[For the entire collection see Zbl 0658.00009.]
The author solves the following second order linear coefficient linear difference equation having five essential complex parameters \[ (1.1)\quad X_{n+1}-(z-dn)X_ n+(an^ 2+bn+c)X_{n-1}=0,\quad n\geq 0 \] in terms of the hypergeometric function \({}_ 2F_ 1\) and its limiting forms \({}_ 1F_ 1\), \(\psi\), \({}_ 0F_ q\) and \(D_{\lambda}.\)
For suitable values of parameters, equation (1.1) provides a generic link between two families of Jacobi matrices, their associated moment problems, continued fractions and their corresponding classical orthogonal polynomials due to R. Askey and M. Ismail [Mem. Am. Math. Soc. 300 (1984; Zbl 0548.33001)]. The method also yields a new set of polynomials orthogonal with respect to a discrete measure determined by Bessel functions called “Bessel order” polynomials.
The solutions to (1.1) tie together a rich variety of classical orthogonal polynomials P(x) and their associated positive measure \(d\sigma\) (x). In this connection there are actually two distinct families which can be obtained by taking the parameters a, b, c, d real and having either \[ (I)\quad an^ 2+bn+c>0,\quad n=1,2,... \] or \[ (II)\quad (an^ 2+bn+c)/(e-dn)(e+d-dn)>0,\quad n=1,2,... \] where for (II), z-dn in (1.1) has been replaced by \(z(e-dn)+f\) with e, f real and d, e-dn, \(e+d-dn\neq 0.\)
The family (II) includes Jacobi polynomials \(P_ n^{(\alpha,\beta)}(x)\) for \(\alpha =\beta\), their generalizations due to F. Pollaczek [C. R. Acad. Sci. Paris 228, 1363-1365 (1949; Zbl 0041.035); Mem. Sci. Math. 131 (Gauthier Villars, Paris, 1956; MR 17,730)] and J. Szegö [Proc. Am. Math. Soc. 1, 731-737 (1950; Zbl 0041.392)], the more recently examined, discretely orthogonal polynomials of T. S. Chichara and M. Ismail [Adv. Appl. Math. 3, 441-462 (1982; Zbl 0504.60094)] and the Bessel related polynomials of M. Ismail [J. Math. Anal. Appl. 86, 1-19 (1982; Zbl 0483.33004)] and J. Wimp [SIAM J. Math. Anal. 16, 887-895 (1985; Zbl 0587.41012)].
Family (I) is more familiar. Its all the results can however be translated over to family (II). Family (I) contains the polynomials due to J. Meixner [J. Lond. Math. Soc. 9, 6-13 (1934; Zbl 0008.16205)], F. Pollaczek [C. R. Acad. Sci. Paris 230, 2254-2256 (1950; Zbl 0038.224)]. Laguerre, Hermite and Charlier polynomials [A. Erdélyi, Higher Transcendental Functions, V.I.II (1981; Zbl 0505.33001)].
Family (I) also includes the neglected case of Bessel order polynomials and associated Laguerre and Hermite polynomials due to R. Askey and J. Wimp [Proc. R. Soc. Edinb., Sect. A 96, 15-17 (1984; Zbl 0547.33006)].