The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One

We investigate the quadratic decomposition and duality to classify symmetrical Hq-semiclassical orthogonal q-polynomials of class one where Hq is the Hahn's operator. For any canonical situation, the recurrence coefficients, the q-analog of the distributional equation of Pearson type, the momen...

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Дата:	2009
Автори:	Ghressi, A., Khériji, L.
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Мова:	English
Опубліковано:	Інститут математики НАН України 2009
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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Цитувати:	The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One / A. Ghressi, L. Khériji // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 35 назв. — англ.

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spelling	irk-123456789-1491362019-02-20T01:26:29Z The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One Ghressi, A. Khériji, L. We investigate the quadratic decomposition and duality to classify symmetrical Hq-semiclassical orthogonal q-polynomials of class one where Hq is the Hahn's operator. For any canonical situation, the recurrence coefficients, the q-analog of the distributional equation of Pearson type, the moments and integral or discrete representations are given. 2009 Article The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One / A. Ghressi, L. Khériji // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 35 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 33C45; 42C05 http://dspace.nbuv.gov.ua/handle/123456789/149136 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We investigate the quadratic decomposition and duality to classify symmetrical Hq-semiclassical orthogonal q-polynomials of class one where Hq is the Hahn's operator. For any canonical situation, the recurrence coefficients, the q-analog of the distributional equation of Pearson type, the moments and integral or discrete representations are given.
format	Article
author	Ghressi, A. Khériji, L.
spellingShingle	Ghressi, A. Khériji, L. The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Ghressi, A. Khériji, L.
author_sort	Ghressi, A.
title	The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One
title_short	The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One
title_full	The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One
title_fullStr	The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One
title_full_unstemmed	The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One
title_sort	symmetrical hq-semiclassical orthogonal polynomials of class one
publisher	Інститут математики НАН України
publishDate	2009
url	http://dspace.nbuv.gov.ua/handle/123456789/149136
citation_txt	The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One / A. Ghressi, L. Khériji // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 35 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
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first_indexed	2025-07-12T21:28:24Z
last_indexed	2025-07-12T21:28:24Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 076, 22 pages The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One Abdallah GHRESSI † and Lotf i KHÉRIJI ‡ † Faculté des Sciences de Gabès, Route de Mednine 6029 Gabès, Tunisia E-mail: Abdallah.Ghrissi@fsg.rnu.tn ‡ Institut Supérieur des Sciences Appliquées et de Technologies de Gabès, Rue Omar Ibn El-Khattab 6072 Gabès, Tunisia E-mail: Lotfi.Kheriji@issatgb.rnu.tn Received December 12, 2008, in final form July 07, 2009; Published online July 22, 2009 doi:10.3842/SIGMA.2009.076 Abstract. We investigate the quadratic decomposition and duality to classify symmetrical Hq-semiclassical orthogonal q-polynomials of class one where Hq is the Hahn’s operator. For any canonical situation, the recurrence coefficients, the q-analog of the distributional equation of Pearson type, the moments and integral or discrete representations are given. Key words: quadratic decomposition of symmetrical orthogonal polynomials; semiclassical form; integral representations; q-difference operator; q-series representations; the q-analog of the distributional equation of Pearson type 2000 Mathematics Subject Classification: 33C45; 42C05 1 Introduction Orthogonal polynomials (OP) have been a subject of research in the last hundred and fifty years. The orthogonality considered in our contribution is related to a form (regular linear functional) [8, 24] and not only to a positive measure. By classical orthogonal polynomials sequences (OPS), we refer to Hermite, Laguerre, Bessel and Jacobi polynomials. In the liter- ature, the extension of classical (OPS) can be done from different approaches such that the hypergeometric character [7, 8, 11, 18, 22] and the distributional equation of Pearson type [6, 8, 20, 29, 32]. A natural generalization of the classical character is the semiclassical one introduced by J.A. Shohat in [35]. This theory was developed by P. Maroni and extensively studied by P. Maroni and coworkers in the last decade [1, 24, 26, 28, 32]. Let Φ monic and Ψ be two polynomials, deg Φ = t ≥ 0, deg Ψ = p ≥ 1. We suppose that the pair (Φ,Ψ) is admissible, i.e., when p = t − 1, writing Ψ(x) = apx p + · · · , then ap 6= n + 1, n ∈ N. A form u is called semiclassical when it is regular and satisfies the distributional equation of Pearson type D(Φu) + Ψu = 0, (1.1) where the pair(Φ,Ψ) is admissible and D is the derivative operator. The corresponding monic orthogonal polynomials sequence (MOPS) {Bn}n≥0 is called semiclassical. Moreover, if u is semiclassical satisfying (1.1), the class of u, denoted s is defined by s := min ( max(deg Φ− 2,deg Ψ− 1) ) ≥ 0, where the minimum is taken over all pairs (Φ,Ψ) satisfying (1.1). In particular, when s = 0 the classical case is recovered. Symmetrical semiclassical forms of class one are well described in [1], see also [6]; there are three canonical situations: mailto:Abdallah.Ghrissi@fsg.rnu.tn mailto:Lotfi.Kheriji@issatgb.rnu.tn http://dx.doi.org/10.3842/SIGMA.2009.076 2 A. Ghressi and L. Khériji 1) The generalized Hermite formH(µ) (µ 6= 0, µ 6= −n− 1 2 , n ≥ 0) satisfying the distributional equation of Pearson type D(xH(µ)) + ( 2x2 − (2µ+ 1) ) H(µ) = 0. (1.2) 2) The generalized Gegenbauer G(α, β) (α 6= −n− 1, β 6= −n− 1, β 6= −1 2 , α+ β 6= −n− 1, n ≥ 0) satisfying the distributional equation of Pearson type D ( x ( x2 − 1 ) G(α, β) ) + ( − 2(α+ β + 2)x2 + 2(β + 1) ) G(α, β) = 0. (1.3) For further properties of the generalized Hermite and the generalized Gegenbauer polyno- mials see [8, 15, 26]. 3) The form B[ν] of Bessel kind (ν 6= −n − 1, n ≥ 0) [1, 31] satisfying the distributional equation of Pearson type D ( x3B[ν] ) − ( 2(ν + 1)x2 + 1 2 ) B[ν] = 0. (1.4) For an integral representation of B[ν] and some additional features of the associated (MOPS) see [14]. Other families of semiclassical orthogonal polynomials of class greater than one were discov- ered by solving functional equations of the type P (x)u = Q(x)v, where P , Q are two polynomials cunningly chosen and u, v two linear forms [19, 26, 27, 34]. For other relevant works in the semi- classical case see [5, 23]. In [21], instead of the derivative operator, the q-difference one is used to establish the theo- ry and characterizations of Hq-semiclassical orthogonal q-polynomials. Some examples of Hq- semiclassical orthogonal q-polynomials are given in [2, 13]. The Hq-classical case is exhaustively described in [20, 32]. Moreover, in [30] the symmetrical Dω-semiclassical orthogonal polynomials of class one are completely described by solving the system of their Laguerre–Freud equations where Dw is the Hahn’s operator. So, the aim of this paper is to present the classification of the symmetrical Hq-semiclassical orthogonal q-polynomials of class one by investigating the quadratic operator σ, the q-analog of the distributional equation of Pearson type satisfied by the corresponding form and some Hq- classical situations (see Tables 1 and 2) in connection with our problem. Among the obtained canonical cases, three are well known: two symmetrical Brenke type (MOPS) [8, 9, 10] and a symmetrical case of the Al-Salam and Verma (MOPS) [2]. Also, q-analogues of H(µ), G(α, β) and B[ν] appear. In [3, 33], the authors have established, up a dilation, a q-analogues of H(µ) and B[ν] using other methods. For any canonical case, we determine the recurrence coefficient, the q-analog of the distributional equation of Pearson type, the moments and a discrete measure or an integral representation. 2 Preliminary and first results 2.1 Preliminary and notations Let P be the vector space of polynomials with coefficients in C and let P ′ be its topological dual. We denote by 〈u, f〉 the effect of u ∈ P ′ on f ∈ P. In particular, we denote by (u)n := 〈u, xn〉, n ≥ 0 the moments of u. Moreover, a form (linear functional) u is called symmetric if (u)2n+1 = 0, n ≥ 0. Let us introduce some useful operations in P ′. For any form u, any polynomial g and any (a, b, c) ∈ (C \ {0})× C2, we let Hqu, gu, hau, τbu, (x− c)−1u and δc, be the forms defined by The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One 3 Table 1. Canonical cases. Hq-classical linear form case 1.1 U β̂n = {1− (1 + q)qn}qn−1, n ≥ 0, γ̂n+1 = (qn+1 − 1)q3n, n ≥ 0, Hq(xU)− (q − 1)−1(x+ 1)U = 0, (U)n = (−1)nq 1 2 n(n−1), n ≥ 0, U = ∞∑ k=0 (−1)k q−k2 s(k) (q−1; q−1)k δ−qk , q > 1, where s(k) = ∞∑ m=0 q−( 1 2 m(m+1)+km) (q−1; q−1)k εm+k, ε2k = (q − 1)k, k ≥ 0 and ε2k+1 = 0, k ≥ 0 case 1.2 little q-Laguerre L(a, q) ( a 6= 0, a 6= q−n−1, n ≥ 0 ) β̂n = {1 + a− a(1 + q)qn}qn, n ≥ 0, γ̂n+1 = a(1− qn+1)(1− aqn+1)q2n+1, n ≥ 0, Hq(xL(a, q))− (aq)−1(q − 1)−1{x− 1 + aq}L(a, q) = 0, (L(a, q))n = (aq; q)n, n ≥ 0, L(a, q) = (aq; q)∞ ∞∑ k=0 (aq)k (q; q)k δqk , 0 < q < 1, 0 < a < q−1, 〈L(a, q), f〉 = K ∫ q−1 0 x ln a ln q (qx; q)∞f(x)dx, f ∈ P, 0 < q < 1, 0 < a < q−1, where K−1 = q− ln a ln q−1 ∫ 1 0 x ln a ln q (x; q)∞dx, L(a, q) = 1 (a;q−1)∞ ∞∑ k=0 q− 1 2 k(k−1) (q−1; q−1)k (−a)kδqk , q > 1, a < 0 case 1.3 Wall W(b, q) ( b 6= 0, b 6= q−n, n ≥ 0 ) β̂n = {b+ q − b(1 + q)qn}qn, n ≥ 0, γ̂n+1 = b(1− qn+1)(1− bqn)q2n+2, n ≥ 0, Hq(xW(b, q))− b−1(q − 1)−1(q−1x+ b− 1)W(b, q) = 0, (W(b, q))n = qn(b; q)n, n ≥ 0, 〈W(b, q), f〉 = (b;q)∞ 2 ∞∑ k=0 bk (q; q)k 〈δq1+k , f〉+ K 2 ∫ 1 0 x ln b ln q−1(x; q)∞f(x)dx, where K−1 = ∫ 1 0 x ln b ln q−1(x; q)∞dx, f ∈ P, 0 < q < 1, 0 < b < 1, 〈W(b, q), f〉 = 1 (bq−1;q−1)∞ ∞∑ k=0 q− 1 2 k(k+1) (q−1; q−1)k (−b)k〈δq1+k , f〉, f ∈ P, q > 1, b 6= q±k, k ≥ 0 duality 〈Hqu, f〉 := −〈u,Hqf〉, 〈gu, f〉 := 〈u, gf〉, 〈hau, f〉 := 〈u, haf〉, f ∈ P, 〈τbu, f〉 := 〈u, τ−bf〉, 〈(x− c)−1u, f〉 := 〈u, θcf〉, 〈δc, f〉 := f(c), f ∈ P, where (Hqf)(x) = f(qx)−f(x) (q−1)x , q ∈ C̃ := { z ∈ C, z 6= 0, zn 6= 1, n ≥ 1 } [16, 18], (haf)(x) = f(ax), (τ−bf)(x) = f(x+ b), (θcf)(x) = f(x)−f(c) x−c [24] and it’s easy to see that [20, 26] Hq(fu) = (hq−1f)Hqu+ q−1(Hq−1f)u, f ∈ P, u ∈ P ′, (2.1) (x− c)((x− c)−1u) = u, (x− c)−1((x− c)u) = u− (u)0δc. 4 A. Ghressi and L. Khériji Continuation of Table 1. case 1.4 Generalized q−1-Laguerre U (α)(b, q) ( b 6= 0, b 6= qn+1+α, n ≥ 0 ) β̂n = {1− q−n−1 + q−1(1− bq−n−α)}q2n+α+1, n ≥ 0, γ̂n+1 = (1− q−n−1)(1− bq−n−1−α)q4n+2α+3, n ≥ 0, Hq(xU (α)(b, q)) + (q − 1)−1q−α−1(x+ b− qα+1)U (α)(b, q) = 0, (U (α)(b, q))n = (−b)n(b−1qα+1; q)n, n ≥ 0, 〈U (α)(b, q), f〉 = (b−1qα+1; q)∞ ∞∑ k=0 (b−1qα+1)k (q; q)k 〈δ−bqk , f〉, f ∈ P, 0 < q < 1, b > qα+1, α ∈ R, 〈U (α)(b, q), f〉 = 1 2(b−1qα;q−1)∞ ∞∑ k=0 q− 1 2 k(k−1)(−b−1qα)k (q−1; q−1)k 〈δ−bqk , f〉 + K 2 ∫ ∞ 0 xα− ln b ln q (−b−1x; q−1)∞ f(x)dx, f ∈ P, q > 1, qα < b < qα+1, α ∈ R, where K−1 = ∫ ∞ 0 xα− ln b ln q (−b−1x; q−1)∞ dx is given by (2.4) case 1.5 Alternative q-Charlier A(a, q) ( a 6= 0, a 6= −q−n, n ≥ 0 ) β̂n = 1 + aqn−1 + aqn − aq2n (1 + aq2n−1)(1 + aq2n+1) qn, n ≥ 0, γ̂n+1 = aq3n+1 (1− qn+1)(1 + aqn) (1 + aq2n)(1 + aq2n+1)2(1 + aq2n+2) , n ≥ 0, Hq(x2A(a, q))− (aq)−1(q − 1)−1{(1 + aq)x− 1}A(a, q) = 0, (A(a, q))n = 1 (−aq; q)n , n ≥ 0, 〈A(a, q), f〉 = 1 2(−aq; q)∞ ∞∑ k=0 q 1 2 k(k+1)ak (q; q)k 〈δqk , f〉+ q 1 2 ( ln a ln q + 1 2 )2 (−a−1; q)∞ 2 √ 2π ln q−1 × ∫ ∞ 0 x ln a ln q− 1 2 (qx; q)∞ exp ( − ln2 x 2 ln q−1 ) f(x)dx, f ∈ P, 0 < q < 1, a > 0 Now, we introduce the operator σ : P −→ P defined by (σf)(x) := f(x2) for all f ∈ P. Consequently, we define σu by duality [8, 25] 〈σu, f〉 := 〈u, σf〉, f ∈ P, u ∈ P ′. We have the well known formula [25] f(x)σu = σ ( f ( x2 ) u ) . (2.2) Let {Bn}n≥0 be a sequence of monic polynomials with degBn = n, n ≥ 0, the form u is called regular if we can associate with it a sequence of polynomials {Bn}n≥0 such that 〈u,BmBn〉 = rnδn,m, n,m ≥ 0; rn 6= 0, n ≥ 0. The sequence {Bn}n≥0 is then said orthogonal with respect to u. {Bn}n≥0 is an (OPS) and it can be supposed (MOPS). The sequence {Bn}n≥0 fulfills the recurrence relation B0(x) = 1, B1(x) = x− β0, Bn+2(x) = (x− βn+1)Bn+1(x)− γn+1Bn(x), γn+1 6= 0, n ≥ 0. (2.3) When u is regular, {Bn}n≥0 is a symmetrical (MOPS) if and only if βn = 0, n ≥ 0. The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One 5 Continuation of Table 1. case 1.6 little q-Jacobi U(a, b, q) ( ab 6= 0, a 6= q−n−1, b 6= q−n−1, ab 6= q−n, n ≥ 0 ) β̂n = (1 + a)(1 + abq2n+1)− a(1 + b)(1 + q)qn (1− abq2n)(1− abq2n+2) qn, n ≥ 0, γ̂n+1 = aq2n+1 (1− qn+1)(1− aqn+1)(1− bqn+1)(1− abqn+1) (1− abq2n+1)(1− abq2n+2)2(1− abq2n+3) , n ≥ 0, Hq(x(x− b−1q−1)U(a, b, q)) + (abq2(q − 1))−1{(1− abq2)x+ aq − 1}U(a, b, q) = 0, (U(a, b, q))n = (aq; q)n (abq2; q)n , n ≥ 0, 〈U(a, b, q), f〉 = (aq; q)∞ 2(abq2; q)∞ ∞∑ k=0 (bq; q)k (q; q)k (aq)k〈δqk , f〉+ K 2 ∫ q−1 0 x ln a ln q (qx; q)∞ (bqx; q)∞ f(x)dx, f ∈ P, 0 < q < 1, 0 < a < q−1, b ∈]−∞, 1] \ {0}, where K−1 = ∫ q−1 0 x ln a ln q (qx; q)∞ (bqx; q)∞ dx, 〈U(a, b, q), f〉 = (a−1q−1; q−1)∞ 2(a−1b−1q−2; q−1)∞ ∞∑ k=0 (b−1q−1; q−1)k (q−1; q−1)k (aq)−k〈δb−1q−k−1 , f〉 + K 2 ∫ b−1 0 x ln a ln q (bx; q−1)∞ (x; q−1)∞ f(x)dx, f ∈ P, q > 1, a > q−1, b ≥ 1 where K−1 = ∫ b−1 0 x ln a ln q (bx; q−1)∞ (x; q−1)∞ dx case 1.7 q-Charlier-II-U(µ, q) ( µ 6= 0, µ 6= q−n, n ≥ 0 ) β̂n = 1− (1 + q)qn + µq2n (1− µq2n−1)(1− µq2n+1) qn−1, n ≥ 0, γ̂n+1 = −q3n (1− qn+1)(1− µqn) (1− µq2n)(1− µq2n+1)2(1− µq2n+2) , n ≥ 0, Hq(x(x− µ−1q−1)U(µ, q))− (µq(q − 1))−1{(µq − 1)x− 1}U(µ, q) = 0, (U(µ, q))n = (−1)n q 1 2 n(n−1) (µq; q)n , n ≥ 0, 〈U(µ, q), f〉 = 1 (µ−1q−1; q−1)∞ ∞∑ k=0 q− 1 2 k(k+1) (q−1; q−1)k (−µ−1)k〈δµ−1q−k−1 , f〉, f ∈ P, q > 1, µ < 0 Lastly, let us recall the following standard expressions [8, 11, 20] (a; q)0 := 1, (a; q)n := n∏ k=1 ( 1− aqk−1 ) , n ≥ 1, (a; q)∞ := ∞∏ k=0 ( 1− aqk ) , \|q\| < 1, the q-binomial theorem [4, 17] ∞∑ k=0 (a; q)k (q; q)k zk = (az; q)∞ (z; q)∞ , \|z\| < 1, \|q\| < 1,∫ ∞ 0 tx−1 (−at; q)∞ (−t; q)∞ dt =  π sin(πx) (a; q)∞ (aq−x; q)∞ (q1−x; q)∞ (q; q)∞ , x ∈ R+ \ N, \|a\| < qx, 0 < q < 1, (−q)m 1− qm (q−1; q−1)m (aq−1; q−1)m ln(q−1), x = m ∈ N?, \|a\| < qm, 0 < q < 1. (2.4) 6 A. Ghressi and L. Khériji Continuation of Table 1. case 1.8 Generalized Stieltjes–Wigert S(ω, q) ( ω 6= q−n, n ≥ 0 ) β̂n = {(1 + q)q−n − q − ω}q−n− 3 2 , n ≥ 0, γ̂n+1 = (1− qn+1)(1− ωqn)q−4n−4, n ≥ 0, Hq(x(x+ ωq− 3 2 )S(ω, q))− (q − 1)−1 { x+ (ω − 1)q− 3 2 } S(ω, q) = 0, (S(ω, q))n = q− 1 2 n(n+2)(ω; q)n, n ≥ 0, S(ω, q) = (ω−1; q−1)∞ ∞∑ k=0 ω−k (q−1; q−1)k δ −ωq−k− 3 2 , q > 1, ω > 1, 〈S(ω, q), f〉 = K ∫ ∞ 0 x ln ω ln q −1 (−q 3 2ω−1x; q)∞ f(x)dx, f ∈ P, 0 < q < 1, 0 < ω < 1, where K−1 = ∫ ∞ 0 x ln ω ln q −1 (−q 3 2ω−1x; q)∞ dx is given by (2.4), 〈S(ω, q), f〉 = Kω ∫ ∞ q− 1 2 \|ω\| (−q− 1 2 \|ω\|x−1; q)∞ exp ( − ln2 x 2 ln q−1 ) f(x)dx, f ∈ P, 0 < q < 1, ω ≤ 0, where K−1 ω = ∫ ∞ q− 1 2 \|ω\| (−q− 1 2 \|ω\|x−1; q)∞ exp ( − ln2 x 2 ln q−1 ) dx, in particular K0 = √ q 2π ln q−1 Table 2. Limiting cases. Hq-classical linear form case 2.1 q-analogue of Laguerre L(α, q) (α 6= −[n]q − 1, n ≥ 0) β̂n = qn { (1 + q−1)[n]q + 1 + α } , n ≥ 0, γ̂n+1 = q2n[n+ 1]q { [n]q + 1 + α } , n ≥ 0, Hq(xL(α, q)) + (x− 1− α)L(α, q) = 0 case 2.2 q-analogue of Bessel B(α, q) (α 6= 1 2 (q − 1)−1, α 6= − 1 2 [n]q, n ≥ 0) β̂n = −2qn 2α+ (1 + q−1)[n− 1]q − q−1[2n]q (2α+ [2n− 2]q)(2α+ [2n]q) , n ≥ 0, γ̂n+1 = −4q3n [n+ 1]q(2α+ [n− 1]q) (2α+ [2n− 1]q)(2α+ [2n]q)2(2α+ [2n+ 1]q) , n ≥ 0, Hq(x2B(α, q))− 2(αx+ 1)B(α, q) = 0 case 2.3 q-analogue of Jacobi J(α, β, q) (α+ β 6= 3−2q q−1 , α+ β 6= −[n]q − 2, n ≥ 0, β 6= −[n]q − 1, n ≥ 0 et α+ β + 2− (β + 1)qn + [n]q 6= 0, n ≥ 0) β̂n = qn−1 (1 + q)(α+ β + 2 + [n− 1]q)(β + 1 + [n]q)− (β + 1)(α+ β + 2 + [2n]q) (α+ β + 2 + [2n− 2]q)(α+ β + 2 + [2n]q) , n ≥ 0, γ̂n+1 = q2n [n+ 1]q(α+ β + 2 + [n− 1]q)([n]q + β + 1)(α+ β + 2− (β + 1)qn + [n]q) (α+ β + 2 + [2n− 1]q)(α+ β + 2 + [2n]q)2(α+ β + 2 + [2n+ 1]q) , n ≥ 0, Hq(x(x− 1)J(α, β, q))− ((α+ β + 2)x− (β + 1))J(α, β, q) = 0 2.2 Some results about the Hq-semiclassical character A form u is called Hq-semiclassical when it is regular and there exist two polynomials Φ and Ψ, Φ monic, deg Φ = t ≥ 0, deg Ψ = p ≥ 1 such that Hq(Φu) + Ψu = 0, (2.5) the corresponding orthogonal polynomial sequence {Bn}n≥0 is called Hq-semiclassical [21]. The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One 7 The Hq-semiclassical character is kept by a dilation [21]. In fact, let {a−n(haBn)}n≥0, a 6= 0; when u satisfies (2.5), then ha−1u fulfills the q-analog of the distributional equation of Pearson type Hq ( a−tΦ(ax)ha−1u ) + a1−tΨ(ax)ha−1u = 0, and the recurrence coefficients of (2.3) are βn a , γn+1 a2 , n ≥ 0. Also, the Hq-semiclassical form u is said to be of class s = max(p − 1, t − 2) ≥ 0 if and only if [21] ∏ c∈ZΦ {∣∣q(hqΨ)(c) + (HqΦ)(c) ∣∣ + ∣∣〈u, q(θcqΨ) + (θcq ◦ θcΦ)〉 ∣∣} > 0, (2.6) where ZΦ is the set of zeros of Φ. In particular, when s = 0 the form u is usually called Hq-classical (Al-Salam–Carlitz, big q-Laguerre, q-Meixner, Wall, . . . ) [20]. Lemma 1 ([21]). Let u be a symmetrical Hq-semiclassical form of class s satisfying (2.5). The following statements holds i) If s is odd then the polynomial Φ is odd and Ψ is even. ii) If s is even then the polynomial Φ is even and Ψ is odd. In the sequel we are going to use some Hq-classical forms [20], resumed in Table 1 (canonical cases: 1.1–1.8) and Table 2 (limiting cases: 2.1–2.3). In fact, when q → 1 in results of Table 2, we recover the classical Laguerre L(α), Bessel B(α) and h− 1 2 ◦ τ−1J (α, β) respectively where J (α, β) is the Jacobi classical form [24]. Moreover in what follows we are going to use the logarithmic function denoted by Log : C \ {0} −→ C defined by Log z = ln \|z\|+ iArg z, z ∈ C \ {0}, −π < Arg z ≤ π, Log is the principal branch of log and includes ln : R+\{0} −→ R as a special case. Consequently, the principal branch of the square root is √ z = √ \|z\| ei Arg z 2 , z ∈ C \ {0}, −π < Arg z ≤ π. 2.3 On quadratic decomposition of a symmetrical regular form Let u be a symmetrical regular form and {Bn}n≥0 be its MOPS satisfying (2.3) with βn = 0, n ≥ 0. It is very well known (see [8, 25]) that B2n(x) = Pn ( x2 ) , B2n+1(x) = xRn ( x2 ) , n ≥ 0, where {Pn}n≥0 and {Rn}n≥0 are the two MOPS related to the regular form σu and xσu respec- tively. In fact, [8, 25] u is regular ⇔ σu and xσu are regular, u is positive definite ⇔ σu and xσu are positive definite. 8 A. Ghressi and L. Khériji Furthermore, taking P0(x) = 1, P1(x) = x− βP 0 , Pn+2(x) = ( x− βP n+1 ) Pn+1(x)− γP n+1Pn(x), γP n+1 6= 0, n ≥ 0, and R0(x) = 1, R1(x) = x− βR 0 , Rn+2(x) = ( x− βR n+1 ) Rn+1(x)− γR n+1Rn(x), γR n+1 6= 0, n ≥ 0, we get [8, 25] βP 0 = γ1, βP n+1 = γ2n+2 + γ2n+3 , n ≥ 0, γP n+1 = γ2n+1γ2n+2, n ≥ 0, (2.7) and βR n = γ2n+1 + γ2n+2, n ≥ 0, γR n+1 = γ2n+2γ2n+3, n ≥ 0. (2.8) Consequently, γ1 = βP 0 , γ2 = γP 1 βP 0 , γ2n+1 = βP 0 n∏ k=1 γR k n∏ k=1 γP k , γ2n+2 = 1 βP 0 n+1∏ k=1 γP k n∏ k=1 γR k , n ≥ 1. (2.9) Proposition 1. Let u be a symmetrical regular form. (i) The moments of u are (u)2n = (σu)n, (u)2n+1 = 0, n ≥ 0. (2.10) (ii) If σu has the discrete representation σu = ∞∑ k=0 ρkδτk , ∞∑ k=0 ρk = 1, (2.11) then a possible discrete measure of u is u = ∞∑ k=0 ρk δ√τk + δ− √ τk 2 . (2.12) (iii) If u is positive definite and σu has the integral representation 〈σu, f〉 = ∫ ∞ 0 V (x)f(x)dx, f ∈ P, ∫ ∞ 0 V (x)dx = 1, (2.13) then, a possible integral representation of u is 〈u, f〉 = ∫ ∞ −∞ \|x\|V (x2)f(x)dx, f ∈ P. (2.14) The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One 9 Proof. (i) is a consequence from the definition of the quadratic operator σ. For (ii) taking into account (2.10), (2.11) we get (u)2n = (σu)n = ∞∑ k=0 ρk( √ τk)2n = ∞∑ k=0 ρk ( √ τk)2n + (− √ τk)2n 2 . But (u)2n+1 = 0 = ∞∑ k=0 ρk ( √ τk)2n+1 + (− √ τk)2n+1 2 . Hence the desired result (2.12) holds. For (iii) consider f ∈ P and let us split up the polynomial f accordingly to its even and odd parts f(x) = f e ( x2 ) + xfo ( x2 ) . (2.15) Therefore since u is a symmetrical form 〈u, f(x)〉 = 〈u, f e ( x2 ) 〉 = 〈σu, f e(x)〉. (2.16) From (2.15) we get f e(x) = f( √ x) + f(− √ x) 2 , x ∈ R+. (2.17) By (2.13) and according to (2.16), (2.17) we recover the representation in (2.14). � 3 Symmetrical H√ q-semiclassical orthogonal polynomials of class one Lemma 2. We have σ(Hqu) = (q + 1)Hq2(σ(xu)), u ∈ P ′. (3.1) Proof. From the definition of Hq we get (Hq(σf))(x) = (q + 1)x(σ(Hq2f))(x), f ∈ P. Therefore, ∀ f ∈ P, 〈σ(Hqu), f〉 = 〈Hqu, σf〉 = −〈u, (q + 1)xσ(Hq2f)〉 = −〈(q + 1)σ(xu),Hq2f〉 = 〈(q + 1)Hq2(σ(xu)), f〉. Thus the desired result. � Lemma 3. Let u be a symmetrical H√ q-semiclassical form of class one. There exist two poly- nomials ϕ and ψ, ϕ monic, with degϕ ≤ 1 and degψ = 1, such that H√ q ( xϕ ( x2 ) u ) + ψ ( x2 ) u = 0. (3.2) Proof. The result is a consequence from the definition of the class and Lemma 1. � 10 A. Ghressi and L. Khériji Corollary 1. Let u be a symmetrical H√ q-semiclassical form of class one satisfying (3.2); then σu et xσu are Hq-classical satisfying respectively the following q-analog of the distributional equation of Pearson type Hq(xϕ(x)σu) + 1 √ q + 1 ψ(x)σu = 0, (3.3) Hq(xϕ(x)(xσu)) + q−1 ( 1 √ q + 1 ψ(x)− ϕ(x) ) (xσu) = 0. (3.4) Proof. First, σu and xσu are regular because u is symmetrical and regular. Applying the quadratic operator σ to (3.2) and taking into account (3.1) we get ( √ q + 1)Hq ( σ ( x2ϕ ( x2 ) u )) + σ ( ψ ( x2 ) u ) = 0. By (2.2) we get (3.3). Now, multiplying both sides of (3.3) by q−1x, using the identity in (2.1), this yields to (3.4). � Regarding Table 1 (cases 1.1–1.8), Table 2 (cases 2.1–2.3) and the q-analog of the distribu- tional equation of Pearson type (3.3), (3.4), we consider the following situations for the polyno- mial ϕ in order to get a H√ q-semiclassical form from a Hq-classical A. ϕ(x) = 1 (cases 1.1, 1.2, 1.3, 1.4, 2.1); B. ϕ(x) = x (cases 1.5, 2.2); C. ϕ(x) = x− 1 (case 2.3); D. ϕ(x) = x− b−1q−1 (case 1.6); E. ϕ(x) = x− µ−1q−1 (case 1.7); F. ϕ(x) = x+ ωq− 3 2 (case 1.8). A. In the case ϕ(x) = 1 the q-analog of the distributional equation of Pearson type (3.3), (3.4) are Hq(xσu) + 1 √ q + 1 ψ(x)σu = 0, (3.5) Hq(x(xσu)) + q−1 ( 1 √ q + 1 ψ(x)− 1 ) (xσu) = 0. (3.6) A1. If ψ(x) = ( √ q+1)(x−1−α) the q-analogue of the Laguerre form L(α, q), α 6= −[n]q−1, n ≥ 0 (case 2.1 in Table 2) satisfying Hq(xL(α, q)) + (x− 1− α)L(α, q) = 0. Comparing with (3.5), (3.6) we get σu = L(α, q), α 6= −[n]q − 1, n ≥ 0, (3.7) and xσu = (1 + α)L ( q−1(α+ 2)− 1, q ) , α 6= −[n]q − 1, n ≥ 0. (3.8) Taking into account the recurrence coefficients (see case 2.1 in Table 2), by virtue of (3.7), (3.8) and (2.7), (2.8) we get for n ≥ 0 βP n = qn {( 1 + q−1 ) [n]q + 1 + α } , γP n+1 = q2n[n+ 1]q{[n]q + 1 + α}, βR n = qn+1 {( 1 + q−1 ) [n]q + q−1(2 + α) } , γR n+1 = q2n+2[n+ 1]q { [n]q + q−1(2 + α) } . The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One 11 With the relation [k − 1]q = q−1[k]q − q−1, k ≥ 1 the system (2.9) becomes for n ≥ 0 γ2n+1 = qn([n]q + 1 + α), γ2n+2 = qn[n+ 1]q. (3.9) Writing α = µ − 1 2 , µ 6= −[n]q − 1 2 , n ≥ 0 and denoting the symmetrical form u by H(µ, q) we get the following result: Proposition 2. The symmetrical form H(µ, q) satisfies the following properties: 1) The recurrence coefficient γn+1 satisfies (3.9). 2) H(µ, q) is regular if and only if µ 6= −[n]q − 1 2 , n ≥ 0. 3) H(µ, q) is positive definite if and only if q > 0, µ > −1 2 . 4) H(µ, q) is a H√ q-semiclassical form of class one for µ 6= 1√ q( √ q+1) − 1 2 , µ 6= −[n]q − 1 2 , n ≥ 0 satisfying the q-analog of the distributional equation of Pearson type H√ q(xH(µ, q)) + ( √ q + 1) ( x2 − µ− 1 2 ) H(µ, q) = 0. (3.10) Proof. The results in 1), 2) and 3) are straightforward from (3.9). For 4), it is clear that H(µ, q) satisfies (3.10); in this case and by virtue of (2.6), we are going to prove that the class of H(µ, q) is exactly one for µ 6= 1√ q( √ q+1) − 1 2 , µ 6= −[n]q − 1 2 , n ≥ 0. Denoting Φ(x) = x, Ψ(x) = ( √ q + 1) ( x2 − µ− 1 2 ) , we have accordingly to (2.6), on one hand √ q ( h√qΨ ) (0) + ( H√ qΦ ) (0) = 1−√ q( √ q + 1) ( µ+ 1 2 ) 6= 0, and on the other hand by (θ0Ψ)(x) = ( √ q + 1)x and (θ2 0Φ)(x) = 0, 〈H(µ, q), √ qθ0Ψ + θ2 0Φ〉 = 0, taking into account that u is a symmetrical form. � Remark 1. The symmetrical form H(µ, q), µ 6= 1√ q( √ q+1) − 1 2 , µ 6= −[n]q − 1 2 , n ≥ 0 is the q-analogue of the generalized Hermite one [12] (when q → 1 we recover the generalized Hermite formH(µ) (see (1.2)) which is a symmetrical semiclassical form of class one for µ 6= 0, µ 6= −n− 1 2 , n ≥ 0 [1, 8, 15, 26]). A2. If ψ(x) = −( √ q−1)−1(x+1) the form U that satisfies the q-analog of the distributional equation of Pearson type (see case 1.1 in Table 1) Hq(xU)− (q − 1)−1(x+ 1)U = 0. Comparing with (3.5), (3.6) we get σu = U , (3.11) and xσu = −hqU . (3.12) Taking into account (3.11), (3.12), (2.7), (2.8) and the case 1.1 in Table 1 we obtain for n ≥ 0 βP n = { 1− (1 + q)qn } qn−1, γP n+1 = ( qn+1 − 1 ) q3n, βR n = { 1− (1 + q)qn } qn, γR n+1 = ( qn+1 − 1 ) q3n+2. Consequently, the system (2.9) becomes for n ≥ 0 γ2n+1 = −q2n, γ2n+2 = ( 1− qn+1 ) qn. (3.13) 12 A. Ghressi and L. Khériji Proposition 3. The symmetrical form u satisfies the following properties: 1) The recurrence coefficient γn+1 satisfies (3.13). 2) u is regular for any q ∈ C̃. 3) u is a H√ q-semiclassical form of class one satisfying H√ q(xu)− ( √ q − 1)−1 ( x2 + 1 ) u = 0. (3.14) 4) The moments of u are (u)2n = (−1)nq 1 2 n(n−1), (u)2n+1 = 0, n ≥ 0. 5) we have the following discrete representation u = ∞∑ k=0 (−1)kq−k2 s(k) (q−1; q−1)k δ iq k 2 + δ −iq k 2 2 , q > 1. Proof. The results in 1), 2) are obvious from (3.13). For 3), it is clear that u satisfies (3.14). Denoting Φ(x) = x, Ψ(x) = −( √ q − 1)−1 ( x2 + 1 ) , we have (2.6) √ q ( h√qΨ ) (0) + ( H√ qΦ ) (0) = 1 1−√ q 6= 0, 〈u,√qθ0Ψ + θ2 0Φ〉 = 0. Therefore, u is of class one. The results in 4) and 5) are consequence from (2.10)–(2.12) and those for U (case 1.1 in Table 1). � A3. If ψ(x) = −(aq)−1( √ q − 1)−1(x − 1 + aq) the little q-Laguerre form L(a, q), a 6= 0, a 6= q−n−1, n ≥ 0 (case 1.2 in Table 1) satisfying Hq(xL(a, q))− (aq)−1(q − 1)−1(x− 1 + aq)L(a, q) = 0. With (3.5), (3.6) we obtain σu = L(a, q), a 6= 0, a 6= q−n−1, n ≥ 0, (3.15) and xσu = (1− aq)L(aq, q), a 6= 0, a 6= q−n−1, n ≥ 0. (3.16) By virtue of the recurrence coefficients of little q-Laguerre polynomials in Table 1, case 1.2, the relations in (3.15), (3.16) and (2.7), (2.8) we get for n ≥ 0 βP n = { 1 + a− a(1 + q)qn } qn, γP n+1 = a ( 1− qn+1 )( 1− aqn+1 ) q2n+1, βR n = { 1 + aq − a(1 + q)qn+1 } qn, γR n+1 = a ( 1− qn+1 )( 1− aqn+2 ) q2n+2. Therefore (2.9) becomes for n ≥ 0 γ2n+1 = qn ( 1− aqn+1 ) , γ2n+2 = aqn+1 ( 1− qn+1 ) . (3.17) Comparing with [2], u is a symmetrical case of the Al-Salam–Verma form, u := SV(a, q). From (3.17), it is easy to see that SV(a, q) is regular if and only if a 6= 0, a 6= q−n−1, n ≥ 0. Also, SV(a, q) is positive definite if and only if 0 < q < 1, 0 < a < q−1 or q > 1, a < 0. The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One 13 Proposition 4. The form SV(a, q) is a H√ q-semiclassical form of class one for a 6= 0, a 6= q− 1 2 , a 6= q−n−1, n ≥ 0 satisfying H√ q(xSV(a, q))− (aq)−1( √ q − 1)−1 ( x2 − 1 + aq ) SV(a, q) = 0. (3.18) The moments are (SV(a, q))2n = (aq; q)n, (SV(a, q))2n+1 = 0, n ≥ 0, (3.19) and the orthogonality relation can be represented 〈SV(a, q), f〉 = (aq; q)∞ 2 ∞∑ k=0 (aq)k (q; q)k 〈δ q k 2 + δ −q k 2 2 , f 〉 + K 2 ∫ q− 1 2 −q− 1 2 \|x\|2 ln a ln q +1(qx2; q)∞f(x)dx, f ∈ P, 0 < q < 1, 0 < a < q−1, (3.20) with K−1 = q − ln a ln q −1 ∫ 1 0 x ln a ln q (x; q)∞dx, and SV(a, q) = 1 (a; q−1)∞ ∞∑ k=0 q− 1 2 k(k−1)(−a)k (q−1; q−1)k δ q k 2 + δ −q k 2 2 , q > 1, a < 0. (3.21) Proof. It is direct that the form SV(a, q) satisfies the q-analog of the distributional equation of Pearson type (3.18). Denoting Φ(x) = x, Ψ(x) = −(aq)−1( √ q−1)−1 ( x2−1+aq ) , we have (2.6) √ q ( h√qΨ ) (0) + ( H√ qΦ ) (0) = a−1q− 1 2 − 1 √ q − 1 6= 0, 〈SV(a, q), √ qθ0Ψ + θ2 0Φ〉 = 0, from which we get that SV(a, q) is of class one because a 6= 0, a 6= q− 1 2 , a 6= q−n−1, n ≥ 0. The results mentioned in (3.19)–(3.21) are easily obtained from those well known the properties of the little q-Laguerre from (case 1.2 in Table 1) and (2.10)–(2.14). � Remark 2. The regular form SV(q− 1 2 , q) is the discrete √ q-Hermite form which is H√ q- classical [20]. A4. If ψ(x) = −b−1( √ q − 1)−1 ( q−1x+ b − 1 ) the Wall form W(b, q), b 6= 0, b 6= q−n, n ≥ 0 (case 1.3 in Table 1) that satisfies Hq(xW(b, q))− b−1(q − 1)−1(q−1x+ b− 1)W(b, q) = 0. In accordance of (3.5), (3.6) we get σu = W(b, q), b 6= 0, b 6= q−n, n ≥ 0, and xσu = q(1− b)W(bq, q), b 6= 0, b 6= q−n, n ≥ 0. We recognize the Brenke type symmetrical regular form Y(b, q) [8, 9, 10]. In [13] it is proved that Y(b, q) is H√ q-semiclassical of class one for b 6= 0, b 6= √ q, b 6= q−n, n ≥ 0 satisfying H√ q(xY(b, q))− b−1 ( q 1 2 − 1 )−1{ q−1x2 + b− 1 } Y(b, q) = 0. (3.22) Also in that work, moments, discrete and integral representations are established. 14 A. Ghressi and L. Khériji Remark 3. Likewise, from (3.22) it is easy to see that h 1√ q Y( √ q, q) is the H√ q-classical discrete √ q-Hermite form [20]. A5. If ψ(x) = ( √ q − 1)−1q−α−1 ( x+ b− qα+1 ) the generalized q−1-Laguerre U (α)(b, q) form, b 6= 0, b 6= qn+1+α, n ≥ 0 and its q-analog of the distributional equation of Pearson type (case 1.4 in Table 1) Hq(xU (α)(b, q)) + (q − 1)−1q−α−1(x+ b− qα+1)U (α)(b, q) = 0. By (3.5), (3.6) we deduce the following relationships σu = U (α)(b, q), b 6= 0, b 6= qn+1+α, n ≥ 0, (3.23) xσu = ( qα+1 − b ) U (α+1)(b, q), b 6= 0, b 6= qn+1+α, n ≥ 0. (3.24) From Table 1, case 1.4, the relations in (3.23), (3.24) and (2.7), (2.8) we get for n ≥ 0 βP n = { 1− q−n−1 + q−1 ( 1− bq−n−α )} q2n+α+1, γP n+1 = ( 1− q−n−1 )( 1− bq−n−1−α ) q4n+2α+3, βR n = { 1− q−n−1 + q−1 ( 1− bq−n−α−1 )} q2n+α+2, γR n+1 = ( 1− q−n−1 )( 1− bq−n−2−α ) q4n+2α+5. Thus, for n ≥ 0 γ2n+1 = ( 1− bq−n−1−α ) q2n+α+1, γ2n+2 = ( 1− q−n−1 ) q2n+α+2. Consequently, the symmetrical form u := u(α, b, q) is regular if and only if b 6= 0, b 6= qn+1+α, n ≥ 0. It is positive definite for α ∈ R, q > 1, b < qα+1. Proposition 5. The symmetrical form u is a H√ q-semiclassical form of class one for b 6= 0, b 6= qn+1+α, n ≥ 0, α ∈ R satisfying H√ q(xu) + q−α−1 ( q 1 2 − 1 )−1{ x2 + b− qα+1 } u = 0. Moreover, we have the following identities (u)2n = (−b)n ( b−1qα+1; q ) n , (u)2n+1 = 0, n ≥ 0, (3.25) 〈u, f〉 = K ∫ ∞ −∞ \|x\|2α−2 ln b ln q +1 (−b−1x2; q−1)∞ f(x)dx, (3.26) for f ∈ P, α ∈ R, q > 1, 0 < b < qα+1, with K−1 = ∫ ∞ 0 x α− ln b ln q (−b−1x; q−1)∞ dx is given by (2.4), u = 1 (b−1qα; q−1)∞ ∞∑ k=0 q− 1 2 k(k−1) (q−1; q−1)k (−b−1qα)k δ√ −bq k 2 + δ − √ −bq k 2 2 , (3.27) for α ∈ R, q > 1, b < 0, and u = ( b−1qα+1; q ) ∞ ∞∑ k=0 (b−1qα+1)k (q; q)k δ i √ bq k 2 + δ −i √ bq k 2 2 , (3.28) for α ∈ R, 0 < q < 1, b > qα+1. The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One 15 Proof. First, let us obtain the class of the form; denoting Φ(x) = x, Ψ(x) = ( √ q − 1)−1q−α−1 ( x2 + b− qα+1 ) , we have √ q ( h√qΨ ) (0) + ( H√ qΦ ) (0) = bq−α− 1 2 − 1 √ q − 1 6= 0, 〈u,√qθ0Ψ + θ2 0Φ〉 = 0, for b 6= 0, b 6= qn+1+α, n ≥ 0, α ∈ R. Thus, u is of class one. The identities given in (3.25)–(3.28) are easily obtained from the properties of the generalized q−1-Laguerre U (α)(b, q) form (Table 1, case 1.4) and (2.10)–(2.14). � B. In the case ϕ(x) = x the q-analog of the distributional equation of Pearson type (3.3), (3.4) are Hq ( x2σu ) + 1 √ q + 1 ψ(x)σu = 0, (3.29) Hq ( x2(xσu) ) + q−1 { 1 √ q + 1 ψ(x)− x } (xσu) = 0. (3.30) B1. If ψ(x) = −2( √ q + 1)(αx + 1) the q-analogue of the Bessel form (case 2.2 in Table 2), the form B(α, q), α 6= 1 2(q − 1)−1, α 6= −1 2 [n]q, n ≥ 0 satisfying Hq ( x2B(α, q) ) − 2(αx+ 1)B(α, q) = 0. Thus, comparing with (3.29), (3.30), we get σu = B(α, q), α 6= 1 2 (q − 1)−1, α 6= −1 2 [n]q, n ≥ 0, and xσu = −α−1hq−1B(q−1(α+ 1 2 ), q), α 6= 1 2 (q − 1)−1, α 6= −1 2 [n]q, n ≥ 0. By the recurrence coefficients in case 2.2 of Table 2, the relations in (3.29), (3.30) and (2.7), (2.8) we get for n ≥ 0 βP n = −2qn 2α+ (1 + q−1)[n− 1]q − q−1[2n]q (2α+ [2n− 2]q)(2α+ [2n]q) , γP n+1 = −4q3n [n+ 1]q(2α+ [n− 1]q) (2α+ [2n− 1]q)(2α+ [2n]q)2(2α+ [2n+ 1]q) , βR n = −2qn−1 (2α+ 1)q−1 + (1 + q−1)[n− 1]q − q−1[2n]q ((2α+ 1)q−1 + [2n− 2]q)((2α+ 1)q−1 + [2n]q) , γR n+1 = −4q3n−2 [n+ 1]q((2α+ 1)q−1 + [n− 1]q) ((2α+ 1)q−1+ [2n− 1]q)((2α+ 1)q−1+ [2n]q)2((2α+ 1)q−1+ [2n+ 1]q) . By the relation [k − 1]q = q−1[k]q − q−1, k ≥ 1, (2.9) leads to for n ≥ 0 γ1 = − 1 α , γ2n+2 = 2q2n [n+ 1]q (2α+ [2n]q)(2α+ [2n+ 1]q) , γ2n+3 = −2qn+1 (2α+ [n]q) (2α+ [2n+ 1]q)(2α+ [2n+ 2]q) . (3.31) 16 A. Ghressi and L. Khériji We put α = ν+1 2 , ν 6= 2−q q−1 , ν 6= −[n]q − 1, n ≥ 0 and denote the symmetrical form u by B[ν, q]. From (3.31) the form B[ν, q] is regular if and only if ν 6= 2−q q−1 , ν 6= −[n]q − 1, n ≥ 0. Also, it is quite straightforward to deduce that the symmetrical form B[ν, q] is H√ q-semiclassical of class one for ν 6= 2−q q−1 , ν 6= −[n]q − 1, n ≥ 0 satisfying the q-analog of the distributional equation of Pearson type H√ q ( x3B[ν, q] ) − 2( √ q + 1) ( ν + 1 2 x2 + 1 ) B[ν, q] = 0. Remark 4. The symmetrical form h(2 √ 2)−1B[ν, q], ν 6= 2−q q−1 , ν 6= −[n]q − 1, n ≥ 0 is the q-analogue of the symmetrical form B[ν] [14] (when q → 1 we recover the symmetrical semiclas- sical B[ν], ν 6= −n− 1, n ≥ 0 of class one, see (1.4)). Also, for any parameter α 6= −n− 1, n ≥ 0 the symmetrical form h (2 √ 1+ √ q)−1B[− q−α−1−1 q−1 − 1, q] appears in [33]. B2. If ψ(x) = −(aq)−1( √ q − 1)−1((1 + aq)x − 1) the Alternative q-Charlier A(a, q) form with a 6= 0, a 6= −q−n, n ≥ 0 that satisfies (case 1.5 in Table 1) Hq ( x2A(a, q) ) − (aq)−1(q − 1)−1 ( (1 + aq)x− 1 ) A(a, q) = 0. Thus σu = A(a, q), a 6= 0, a 6= −q−n, n ≥ 0, and xσu = 1 1 + aq A(aq, q), a 6= 0, a 6= −q−n, n ≥ 0. The systems (2.7), (2.8) are for n ≥ 0 βP n = qn 1 + aqn−1 + aqn − aq2n (1 + aq2n−1)(1 + aq2n+1) , γP n+1 = aq3n+1 (1− qn+1)(1 + aqn) (1 + aq2n)(1 + aq2n+1)2(1 + aq2n+2) , βR n = qn 1 + aqn + aqn+1 − aq2n+1 (1 + aq2n)(1 + aq2n+2) , γR n+1 = aq3n+2 (1− qn+1)(1 + aqn+1) (1 + aq2n+1)(1 + aq2n+2)2(1 + aq2n+3) , from which we get for n ≥ 0 γ2n+1 = qn 1 + aqn (1 + aq2n)(1 + aq2n+1) , γ2n+2 = aq2n+1 1− qn+1 (1 + aq2n+1)(1 + aq2n+2) . Consequently, the symmetrical form u = u(a, q) is regular if and only if a 6= 0, a 6= −q−n, n ≥ 0. It is positive definite for 0 < q < 1, a > 0. Also, u is H√ q-semiclassical of class one for a 6= 0, a 6= −q−n, n ≥ 0 satisfying the q-analog of the distributional equation of Pearson type H√ q ( x3u ) − (aq)−1( √ q − 1)−1 ( (1 + aq)x2 − 1 ) u = 0. After some straightforward computations, we get the following representations for the moments and the orthogonality (u)2n = 1 (−aq; q)n , (u)2n+1 = 0, n ≥ 0, 〈u, f〉 = q 1 2 ( ln a ln q + 1 2 )2 (−a−1; q)∞√ 2π ln q−1 ∫ ∞ −∞ \|x\|2 ln a ln q (qx2; q)∞ exp ( −2 ln2 \|x\| ln q−1 ) f(x)dx, The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One 17 for f ∈ P, 0 < q < 1, a > 0, and u = 1 (−aq; q)∞ ∞∑ k=0 akq 1 2 k(k+1) (q; q)k δ −q k 2 + δ q k 2 2 , 0 < q < 1, a > 0. C. In the case ϕ(x) = x−1 the q-analogue of Jacobi form (case 2.3 in Table 2), therefore the q-analog of the distributional equation of Pearson type (3.3), (3.4) become Hq(x(x− 1)σu)− ( (α+ β + 2)x− (β + 1) ) σu = 0, and Hq ( x(x− 1)(xσu) ) − q−1 ( (α+ β + 3)x− (β + 2) ) (xσu) = 0. Consequently, σu = J(α, β, q), (3.32) xσu = β + 1 α+ β + 2 J ( q−1(α+ 1)− 1, q−1(β + 2)− 1, q ) (3.33) with the constraints α+ β 6= 3− 2q q − 1 , α+ β 6= −[n]q − 2, β 6= −[n]q − 1, α+ β + 2− (β + 1)qn + [n]q 6= 0, n ≥ 0. (3.34) By Table 2 and (3.32), (3.33), the systems (2.7), (2.8) give for n ≥ 0 βP n = qn−1 (1 + q)(α+ β + 2 + [n− 1]q)(β + 1 + [n]q)− (β + 1)(α+ β + 2 + [2n]q) (α+ β + 2 + [2n− 2]q)(α+ β + 2 + [2n]q) , γP n+1 = q2n [n+ 1]q(α+ β + 2 + [n− 1]q)([n]q + β + 1)(α+ β + 2− (β + 1)qn + [n]q) (α+ β + 2 + [2n− 1]q)(α+ β + 2 + [2n]q)2(α+ β + 2 + [2n+ 1]q) , βR n = qn−1 (1 + q)(α+ β + 2 + [n]q)(β + 1 + [n+ 1]q)− (β + 2)(α+ β + 2 + [2n+ 1]q) (α+ β + 2 + [2n− 1]q)(α+ β + 2 + [2n+ 1]q) , γR n+1 = q2n+1 [n+1]q(α+β+2+[n]q)([n+ 1]q+β+1)(α+β+2−(β + 2)qn+[n+ 1]q) (α+ β + 2 + [2n]q)(α+ β + 2 + [2n+ 1]q)2(α+ β + 2 + [2n+ 2]q) . Using the above results and the relations [k − 1]q = q−1[k]q − q−1, [k]q = qk−1 + [k − 1]q, k ≥ 1 we deduce from (2.9) for n ≥ 0 γ2n+1 = qn (α+ β + 2 + [n− 1]q)(β + 1 + [n]q) (α+ β + 2 + [2n− 1]q)(α+ β + 2 + [2n]q) , γ2n+2 = qn[n+ 1]q α+ β + 2− (β + 1)qn + [n]q (α+ β + 2 + [2n]q)(α+ β + 2 + [2n+ 1]q) . (3.35) We denote the symmetrical form u by G(α, β, q). From (3.35) the symmetrical form G(α, β, q) is regular if and only if the conditions in (3.34) hold. It is H√ q-semiclassical of class one for α + β 6= 3−2q q−1 , α + β 6= −[n]q − 2, β 6= −[n]q − 1, α + β + 2 − (β + 1)qn + [n]q 6= 0, n ≥ 0, β 6= 1√ q( √ q+1) − 1 satisfying Hq ( x(x2 − 1)G(α, β, q) ) − ( √ q + 1) ( (α+ β + 2)x2 − (β + 1) ) G(α, β, q) = 0. 18 A. Ghressi and L. Khériji Remark 5. The symmetrical form G(α, β, q) is the q-analogue of the symmetrical generalized Gegenbauer G(α, β) form (see (1.3)) which is semiclassical of class one for α 6= −n−1, β 6= −n−1, β 6= −1 2 , α+ β 6= −n− 1, n ≥ 0 [1, 6]. D. In the case ϕ(x) = x− b−1q−1 the little q-Jacobi U(a, b, q) form (case 1.6 in Table 1). The q-analog of the distributional equation of Pearson type in (3.3), (3.4) become Hq ( x ( x− b−1q−1 ) σu ) + ( abq2(q − 1) )−1((1− abq2 ) x+ aq − 1 ) σu = 0, Hq ( x ( x− b−1q−1 ) (xσu) ) + ( abq3(q − 1) )−1((1− abq3 ) x+ aq2 − 1 ) (xσu) = 0. Hence σu = U(a, b, q), (3.36) xσu = 1− aq 1− abq2 U(aq, b, q) (3.37) with the constraints ab 6= 0, a 6= q−n−1, b 6= q−n−1, ab 6= q−n, n ≥ 0. (3.38) By Table 1 and (3.36), (3.37), the systems (2.7), (2.8) lead to for n ≥ 0 βP n = qn (1 + a)(1 + abq2n+1)− a(1 + b)(1 + q)qn (1− abq2n)(1− abq2n+2) , γP n+1 = aq2n+1 (1− qn+1)(1− aqn+1)(1− bqn+1)(1− abqn+1) (1− abq2n+1)(1− abq2n+2)2(1− abq2n+3) , βR n = qn (1 + aq)(1 + abq2n+2)− a(1 + b)(1 + q)qn+1 (1− abq2n+1)(1− abq2n+3) , γR n+1 = aq2n+2 (1− qn+1)(1− aqn+2)(1− bqn+1)(1− abqn+2) (1− abq2n+2)(1− abq2n+3)2(1− abq2n+4) . Using the above results and (2.9) we get for n ≥ 0 γ2n+1 = qn (1− aqn+1)(1− abqn+1) (1− abq2n+1)(1− abq2n+2) , γ2n+2 = aqn+1 (1− qn+1)(1− bqn+1) (1− abq2n+2)(1− abq2n+3) . Therefore, the symmetrical form u = u(a, b, q) is regular if and only if the conditions in (3.38) are satisfied. Further, the form u is positive definite for 0 < q < 1, 0 < a < q−1, b < 1, b 6= 0 or q > 1, a > q−1, b ≥ 1. Moreover, by virtue of (2.6), the form u is H√ q-semiclassical of class one for ab 6= 0, a 6= q−n−1, b 6= q−n−1, ab 6= q−n, n ≥ 0, a 6= q− 1 2 H√ q ( x ( x2 − b−1q−1 ) u ) + ( abq2( √ q − 1) )−1((1− abq2 ) x2 + aq − 1 ) u = 0. Proposition 1 and the well known representations of the little q-Jacobi form (Table 1) allow us to establish the following results (u)2n = (aq; q)n (abq2; q)n , (u)2n+1 = 0, n ≥ 0. For f ∈ P, 0 < q < 1, 0 < a < q−1, b < 1, b 6= 0, u = (aq; q)∞ (abq2; q)∞ ∞∑ k=0 (aq)k(bq; q)k) (q; q)k δ −q k 2 + δ q k 2 2 , The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One 19 and 〈u, f〉 = K ∫ q− 1 2 −q− 1 2 \|x\|2 ln a ln q +1 (qx2; q)∞ (bqx2; q)∞ f(x)dx, with K−1 = ∫ q−1 0 x ln a ln q (qx; q)∞ (bqx; q)∞ dx. For f ∈ P, q > 1, a > q−1, b ≥ 1 u = (a−1q−1; q−1)∞ (a−1b−1q−2; q−1)∞ ∞∑ k=0 (aq)−k(b−1q−1; q−1)k) (q−1; q−1)k δ − √ b−1q− k+1 2 + δ√ b−1q− k+1 2 2 , and 〈u, f〉 = K ∫ b− 1 2 −b− 1 2 \|x\|2 ln a ln q +1 (bx2; q−1)∞ (x2; q−1)∞ f(x)dx, with K−1 = ∫ b−1 0 x ln a ln q (bx; q−1)∞ (x; q−1)∞ dx. E. In the case ϕ(x) = x− µ−1q−1 the q-Charlier-II-form U(µ, q) (case 1.7 in Table 1). From the above assumption (3.3), (3.4) are Hq ( x ( x− µ−1q−1 ) σu ) − (µq(q − 1))−1 ( (µq − 1)x− 1 ) σu = 0, Hq ( x ( x− µ−1q−1 ) (xσu) ) − (µq(q − 1))−1 (( µq2 − 1 ) q−1x− 1 ) (xσu) = 0. Thus σu = U(µ, q), µ 6= 0, µ 6= q−n, n ≥ 0, xσu = 1 µq − 1 hqU(µq, q), µ 6= 0, µ 6= q−n, n ≥ 0. Consequently, the systems (2.7), (2.8) for n ≥ 0 are βP n = qn−1 1− (1 + q)qn + µq2n (1− µq2n−1)(1− µq2n+1) , γP n+1 = −q3n (1− qn+1)(1− µqn) (1− µq2n)(1− µq2n+1)2(1− µq2n+2) , βR n = qn 1− (1 + q)qn + µq2n+1 (1− µq2n)(1− µq2n+2) , γR n+1 = −q3n+2 (1− qn+1)(1− µqn+1) (1− µq2n+1)(1− µq2n+2)2(1− µq2n+3) . On account of (2.9) we have for n ≥ 0 γ2n+1 = −q2n 1− µqn (1− µq2n)(1− µq2n+1) , γ2n+2 = qn 1− qn+1 (1− µq2n+1)(1− µq2n+2) . 20 A. Ghressi and L. Khériji From the last result, the symmetrical form u = u(µ, q) is regular if and only if µ 6= 0, µ 6= q−n, n ≥ 0. Moreover, by virtue of (2.6), it is clear that u is H√ q-semiclassical of class one for µ 6= 0, µ 6= q−n, n ≥ 0 satisfying H√ q ( x ( x2 − µ−1q−1 ) u ) − (µq( √ q − 1))−1 { (µq − 1)x2 − 1 } u = 0. Furthermore, by the same procedure as in D we get (u)2n = (−1)nq 1 2 n(n−1) (µq; q)n , (u)2n+1 = 0, n ≥ 0, u = 1 (µ−1q−1; q−1)∞ ∞∑ k=0 (−µ−1)kq− 1 2 k(k+1) (q−1; q−1)k δ −i √ −µ−1q− k+1 2 + δ i √ −µ−1q− k+1 2 2 , for q > 1, µ < 0. F. In the case ϕ(x) = x + ωq− 3 2 the generalized Stieltjes–Wigert form S(ω, q) (case 1.8 in Table 1). From (3.3), (3.4) it follows Hq ( x ( x+ ωq− 3 2 ) σu ) − (q − 1)−1 ( x+ (ω − 1)q− 3 2 ) σu = 0, Hq ( x ( x+ ωq− 3 2 ) (xσu) ) − (q − 1)−1 ( x+ (ωq − 1)q− 5 2 ) (xσu) = 0. Thus σu = S(ω, q), ω 6= q−n, n ≥ 0, xσu = (1− ω)q− 3 2hq−1S(ωq, q), ω 6= q−n, n ≥ 0. We obtain for n ≥ 0 βP n = { (1 + q)q−n − q − ω } q−n− 3 2 , γP n+1 = ( 1− qn+1 )( 1− ωqn ) q−4n−4, βR n = { (1 + q)q−n − q(1 + ω) } q−n− 5 2 , γR n+1 = ( 1− qn+1 )( 1− ωqn+1 ) q−4n−6. Thus, (2.9) gives for n ≥ 0 γ2n+1 = q−2n− 3 2 ( 1− ωqn ) , γ2n+2 = q−2n− 5 2 ( 1− qn+1 ) . (3.39) We recognize the Brenke type symmetrical orthogonal polynomials [8, 9, 10] Bn = Tn(·;ω, q), n ≥ 0. We denote u = T (w, q). Taking into consideration (3.39), the symmetrical form T (ω, q) is regular if and only if ω 6= q−n, n ≥ 0, and it is positive definite for 0 < q < 1, ω < 1. Furthermore, it is easy to deduce that T (ω, q) is H√ q-semiclassical of class one for ω 6= √ q, ω 6= q−n, n ≥ 0 satisfying the q-analog of the distributional equation of Pearson type H√ q ( x ( x2 + ωq− 3 2 ) T (ω, q) ) − ( √ q − 1)−1 ( x2 + (ω − 1)q− 3 2 ) T (ω, q) = 0. The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One 21 Finally, with Proposition 1 and the properties of the generalized Stieltjes–Wigert Hq-classical form (Table 1, case 1.8) we deduce the following results (T (ω, q))2n = q− 1 2 n(n+2)(ω; q)n, (T (ω, q))2n+1 = 0, n ≥ 0, T (ω, q) = ( ω−1; q−1 ) ∞ ∞∑ k=0 ω−k (q−1; q−1)k δ −i √ ωq− k 2− 3 4 + δ i √ ωq− k 2− 3 4 2 , q > 1, ω > 1, 〈T (ω, q), f〉 = K ∫ ∞ −∞ \|x\|2 ln ω ln q −1 (−q 3 2ω−1x2; q)∞ f(x)dx, f ∈ P, 0 < q < 1, 0 < ω < 1, with K−1 = ∫ ∞ 0 x ln ω ln q −1 (−q 3 2ω−1x; q)∞ dx is given by (2.4), 〈T (0, q), f〉 = √ q 2π ln q−1 ∫ ∞ −∞ \|x\| exp ( −2 ln2 \|x\| ln q−1 ) f(x)dx, 0 < q < 1. 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[32] Medem J.C., Álvarez-Nodarse R., Marcellan F., On the q-polynomials: a distributional study, J. Comput. Appl. Math. 135 (2001), 157–196. [33] Mejri M., q-extension of some symmetrical and semi-classical orthogonal polynomials of class one, Appl. Anal. Discrete Math. 3 (2009), 78–87. [34] Sghaier M., Alaya J., Semiclassical forms of class s = 2: the symmetric case when Φ(0) = 0, Methods Appl. Anal. 13 (2006), 387–410. [35] Shohat J.A., A differential equation for orthogonal polynomials, Duke Math. J. 5 (1939), 401–407. http://arxiv.org/abs/math.CA/0601303 1 Introduction 2 Preliminary and first results 2.1 Preliminary and notations 2.2 Some results about the Hq-semiclassical character 2.3 On quadratic decomposition of a symmetrical regular form 3 Symmetrical Hq-semiclassical orthogonal polynomials of class one References

The Symmetrical Hq-Semiclassical Orthogonal Polynomials of Class One

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