Symmetry, Integrability and Geometry: Methods and Applications On the Limit from q-Racah Polynomials to Big q-Jacobi Polynomials ⋆

A limit formula from q-Racah polynomials to big q-Jacobi polynomials is given which can be considered as a limit formula for orthogonal polynomials. This is extended to a multi-parameter limit with 3 parameters, also involving (q-)Hahn polynomials, little q-Jacobi polynomials and Jacobi polynomials. Also the limits from Askey-Wilson to Wilson polynomials and from q-Racah to Racah polynomials are given in a more conceptual way.

The book Koekoek, Lesky & Swarttouw [3] is the successor of the report Koekoek & Swarttouw [4], which can be alternatively used as a reference whenever the present paper refers to some formula in [3,Chapters 9 and 14]. For notation of q-hypergeometric series used in this paper the reader is referred to [2]. Throughout it will be assumed that 0 < q < 1, that N is a positive integer and that n ∈ {0, 1, . . . , N } if N is present.
They are indeed polynomials of degree n in x: Thus we have proved our main result: Theorem 1. There is the following limit formula from q-Racah polynomials to big q-Jacobi polynomials: Then the polynomials are orthogonal with respect to positive weights (see [3, (14.2.2)]) on the points q N +1−y a + q y+1 c (y = 0, 1, . . . , N ), which, for certain M depending on N can be written as the union of the increasing sequence of nonpositive points qc + q N +1 a, q 2 c + q N a, . . . , q M c + q N −M +2 a and the decreasing sequence of nonnegative points qa + q N +1 c, q 2 a + q N c, . . . , q N −m+1 a + q M +1 c.
Formally, in the limit for N → ∞ this tends to the union of the sequence of negative points {q k+1 c} k=0,1,... and the sequence of positive points {q k+1 a} k=0,1,... . But indeed, we know that under the constraints (2.3) the big q-Jacobi polynomials are orthogonal with respect to positive weights on this set of points (see [3, (14.5.2)]). Thus the limit formula (2.2) is under the constraints (2.3) on the parameters really a limit formula for orthogonal polynomials.
Remark 2. The limit formula [3, (14.2.15)], which reads P n q −y ; a, b, c; q = lim δ→0 R n q −y + cδq y+1 ; a, b, c, δ | q , (2.4) cannot be considered as a limit formula for orthogonal polynomials. Indeed, for the q-Racah polynomials on the right-hand side it is required that qa or qbδ or qc is equal to q −N for some positive integer N (see [3, (14.2.1)]). Since δ → 0 and a, b, c remain fixed in (2.4), we must have qa or qc equal to q −N . But then we arrive at a limit from q-Racah polynomials to q-Hahn polynomials (see [3, (14.2.16) or (14.2.18)]) rather than big q-Jacobi polynomials.
Remark 3. For c = 0 (2.2) specializes to a limit formula from q-Hahn polynomials to little q-Jacobi polynomials. For the left-hand side of (2.2) use that  Thus for c = 0 (2.2) specializes to the limit formula which is also given in [3, (14.6.13)].

Limit from Askey-Wilson to Wilson
Consider Askey-Wilson polynomials (see [3, (14.1.1)]), putting x = cos θ: Also consider Wilson polynomials (see [3, (9.1.1)]), putting x = y 2 : Remark 4. In [3, (14.1.21)] the following limit from Askey-Wilson polynomials to Wilson polynomials is given: This limit follows immediately by comparing the (q-)hypergeometric expressions in (3.1) and (3.2). However, the limit (3.5) has the draw-back that the rescaled Askey-Wilson polynomial on the left no longer depends polynomially on y. Note that the limit (3.4) can be written more generally, by the same proof, as Then (3.5) is a special case of (3.6), since
In [5] I combined the limits in the Askey scheme (i.e., for q = 1) into a small number of multi-parameter limits. This was done by renormalizing the Racah and Askey-Wilson polynomials on the top level of the scheme as families of orthogonal polynomials depending on four positive parameters such that these extend continuously for nonnegative parameter values, while (renormalized) families lower in the scheme are reached if one or more of the parameters become zero. At the end of [5] the obvious open problem was mentioned to extend this work to the q-Askey scheme including the limits for q ↑ 1. Below I will work this out for the small part of the (q-)Askey scheme in Fig. 1.
By the chosen coefficient on the right these are monic polynomials of degree n, see [3, (14.2.4)].
For the parameters in the arguments of p n we require We will see that the polynomials p n (x; c, N −1 , 1 − q) remain continuous in (c, N −1 , 1 − q) if these three coordinates are also allowed to become zero. For the demonstration we will use the same tool as in [5]. We will see that the coefficients in the three-term recurrence relation for the orthogonal polynomials (5.2) depend continuously on (c, N −1 , 1 − q) for values of these coordinates as in (5.3) or equal to zero.
The various limits are collected in Fig. 2.