The q-Racah-Krall-type polynomials

arXiv:1107.2427v3 [math.CA] 9 Mar 2012 THE q-RACAH-KRALL-TYPE POLYNOMIALS R. ´ ALVAREZ-NODARSE AND R. SEVINIK ADIG¨ UZEL Abstract. In this paper the Krall-type polynomials obtained via the addition of two mass points to the weight function of the standard q-Racah polynomials are introduced. Several algebraic properties of these polynomials are obtained and some of their limit cases are discussed. 1. Introduction In the last decades the study of the discrete analogues of the classical special functions and, in particular, the orthogonal polynomials, has received an increasing interest (for a review see [2, 16, 14]). Special emphasis was given to the q-analogues of the orthogonal polynomials or q-polynomials, which are closely related with different topics in other fields of actual science: Mathematics and Physics (see e.g., [3, 4, 14, 17, 18] and references therein). One of the possible extensions of the classical polynomials are the so-called Krall-type polynomials. The Krall-type polynomials are polynomials which are orthogonal with respect to a linear functional euobtained from a quasi-definite functional u:P7→ C(P, denotes the space of complex polynomials with complex coefficients) via the addition of delta Dirac measures. In the last years the study of such polynomials have attracted an increasing interest with a special emphasis on the case when the starting functional uis a classical continuous, discrete or qlinear functional (for more details see [7] and references therein). These kind of polynomials appear as eigenfunctions of a fourth order linear differential operator with polynomial coefficients that do not depend on the degree of the polynomials. They were firstly considered by Krall in 1940 (see e.g. [13, chapter XV]) and further studied by several authors (for more details see [7] and references therein). For the case of the discrete lattice A. Dur´an has discovered very recently [10, 11] a method for obtaining the orthogonal polynomials satisfying higher order differential and difference equations. In two recent papers [7, 8] a general theory of the Krall-type polynomials on non-uniform lattices was developed. In fact, in [7, 8] the authors studied the polynomials e Pn(s)qwhich are orthogonal with respect to the linear functionals eu=u+PN k=1 Akδxkdefined on the q-quadratic lattice x(s) = c1qs+c2q−s+c3. For these polynomials the following expression have been found [7, Eq. (13)] (1) e Pn(s)q=Pn(s)q− M X i=1 Aie Pn(ai)qKn−1(s, ai), 2000 Mathematics Subject Classification. 33D45, 33C45, 42C05. Key words and phrases. Krall-type polynomials, second order linear difference equation, q-polynomials, basic hypergeometric series. The first author was partially supported by DGES grants MTM2009-12740-C03; PAI grant FQM-0262 and Junta de Andaluc´ıa grant P09-FQM-4643. 1 2 R. ´ ALVAREZ-NODARSE AND R. SEVINIK ADIG¨ UZEL provided that the sequence of monic polynomials ( e Pn)nexist. Here, (Pn)nare the polynomials orthogonal with respect to u,Ai∈Cand Kn(x, y) are the n-th reproducing kernels given by the Christoffel-Darboux formula Kn(s1, s2) := n X k=0 Pk(s1)qPk(s2)q d2 k =αn d2 n Pn+1(s1)qPn(s2)q−Pn+1(s2)qPn(s1)q x(s1)−x(s2). Moreover, in [7], we have studied the modifications of the non-standard q-Racah polynomials defined on the lattice x(s) = [s]q[s+ 1]q[6, 14], where [s]qdenotes the symmetric q-numbers [s]q= (qs/2−q−s/2)/(q1/2−q−1/2), whereas in [8] we developed the theory for the Askey-Wilson-Krall polynomials. Here, in this paper we will consider the “standard” (or classical) q-Racah-Krall polynomials defined on the lattice x(s) = q−s+δq−Nqsintroduced in [9] (see also [16, page 422]). For doing that we will follow the general results obtained in [7, section 2, 3]. The structure of the paper is as follows. In Section 2, the standard q-Racah-Krall polynomials are introduced and all their main characteristics are studied in detail. Finally in Section 3, some important limit cases are considered. 2. Standard q-Racah-Krall polynomials In this section, we study the standard q-Racah-Krall polynomials, i.e., the polynomials obtained from the standard q-Racah polynomials Rα,β n(s)qvia the addition of two mass points. In the following we follow the notations introduced in [7]. 2.1. Preliminaries. The standard q-Racah Rα,β n(s)q:= Rn(x(s), α, β, γ, δ|q) are polynomials on the q-quadratic lattice x(s) = q−s+δγqs+1 introduced in [9] (see also the more recent book [16, page 422]). They are defined by the following basic series (for properties of basic series see [12]) Rn(x(s), α, β, γ, δ|q) = (αq, βδq, γq;q)n (αβqn+1;q)n4ϕ3q−n, αβqn+1, q−s, δγqs+1 αq, βδq, γq q;q,(2) where x(s) = q−s+δγqs+1, and αq =q−N,βδq =q−N,γq =q−N,N, a nonnegative integer. Their main data can be found in table 1. Notice from (2) that we have Rα,β n(0)q:= Rn(x(0), α, β, γ, δ|q) = (αq, βδq, γq;q)n (αβqn+1;q)n , and Rα,β n(N)q:= Rn(x(N), α, β, γ, δ|q) Rα,β n(N)q= (δγqN+1)n(q−N, βγ−1q−N, δ−1q−N;q)n (βqn−N;q)n ,if αq =q−N, Rα,β n(N)q=δn(q−N, βq, δ−1αq;q)n (αβqn+1;q)n ,if γq =q−N, Rα,β n(N)q= (β−1γ)n(q−N, βq, δ−1γ−1αq−N;q)n (αβqn+1;q)n ,if βδq =q−N. We also need the following identity for the standard q-Racah polynomials which follows from [7, Eq. (7)] (3) Rα,β n−1(s)q= Θ(s, n)Rα,β n(s)q+ Ξ(s, n)Rα,β n(s+ 1)q, THE q-RACAH-KRALL-TYPE POLYNOMIALS 3 Table 1. Main data of the monic q-Racah polynomials Pn(s)Rn(x(s), α, β, γ, δ|q), x(s) = q−s+δγqs+1 , ∆x(s) = q−s(1 −δγq2s+2)(q−1−1) (a, b) [0, N] ρ(s)(αβ)−s (q−1/2−q1/2) (α−1β−1q−1, α−1γδq, β−1γq, δq;q)∞ (α−1β−1γ, α−1δ, β−1, γδq2;q)∞ (δγq, αq, βδq, γq;q)s (q, α−1δγq, β−1γq, δq;q)s σ(s)δ2q−2N(q1/2−q−1/2)2q−2s(qs−1)(qs−δ−1)(qs−βγ−1)(qs−αδ−1γ−1) Φ(s) (q1/2−q−1/2)2q−2s(1 −αqs+1)(1 −βδqs+1)(1 −γqs+1)(1 −δγqs+1) τ(s) (q−1/2−q1/2)q−s (1−γδq2s+1 )h(1 −αqs+1)(1 −βδqs+1)(1 −γqs+1)(1 −δγqs+1)−(δγq)2(qs−1) ×(qs−δ−1)(qs−βγ−1)(qs−αδ−1γ−1)i τn(s) −q−n(q1/2−q−1/2)n(1 −αβq2n+2)x(s+n 2) + q−n/2h(1 −αqn+1)(1 −βδqn+1)(1 −γqn+1) −(1 + δγqn+1)(1 −αβq2n+2)io λn−q−n+1 2(1 −qn)(1 −αβqn+1) d2 n (1 −αβq)(δγq)n (1 −αβq2n+1) (αq, βδq, γq, q, qαβγ−1, αδ−1q, βq;q)n (αβq, αβqn+1, αβqn+1;q)n βn 1 + δγq −(1 −αqn+1)(1 −αβqn+1)(1 −βδqn+1)(1 −γqn+1) (1 −αβq2n+1)(1 −αβq2n+2) −q(1 −qn)(1 −βqn)(γ−αβqn)(δ−αqn) (1 −αβq2n)(1 −αβq2n+1) γn (1 −αqn)(1 −αβqn)(1 −βδqn)(1 −γqn) (1 −αβq2n−1)(1 −αβq2n) q(1 −qn)(1 −βqn)(γ−αβqn)(δ−αqn) (1 −αβq2n)(1 −αβq2n+1) αn=bαnq−n+1 2(q−1/2−q1/2)(1 −αβq2n+1) βn(s)q−n/2+1/2(q1/2−q−1/2)(1 −αβqn+1) (1 −αβq2n+2)n(1 −αβq2n+2)x(s+n 2) + q−n/2h(1 −αqn+1) ×(1 −βδqn+1)(1 −γqn+1)−(1 + δγqn+1)(1 −αβq2n+2)io b βn(s)βn(s)−q−s−n+1 2(q1/2−q−1/2)(1 −qn)(1 −αβqn+1)(1 −δγq2s+1) where Θ(s, n) = −Ξ(s, n) + (1 −αβq2n−1)(1 −αβq2n)2 (1 −αqn)(1 −αβqn)(1 −βδqn)(1 −γqn)(1 −qn)(1 −βqn) ×1 (γ−αβqn)(δ−αqn)h(q−s−1)(1 −δγqs+1) + (1 −αqn+1)(1 −αβqn+1) (1 −αβq2n+1)(1 −αβq2n+2) ×(1 −βδqn+1)(1 −γqn+1) + q(1 −qn)(1 −βqn)(γ−αβqn)(δ−αqn) (1 −αβq2n)(1 −αβq2n+1)i −(1 −αβq2n−1)(1 −αβq2n)2 (1 −αqn)(1 −αβqn)(1 −βδqn)(1 −γqn)(1 −qn)(1 −βqn)(γ−αβqn) ×1 (δ−αqn)(qn/2(1 −αβqn+1) (1 −αβq2n+2)n(1 −αβq2n+2)xn(s) + q−n/2h(1 −αqn+1) ×(1 −βδqn+1)(1 −γqn+1)−(1 + δγqn+1)(1 −αβq2n+2)io−q−s(1 −qn) ×(1 −αβqn+1)(1 −δγq2s+1)), 4 R. ´ ALVAREZ-NODARSE AND R. SEVINIK ADIG¨ UZEL Ξ(s, n) = −q−s−1+n(1 −αβq2n−1)(1 −αβq2n)2(1 −αqs+1) (1 −αqn)(1 −αβqn)(1 −βδqn)(1 −γqn)(1 −qn)(1 −βqn) ×(1 −βδqs+1)(1 −q−N+s)(1 −δγqs+1) (γ−αβqn)(δ−αqn)(1 −δγq2s+2). 2.2. The case of two mass point. Let us now consider the modification of the standard q-Racah polynomials defined in (2). Let ube the functional with respect to which the standard q-Racah polynomials (2) are orthogonal. Then we study the orthogonal polynomials that are orthogonal with respect to the following linear functional eu=u+ Aδ(x(s)−x(0)) + Bδ(x(s)−x(N)). In other words, the q-Racah-Krall polynomials, Rα,β,A,B n(s)q:= RA,B n(x(s), α, β, γ, δ|q) satisfying the following orthogonality relation N X s=0 Rα,β,A,B n(s)qRα,β,A,B m(s)qρ(s)∆x(s−1/2) + ARα,β,A,B n(0)qRα,β,A,B m(0)q +BRα,β,A,B n(N)qRα,β,A,B m(N)q=δn,m e d2 n, (4) where ρis the standard q-Racah weight function (see table 1 (as in [7] we have chosen ρ(s) to be a probability measure). From [7, Eqs. (31)-(32)] it follows that (5) Rα,β,A,B n(0)q=(1 + BKα,β n−1(N, N))Rα,β n(0)q−BKα,β n−1(0, N)Rα,β n(N)q κα,β n−1(0, N), Rα,β,A,B n(N)q=−AKα,β n−1(N, 0)Rα,β n(0)q+ (1 + AKα,β n−1(0,0))Rα,β n(N)q κα,β n−1(0, N), e d2 n=d2 n+A(Rα,β n(0)q)2{1 + BKα,β n−1(N, N)}+B(Rα,β n(N)q)2{1 + AKα,β n−1(0,0)} κα,β n−1(0, N) −2ABRα,β n(0)qRα,β n(N)qKα,β n−1(0, N) κα,β n−1(0, N), where κα,β m(s, t) = 1 + AKα,β m(s, s) + BKα,β m(t, t) +AB nKα,β m(s, s)Kα,β m(t, t)−(Kα,β m(s, t))2o, (6) where Kα,β m(s, t) are the kernels Kα,β m(s, t) = Pm k=0 Rα,β k(s)qRα,β k(t)q/d2 k, and d2 ndenotes the squared norm of the n-th standard q-Racah polynomials (see table 1). Notice that ∀A, B > 0 and a6=b, Kα,β m(a, b)>0. Thus, by the Proposition 1 in [7] the polynomials Rα,β,A,B n(s)qare well defined for all values A, B > 0. Remark 1. If A, B are in general complex numbers then, according to Proposition 1 in [7], in order that there exists a sequence of orthogonal polynomials (Rα,β,A,B n(s)q)the condition κα,β n−1(0, N)6= 0, where κα,β n−1(0, N)is defined in (6), should be hold for all n∈N,A, B ∈C. THE q-RACAH-KRALL-TYPE POLYNOMIALS 5 Representation formulas for Rα,β,A,B n(s)q.Let us now obtain some explicit formulas for the q-Racah-Krall polynomials. The first formula follows from [7, Eq. (29)] (see also (1) from above) Rα,β,A,B n(s)q=Rα,β n(s)q−ARα,β,A,B n(0)qKα,β n−1(s, 0) −BRα,β,A,B n(N)qKα,β n−1(s, N) (7) where (8) Kα,β n−1(s, 0) = κα,β 0(s, n)Rα,β n−1(s)q+κα,β 0(s, n)∇Rα,β n−1(s)q ∇x(s), κα,β 0(s, n) =(δγq)−n+1(1 −αβqn)(1 −δγqs+n)(αβq, αβqn+1;q)n−1 (1 −αβq)(1 −δγqs+1)(q, αβγ−1q, αδ−1q, βq;q)n−1 , κα,β 0(s, n) =q2−s(δγ)−n+3(αβq, αβqn+1;q)n−1(qs−1)(qs−δ−1) (1 −αβq)(1 −δγq2s+2) ×(qs−βγ−1)(qs−αδ−1γ−1) (q, αβγ−1q, αδ−1q, βq;q)n−1 , (9) and (10) Kα,β n−1(s, N) = κα,β N(s, n)Rα,β n−1(s)q+κα,β N(s, n)∆Rα,β n−1(s)q ∆x(s), κα,β N(s, n) = (γ)−n+1(1 −αβqn)(1 −δqs−n+1)(αβq, αβqn+1;q)n−1 (1 −αβ)(1 −δqs)(β−1δ−1q−n, αq, βδq, q, αβγ−1q;q)n−1 , κα,β N(s, n) =(γ)−n+1q−s(1 −αqs+1)(1 −βδqs+1)(1 −γqs+1)(1 −δγqs+1) (1 −αβq)(1 −δγq2s+2) ×(αβq, αβqn+1;q)n−1 (β−1δ−1q−n, αq, βδq, q, αβγ−1q;q)n−1 . Inserting (8) and (10) into the formula (7) one finds the 1st representation formula Rα,β,A,B n(s)q=Rα,β n(s)q+A(s, n)Rα,β n−1(s)q+B(s, n)∇Rα,β n−1(s)q ∇x(s) +C(s, n)∆Rα,β n−1(s)q ∆x(s) (11) with A(s, n) = −ARα,β,A,B n(0)qκα,β 0(s, n)−BRα,β,A,B n(N)qκα,β N(s, n), B(s, n) = −ARα,β,A,B n(0)qκα,β 0(s, n), C(s, n) = −BRα,β,A,B n(N)qκα,β N(s, n), where Rα,β,A,B n(0)qand Rα,β,A,B n(N)qare given in (5). Notice that Rα,β,A,B n(s)qdefined in (7) is a polynomial of degree nin x(s). However, it is not easy to see that Rα,β,A,B n(s)qis a polynomial of degree nin x(s) by the formula (11) since the functions A,Band Cas well as ∇Rα,β n−1(s)q/∇x(s) and ∆Rα,β n−1(s)q/∆x(s) in (11) are not, in general, polynomials in x(s). The 2nd representation formula of the q-Racah-Krall polynomials follows from [7, section 3, (17)] and has the following form (12) φ(s)Rα,β,A,B n(s)q=A(s, n)Rα,β n(s)q+B(s, n)Rα,β n−1(s)q, 6 R. ´ ALVAREZ-NODARSE AND R. SEVINIK ADIG¨ UZEL φ(s) = (1 −δγqN+1qs)(1 −δγqs+1)(q−s−1)(q−s−q−N), A(s, n) = φ(s)−1 d2 n−1nARα,β,A,B n(0)qRα,β n−1(0)q(q−s−q−N)(1 −δγqN+1qs) +BRα,β,A,B n(N)qRα,β n−1(N)q(1 −δγqs+1)(q−s−1)o, B(s, n) = 1 d2 n−1nARα,β,A,B n(0)qRα,β n(0)q(q−s−q−N)(1 −δγqN+1qs) +BRα,β,A,B n(N)qRα,β n(N)q(1 −δγqs+1)(q−s−1)o, (13) where Rα,β,A,B n(0)qand Rα,β,A,B n(N)qare given in (5). Note that the functions φ(s) and A(s, n) are polynomials of degree 2 in x(s) and B(s, n) is 1st degree polynomial in x(s). Therefore it is obvious from the right hand side of the formula (12) that φ(s)Rα,β,A,B n(s)q is a polynomial of degree n+ 2 in x(s). Another representation formula for the q-Racah-Krall polynomials may be obtained by putting the relation (3) into (12) φ(s)Rα,β,A,B n(s)q=a(s;n)Rα,β n(s)q+b(s;n)Rα,β n(s+ 1)q, where a(s;n) = A(s;n) + B(s;n)Θ(s;n), b(s;n) = B(s;n)Ξ(s;n), and A,Band Θ, Ξ are given by (13) and (3), respectively. We note that from Proposition 3 in [7] it follows that the polynomials Rα,β,A,B n(s)q satisfy a second order linear difference equation where the coefficients can be computed explicitly. Since the expression is large enough we will omit them here. Finally, by use of [7, Eq (33), section 3] one can obtain the TTRR for Rα,β,A,B n(s)q x(s)Rα,β,A,B n(s)q=Rα,β,A,B n+1 (s)q+e βnRα,β,A,B n(s)q+eγnRα,β,A,B n−1(s)q with the following coefficients e βn=βn−A Rα,β,A,B n(0)qRα,β n−1(0)q d2 n−1 −Rα,β,A,B n+1 (0)qRα,β n(0)q d2 n! −B Rα,β,A,B n(N)qRα,β n−1(N)q d2 n−1 −Rα,β,A,B n+1 (N)qRα,β n(N)q d2 n!, eγn=γn 1+∆α,β,A,B n 1+∆α,β,A,B n−1 ,∆α,β,A,B n=ARα,β,A,B n(0)qRα,β n(0)q d2 n +BRα,β,A,B n(N)qRα,β n(N)q d2 n (14) where we use the notations defined in Eqs. (5). Representation of Rα,β,A,B n(s)qin terms of basic series. Let us now introduce the representation of Rα,β,A,B n(s)qin terms of the basic hypergeometric series. To this end, we substitute (2) into the formula (12) which leads to φ(s)Rα,β,A,B n(s)q=(αq;q)n−1(βδq;q)n−1(γq;q)n−1 (αβqn;q)n−1 × ∞ X k=0 (q−n, αβqn, q−s, δγqs+1;q)k (αq, βδq, γq, q;q)k qkΠ1(qk), THE q-RACAH-KRALL-TYPE POLYNOMIALS 7 where φ(s), A(s, n) and B(s, n) are given in (13) and Π1(qk) = A(s, n)(1 −αqn)(1 −βδqn)(1 −γqn)(1 −αβqn+k) (1 −αβq2n−1)(1 −αβq2n)+B(s, n)(1 −q−n+k) (1 −q−n) =−1 1−q−nnαβqnA(s, n)ϑα,β,δ,γ n+B(s, n)q−no(qk−qβ1), (15) where qβ1=A(s, n)ϑα,β,δ,γ n+B(s, n) A(s, n)αβqnϑα,β,δ,γ n+B(s, n)q−n, ϑα,β,δ,γ n=(1 −αqn)(1 −βδqn)(1 −γqn)(1 −q−n) (1 −αβq2n−1)(1 −αβq2n). Moreover, by use of the identity (qk−qz)(q−z;q)k= (1 −qz)(q1−z;q)kwe arrive at the following representation φ(s)Rα,β,A,B n(s)q=Dα,β,β1 n(s)5ϕ4q−n, αβqn, q−s, δγqs+1, q1−β1 αq, βδq, γq, q−β1q, q,(16) where Dα,β,β1 n(s) = −(αq;q)n−1(βδq;q)n−1(γq;q)n−1 (αβqn;q)n−1 (1 −qβ1) ×1 1−q−nnA(s, n)αβqnϑα,β,δ,γ n+B(s, n)q−no. Remark 2. Notice that φ(s)Rα,β,A,B n(s)qin (16) is a polynomial of degree n+ 2 in x(s). To see that formula (16) gives a polynomial of degree n+2 it is sufficient to notice that the function Π1defined in (15) is a polynomial of degree 2in x(s)since A(s, n)and B(s, n) defined in (13) are polynomials of degree 2 and 1 in x(s), respectively. Finally let us mention that the direct substitution of (2) into (12) leads to the following representation formula φ(s)Rα,β,A,B n(s)q=A(s, n)Λα,β,δ,γ n4ϕ3q−n, αβqn+1, q−s, δγqs+1 αq, βδq, γq q , q +B(s, n)Λα,β,δ,γ n−14ϕ3q−n+1, αβqn, q−s, δγqs+1 αq, βδq, γq q , q, where Λα,β,δ,γ n=(αq, βδq, γq;q)n (αβqn;q)n . 2.3. The case of one mass point. In this section we introduce the standard q-Racah polynomials but with one mass point at the value s= 0. All the formulas follow by replacing B= 0 into the ones in section 2.2. For example the first representation formula of Rα,β,A n(s)qis produced by inserting B= 0 into (11), Rα,β,A n(s)q=Rα,β n(s)q+A(s, n)Rα,β n−1(s)q+B(s, n)∇Rα,β n−1(s)q ∇x(s), where A(s, n) = −ARα,β,A n(0)qκα,β 0(s, n), B(s, n) = −ARα,β,A n(0)qκα,β 0(s, n), 8 R. ´ ALVAREZ-NODARSE AND R. SEVINIK ADIG¨ UZEL and (17) Rα,β,A n(0)q=Rα,β n(0)q 1 + AKα,β n−1(0,0) in which κα,β 0(s, n) and κα,β 0(s, n) are defined in (9), respectively. Moreover, the second representation formula follows by evaluating (12) for B= 0 φ(s)Rα,β,A n(s)q=A(s;n)Rα,β n(s)q+B(s;n)Rα,β n−1(s)q, where φ(s) = (q−s−1)(1 −δγqs+1), A(s, n) = φ(s)−A d2 n−1 Rα,β,A n(0)qRα,β n−1(0)q, B(s, n) = A d2 n−1 Rα,β,A n(0)qRα,β n(0)q, (18) and Rα,β,A n(0)qis given in (17). Finally, the third representation formula follows by use of the same idea (19) φ(s)Rα,β,A n(s)q=a(s;n)Rα,β n(s)q+b(s;n)Rα,β n(s+ 1)q, where a(s;n) = A(s;n) + B(s;n)Θ(s;n) and b(s;n) = B(s;n)Ξ(s;n), and A,Band Θ, Ξ are given by (18) and (3), respectively. Notice that, as for two mass points case, from the above representation formula (19) the second order difference equation for Rα,β,A n(s)q follows [1, 7]. Furthermore, one can obtain the coefficients of the TTRR by replacing B= 0 into (14) as the following e βn=βn−A Rα,β,A n(0)qRα,β n−1(0)q d2 n−1 −Rα,β,A n+1 (0)qRα,β n(0)q d2 n!, eγn=γn 1 + ∆α,β,A n 1 + ∆α,β,A n−1 ,∆α,β,A n=ARα,β,A n(0)qRα,β n(0)q d2 n . Notice that putting B= 0 in the basic series representation formulas (16) we obtain the corresponding basic series representations for the q-Racah-Krall polynomials Rα,β,A n(s)q. 2.4. Some limit cases. We first consider the modification of standard dual q-Hahn polynomials defined on the lattice x(s) = q−s+γδqs+1 by Rn(x(s), γ, δ, N|q) = (q−N, γq;q)n3ϕ2q−n, q−s, δγqs+1 q−N, γq q;q, that are related with the q-Racah polynomials by the expression [16] (20) lim β→0Rα,β n(s)q=Rn(s)q. In order to obtain the modification of standard dual q-Hahn polynomials by adding two mass points at the end of the interval of orthogonality, i.e., RA,B n(s)qsatisfying the orthogonality relation N X s=0 RA,B n(s)qRA,B m(s)qρ(s)∆x(s−1/2) + ARA,B n(0)qRA,B m(0)q +BRA,B n(N)qRA,B m(N)q=δn,m e d2 n. (21) THE q-RACAH-KRALL-TYPE POLYNOMIALS 9 where ρ(s) = qNs−(s 2)(γq)N (−γ)s(1 −γδq)(q−1/2−q1/2) (δq;q)N (γδq2;q)N (γq, γδq, q−N;q)s (q, δγqN+2, δq;q)s . Using the same procedure as before it follows that RA,B n(0)q=(1 + BKn−1(N, N))Rn(0)q−BKn−1(0, N)Rn(N)q κn−1(0, N), RA,B n(N)q=−AKn−1(N, 0)Rn(0)q+ (1 + AKn−1(0,0))Rn(N)q κn−1(0, N), e d2 n=d2 n+A(Rn(0)q)2{1 + BKn−1(N, N)}+B(Rn(N)q)2{1+AKn−1(0,0)} κn−1(0, N) −2ABRn(0)qRn(N)qKn−1(0, N) κn−1(0, N). In the above formulas κm(s, t) = 1 + AKm(s, s) + BKm(t, t) + AB Km(s, s)Km(t, t)−(Km(s, t))2, Km(s, t) = Pm k=0 Rn(s)qRn(t)q/d2 k, and d2 ndenote m-th Kernel and the norm of the standard dual q-Hahn polynomials d2 n= (γδq)n(q, δ−1q−N, γq, q−N;q)n. If we now fix αq =q−Nin the orthogonality relation for the standard q-Racah polynomials (4) and take the limit β→0, then using (20) we obtain the orthogonality relation (21), and therefore, it is straightforward to see that lim β→0Rα,β,A,B n(s)q=RA,B n(s)q. Thus, all properties of the modified dual q-Hahn polynomials RA,B n(s)qcan be obtained from the corresponding properties of the modified q-Racah polynomials Rα,β,A,B n(s)qby taking the appropriate limit. To conclude this paper we discuss two other important limit cases of the q-Racah-Krall polynomials Rα,β,A,B n(s)q. The first one is when we take the limit q→1. In fact, as q→1 in (2) we recover the Racah polynomials in the quadratic lattice x(s) = s(s+γ+δ+ 1) [16], i.e., lim q→1Rn(q−s+δγqs+1, qα, qβ, qγ, qδ)q=Rn(s(s+γ+δ+ 1), α, β, γ, δ), where Rα,β n(s) =(α+ 1)n(β+δ+ 1)n(γ+ 1)n (α+β+n+ 1)n × 4F3−n, α +β+n+ 1,−s, s +γ+δ+ 1 α+ 1, β +δ+ 1, γ + 1 1. Straightforward calculations show that all the properties of the standard q-Racah polynomials Rα,β n(s)qbecomes into the properties of the standard Racah ones. In particular, we have the following limit relation lim q→1Rα,β,A,B n(q−s+δγqs+1)q=Rα,β,A,B n(s(s+δ+γ+ 1)), where Rα,β,A,B n(s(s+δ+γ+ 1)) denotes the modification of the Racah polynomials by adding two delta Dirac masses at the points 0 and N. Thus, taking appropriate limits one can construct the analogue of the Askey Tableau but for the Krall type polynomials.