \(L_n^\alpha(x)\). For results analogous to (1) and (4), involving certain general classes of functions, one may refer to the recent works of the author [Ill. J. Math. 15, 64--72 (1971; Zbl 0205.35902)] and of the author and \textit{J. P. Singhal} [Pac. J. Math. 42, 755--762 (1972; Zbl 0217.11201)].