GENERALIZED ?-JACOBI POLYNOMIALS 5 (1.22) with An_\Cn 0, n = 1,2,..., and An,Bn,Cn real, then there is a positive measure da (x) with J oo Pn (x)pm {x)da (x) = 0, m * n - 0 0 ^*- r • -A0 J_oo See Perron [PI], Wintner [W3], Stone [S9] and Favard [Fl]. This usually goes under the name of Favard's theorem, but it is older. The measure in (1.23) may not be uniquely determined, and there is no way offindingthe measure exactly for every recurrence relation. However there are methods that will work for many interesting examples and some general facts about the measure (or measures) can be proven. See Askey-Ismail [A16], Chihara [C3], [C4], [C5], Chihara and Nevai [C7], Freud [F3], Geronimo and Case [G4], Geronimo [G3], Geronimo and Nevai [G5], Karlin and McGregor [Kl], [K2], Nevai [Nl], Pollaczek [P4], Szego [SI2], Sherman [S4a]. The three term recurrence relation for the q-Racah polynomials was given in [A 17]. It continues to hold without the restriction that a denominator parameter is q~N. Translating this recurrence relation into the notation for pn(x) gives (1.24) 2xpn{x) = Anpn+l (x) + BnPn (x) + Cnpn.x{x)y \-abcdan~x V * n (\-abcdqi»-i){\-abcdq*") (1.26) Cn = {\-qn)(\-abqn-x)(\-acqn-x){\-adqn-x) (\-bcqn-l)(\-bdqn-l)(l-cdqn-1) X (\-abcdql"-2)(\-abcdqln-i) (1.27) (l-abcdq2n-2)(l-abcdq2n)Bn = qn-l[(\+abcdq2n-l)(sq+s' abcd)-qn-{{\+q)abcd{s+s' q)], wheres = a + b +c + d,s' = a~l + b~l + c~l + d~x. The initial conditions are P-\(x) = 0 andpoM = 1. For convenience, px(x) = 2(\-abcd)x-(a+b+c+d)+abcd(a-l+b-l+c-l+d-1), pn(x) = 2n(abcdqn~l q)nxn + lower terms.

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 1985 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.