Q-classical orthogonal polynomials: a very classical approach

Electronic Transactions on Numerical Analysis. Volume 9, 1999, pp. 112-127. Copyright 1999, Kent State University. ISSN 1068-9613. ETNA Kent State University [email protected] Q−CLASSICAL ORTHOGONAL POLYNOMIALS: A VERY CLASSICAL APPROACH∗ F. MARCELL´ AN †AND J.C. MEDEM ‡ Abstract. The q−classical orthogonal polynomials defined by Hahn satisfy a Sturm-Liouville type equation in geometric differences. Working with this, we classify the q−classical polynomials in twelve families according to the zeros of the polynomial coefficients of the equation and the behavior concerning to q−1. We determine aq−analogue of the weight function for the twelve families, and we give a representation of its orthogonality relation and its q−integral. Wedescribe this representation in some normal and special cases (indeterminate moment problem and finite orthogonal sequences). Finally, the Sturm-Liouville type equation allows us to establish the correspondence between this classification and the Askey Scheme. Key words. orthogonal q−polynomials, classical polynomials. AMS subject classifications. 33D25. 1. Hahn’s generalization of the classical orthogonal polynomials. The q−classical orthogonal polynomials were introduced by Wolfgang Hahn in connection with the q−derivative [7]: a) They are orthogonal in widespread sense, that is, in the three-term recurrence relation (TTRR) for the monic polynomials xPn=Pn+1 +αnPn+βnPn−1,n≥0,P −1 =0,P 0=1,(1.1) it is required that βn6=0,n≥1or, equivalently, in terms of the corresponding functional, it must be regular, that is, the principal submatrices of the Hankel matrix for the moment sequence are nonsingular. b) Since the classical polynomials are characterized as the only ones whose sequence of derivatives is also orthogonal, Hahn considers the L−derivative and studies the orthogonal polynomials (OPS) whose sequence of L−derivatives is also orthogonal. The L−derivative with parameters qand ωincludes as particular cases the difference operator with step ωand the q−derivative (ϑin the work by Hahn): Lq,ωf(x)=f(qx+ω)−f(x) (q−1)x+ω,L1,ω =4ω, Lq,0=Θ,|q|6=1,Θf(x)=f(qx)−f(x) (q−1)x. (1.2) We get the normal derivative when q→1,ω →0. In this way, Hahn considers the L−classical polynomials as a generalization of the classical polynomials (D−classical polynomials) and discrete classical polynomials (4ω−classical polynomials). Traditionally two OPS are considered equal whenever we can pass from one to another bymeans ofanaffine transformationofthe variable. Theaffinetransformationofthevariable, Aa,bf(x)=f(ax +b), modifies the parameters of the L−derivative. Taking into account ∗Received November 1, 1998. Accepted for publication December 1, 1999. Recommended by R. ´ AlvarezNodarse. This work has been partially supported by the Spanish Direcci´on General de Ense˜nanza Superior (DGES) grant PB-96-0120-C03-01 (F. M.). †Departamento de Matem´aticas. Universidad Carlos III de Madrid. Ave. Universidad 30, 28911, Legan´es, Madrid, Spain. ([email protected]) ‡Departamento de An´alisis Matem´atico. Universidad de Sevilla. Apdo. 1160, 41080, Sevilla, Spain. ([email protected]) 112 ETNA Kent State University [email protected] F. Marcell´an and J.C. Medem 113 the effect of the dilation, Haf(x)=f(ax), and the translation, Tbf(x)=f(x+b),we get: |q|6=1 : T b L q,ω =L q,ω+(q−1)bTb b=ω 1−q =⇒TbL=ΘT b, |q|=1 : H a L q,ω =a−1Lq,a−1ωHa a=ω−1 =⇒HaL=a −1 4 q H a. (1.3) In another way the L−classical polynomials with respect to Lq,w ,|q|6=1could be transformed by means of an appropriate translation in the Θ−classical polynomials (q−classical polynomials). If |q|=1a dilation could transform them into the 4−classical polynomials (discrete classical polynomials); see [5], [11], [12] and references contained therein. The study of the classical and classical discrete polynomials was very complete, so actually it is only necessary to study the q−classical polynomials. Starting from the Sturm-Liouville type equation in geometric differences with polynomial coefficients φand ψ,deg φ≤2and deg ψ=1, from now on denoted q−SL , Hahn obtained the first results for the solutions as q−hypergeometric series. Unfortunately, there was no later publication, where the details were all filled in, according to Tom Koornwinder. Thirty six years later, G. Andrews and R. Askey [1] continued Hahn’s work. Since then, a large literature on classical polynomials from the q−hypergeometric point of view has been generated. So, the q−classical polynomialsare presented as a cascade of q−hypergeometric functions. Starting from two polynomials 4φ3, that are not classical in the sense proposed by Hahn, the rest are obtained by means of special choices and changes of parameters for variables, confluent limits, etc. [9, part 4]. A consequence of this procedure is that there does not exist a general theory for this scheme but a lot of particular cases. Moreover, in this hypergeometric approachis not evident how the manipulationshave an influence on the characteristic elements of each family. A. Nikiforov and V. Uvarov represented another standpoint in the hypergeometricapproach [11], [12]. They developeda theory based on the q−SL equation, but the Nikiforov-Uvarov approach leads in the end to the hypergeometric representation of the OPS. In [2], the authors try to unify both, the q−Askey’s scheme and Nikiforov et al. one. In fact theygive a more general frameworkfor the q−Askey’s scheme based on a q−SL equation. Our approach and classification leads from φand ψto the q−weight functions and to the possible intervals of integration so as to represent the orthogonality relation. The zeros of φand φ?[φ?(x)=q −1 φ(x)+(q −1−1)xψ(x)] give the main information about the orthogonality. Our classification is designed to illustrate how alterations of φand ψ(or φand φ?) have an effect on the orthogonality relation. The class of polynomials defined by Hahn are very varied but not a labyrinth. Our approach follows the standard analytic procedure in the D−classical case. Starting from the Sturm-Liouville equation, φD2Pn+ ψDPn=λnPn, we write it in the self-adjoint form D(φwDPn)=λ n wPn.Thisselfadjoint form, together with the integration by parts and the determination of two different points of the completed real line a, b ∈ R such that (φw)(a)=0=(φw)(b)make it easy to get the integral representation of the orthogonality (λn−λm)Rb aPnPmw=Rb aD(φwDPn)·Pm−Rb aD(φwDPm)·Pn=(1.4) =φwDPn·Pm|b a−Rb aφwDPnDPm−φwDPm·Pn|b a+Rb aφwDPnDPm=0, n6=m=⇒λ n6=λ m=⇒Rb aP n P m w=0. ETNA Kent State University [email protected] 114 q−Classical orthogonal polynomials Finally, to prove Rb aP2 nω6=0,n≥0, we only have to check that wis continuous in [a, b] and nonzero in (a, b). Thus we have to determine the weight function w, characterized as a solution of the Pearson equation D(φw)=ψw [⇐⇒ D w w=ψ− D φ φ]. It is evident that the degree of φand the fact that it has a double zero or simple zeros when the degree is two determines the solutions. In conclusion, the classification of the D−classical orthogonal polynomials is based on these aspects of the polynomial φ. The development of a q−analogue of this procedure, where q−hypergeometric functions are not needed, was started with the contribution by M. Frank [4]. Later, S. H¨acker, [6], applied it to the little q−Jacobi case, and in [10]all the casesfor 0<q<1were considered. Our classical approach to the q−classical polynomialsis presentedas follows. In Section 2, a classification of the q−classical polynomialsin 12families with respectto the q−analogueof the weight function is developed. In Section 3, the determination of the q−weight functions by means of a q−analogue of the Pearson equation is given. In Section 4, the foundations of the orthogonality relationship represented with q−integrals and q−weights and an overview of the determination of the positive definite cases are considered. In Section 5, some cases which yield indeterminate moment problems and finite OPS are analyzed. In Section 6, the equivalences with the Askey Scheme are presented. 2. q−classical polynomials: classification. The q−classical polynomials are orthogonal with respect to linear functionals which satisfy a q−difference equation of first order with polynomial coefficients [10] Θ(φu)=ψu,deg φ≤2,deg ψ=1.(2.1) The operations and action of the operators in the dual space of the polynomials is defined by transposition, except the derivative where there is also a change of sign, i.e., hΘu,x ni= −hu,Θxni. Thus, (2.1), is equivalent to [10] φΘΘ ?P n+ψΘ?P n=λ nP n,n≥1,(2.2) where Θ?is the q−1−derivative operator, (1.2), Θ?f(x)=f(q −1 x)−f(x) (q −1 −1)x. Another formulation equivalent to (2.2) is φ?Θ ? ΘPn+ψΘPn=λ? nPn,φ ? (x)=q −1 φ(x)+(q −1−1)xψ(x),(2.3) This is a well-known fact that has a special significance for us since Θ(φu)=ψu⇐⇒ (2.2) ⇐⇒ (2.3) ⇐⇒ Θ?( φ ?u )=ψu,(2.4) that is, every q−classical functional/OPS is also q−1−classical and vice versa. Maybe this fact has gone unnoticed because when working in an analytical way if 0< q<1we have convergencein many expressionswhereas with q−1>1we have divergence. To see what comes next it is very important to keep (2.4) in mind. In fact we will see the q−classical OPS with a stereoscopic vision as q, q−1−classical. We will refer to everything concerningthe inversebasis as symmetric and we will mark it with ?, for example: ψ?=ψ. Let’s recall the Hahn’s scheme (1.3) ETNA Kent State University [email protected] F. Marcell´an and J.C. Medem 115 L−classical polynomials L:=L q,ω |q|=1 |q|6=1 H a ,a=ω−1T b,b=ω 1−q , , , , , , , , @ @ @ @ @ @ @ @R 4−classical polynomials (discrete classical polynomials) Θ−classical polynomials (q−classical polynomials) When |q|6=1, in order to normalize φ, we only need a dilation and the nonzero constant. The dilation Haacting on the distributional equation of the functional u,Θ(φu)=ψu, with the corresponding MOPS, (Pn), leads us to the normalized equation Θ(e φe u)= e ψ e u,e φ=H a φ, e ψ=aH a ψ, e u=H 1/au,(2.5) and the MOPS corresponding to e u,(e Pn), becomes e Pn=a−nHaPn. The factor callows us to take φmonic. A straightforward consequence is that if the origin is a zero of φ, φ(0) = 0 , the origin will continueto be a zero inthe normalizedpolynomialand those c6=0, φ(c)6=0, will continue also to be a zero distinct of the origin after the dilation. Therefore, in the group of Laguerre and Jacobi polynomials, we will now distinguish among those that have a zero at the origin (0−zero) and those that do not vanish at the origin ( ∅−zero). In general we will distinguish between: ∅−zero families: q−Hermite, ∅−Laguerre, ∅−Jacobi, and 0−zero families: 0−Laguerre, 0−Jacobi, q−Bessel. This is the vision from q. What happens for q−1?Ifφ(x)=bax2+¯ax +˙aand ψ(x)=b bx +¯ b, from (2.3), we get φ?(x)=q −1 φ(x)+(q −1−1)xψ(x)= =(q −1 ba+(q −1−1)b b) |{z } b a? x2+(q −1¯a+(q −1−1)¯ b) |{z } ¯a? x+q−1˙a |{z} ˙a? . (2.6) The immediate consequence is that every q−∅−zero family is a q−1−∅−zero family and vice versa. The same is true for the 0−zero families. Notice that, from (2.6), if ba?=0 ⇐⇒ bb=− b a 1 − q( main singularity),(2.7) the ∅−familiesarethe q−1−Laguerreones, providingthat deg φ?=1, otherwise, if deg φ?= 0,thatis, ¯a ?=0 ⇐⇒ ¯ b=−¯a 1 − q( secondary singularity),(2.8) then they become in a q−1Hermite family. In the 0−families, the framework is different. First, the two singularities cannot appear simultaneously. In fact, ba?=0=¯a ?implies φ?≡0,andsouis not regular. On the other hand, first, the 0−Laguerre cannot have a main singularity, (2.7), since then ba?=q−1·0+(q −1−1)b b=0 =⇒b b=0 =⇒deg ψ<1=⇒uis not regular , ETNA Kent State University [email protected] 116 q−Classical orthogonal polynomials and, second, the q−Bessel cannot have a secondary singularity (2.8) ¯a?=q−1·0+(q −1−1)¯ b=0 =⇒¯ b=0 =⇒ψdividesφ=⇒uis not regular . The following chart shows the situation (double arrow := no singularity, m:= main singularity, s:= secondary singularity) P?P L?L H?HL?L P?P B?B q−view q−1−view q−view q−1−view ∅−families 0−families - - , , , , , , , , @ @ @ @R @ @ @ @R @ @R @ @ @ @R @ @ @ @R A A A A A A A AU - - , , , , @ @ @ @R , , , , , , , ,         ms m s m s Looking at the q−classical polynomials from qand q−1we have 12 different families ∅−Jacobi/?Jacobi q−Bessel/?Jacobi ”/?Laguerre ” /?Laguerre ”/?Hermite 0−Jacobi/?Jacobi ∅−Laguerre/?Jacobi ” /?Laguerre q−Hermite/?Jacobi ” /?Bessel 0−Laguerre/?Jacobi /?Bessel 3. q−classical polynomials: q−weight functions. In this part, it will be justified that the zeros of φand φ?determine the poles and zeros of the q−weight function. The weight function in the D−cases satisfies the equation D(φω)=ψω . For our q−polynomials there is a q−analogue of the Pearson equation Θ?(φw)=qψw , which leads to the q−Sturm-Liouville equation in a self-adjoint form φΘΘ ?P n+ψΘ?P n=λ nP n⇐⇒ ΘH − 1( φw)Θ?Pn=λnwPn. We call waq−weight function, and we get it as the solution of the q−Pearson equation. The equations in qand q−1derivatives are reduced to an equation in qdilations H:=H q, [Hf(x)=f(qx)] Θ?(φw)=qψw ⇐⇒ φw =qHφ?Hw⇐⇒ φ ( x ) w( x )=φ ? (qx)w(qx). We solve these equations by a recurrent procedure ETNA Kent State University [email protected] F. Marcell´an and J.C. Medem 117 w=H n w·qHφ ? φ·H qHφ ? φ·...·H n−1qHφ ? φ |{z } H(n)qHφ? φ=Qn−1 k=0 qφ?(qk+1x) φ(qkx) H2φH2w=H 2 (qHφ ? )H3w ......             1 @ @ @ @ @ @R          1 @ @ @R HφHw=H(qHφ ? )H2w ? H w=H 2 wqHφ ? φH qHφ ? φ φw =qHφ?Hw - ? H w=HwqHφ ? φ Let us see what happens when ntends to infinity. If wis continuous at 0, lim n→∞ Hnw= lim n→∞ w(qnx)=w(0) . In order to deduce limn→∞ H(n)qHφ? φwe need to consider infinite products: (a;q)∞= Q∞ k=0(1 −aqn)and (a, b;q)∞=(a;q) ∞ (b;q) ∞. i) ∅−cases: Since the numerator polynomial and the denominator polynomial have the same nonzero independent term (see e.g. (2.6)), then the infinite product converges to w(x)=w(0)(a?−1 1qx;q)∞(a?−1 2qx;q)∞ (a−1 1x;q)∞(a−1 2x;q)∞ , where a? 1and a? 2are the zeros of φ?and a1,a 2those of φ. For any zero, for example a1, it can be interpreted that deg φ<2=⇒a 1=∞=⇒a −1 1=0 =⇒(a −1 1x;q) ∞=1. The q−weights for the ∅−families are given in table 3. These functions were already known by Hahn ([7], page 30), although he obtained them by another procedure. They are meromorphic functions in the complex plane with zeros in a? iq−n,n≥1and poles in aiq−n,n≥0. ii) 0−cases: If the independent term is zero, several situations appear. α)No q±1−Bessel. This is the simplest case also mentioned by Hahn. If both polynomials have nonzero x−term (0−Jacobi/?Jacobi, 0−Jacobi/?Laguerre, 0−Laguerre/?Jacobi) we eliminate a factor xof the numerator with another of the denominator, and we get a ratio of two polynomials with nonzero independent terms which do not coincide in general. To be able to introduce a factor that corrects this we assume the function wpresents a zero or a pole in the origin introducing the factor |x|α. Then, the q−weights are w(x)=|x| α(a ?−1 1qx;q)∞ (a−1 1x;q)∞ , ETNA Kent State University [email protected] 118 q−Classical orthogonal polynomials TABLE 3.1 The q−weights for the ∅−families ∅−families zeros of φzeros of φ?q−weight function ∅−Jacobi/?Jacobi a? 16=∞6=a ? 2w(x)=(a ?−1 1qx,a?−1 2qx;q)∞ (a−1 1x, a−1 2x;q)∞ /?Laguerre a16=∞6=a 2a ? 16=∞=a ? 2w(x)= (a ?−1 1qx;q)∞ (a−1 1x, a−1 2x;q)∞ /?Hermite a? 1=∞=a? 2w(x)= 1 (a −1 1x, a−1 2x;q)∞ ∅−Laguerre a16=∞=a2w(x)=(a ?−1 1qx,a?−1 2qx;q)∞ (a−1 1x;q)∞ a? 16=∞6=a ? 2 q−Hermite a1=∞=a2w(x)=(a ?−1 1qx,a?−1 2qx;q)∞ where once again deg φ<2implies a1=∞. β)q±1−Bessel.The(α)−procedure can not be applied to the q−Bessel and q−1−Bessel: (β1) When the degree is different (q−Bessel/?Laguerre and 0−Laguerre/?Bessel) we can use the function h:h(x)=√ x logqx−1. This function satisfies Hh(x)=xh(x). In fact H¨acker [6] uses it to solve the q−Bessel/?Laguerre case. The following generalization of h,h(β)(we have not found any references to it in the literature) satisfies Hh(β)(x)=x β h(x),h (β) = p x logqxβ−β, and we have used h(−1) to solve the 0−Laguerre/?Bessel case. In general the function hor its generalization can be used when the degrees of the polynomials are different. Hahn uses hin the 0−Jacobi/?Laguerre case to prove that it corresponds to an indeterminate moment problem (generalizing the Stieltjes-Wigert polynomials). Notice that it was the only result developed with some detail in [7], but a mistake appears. It was corrected in a later article [8]. (β2) Finally, for the case when both polynomials have the same degree (q−Bessel/?Jacobi and 0−Jacobi/?Bessel), the iterative solution using Hleads to divergent expressions. So, we try to solve them using H−1. Thus we get w(x)=|x| α1 (a ? 1 /x;q)∞or w(x)=|x| α (a 1 q/x;q)∞. ETNA Kent State University [email protected] F. Marcell´an and J.C. Medem 119 TABLE 3.2 The q−weights for the 0−families 0−families zeros of φzeros of φ?q−weight function q−Bessel/?Jacobi a? 16={0 ∞α? 2=0 w(x)=|x| α1 (a ? 1 /x;q)∞ (b) a1=0,a 2=0 / ? Laguerre a? 1=∞,a ? 2=0 w(x)=|x| α √ x logqx−1(c) 0−Jacobi/?Jacobi a? 16={0 ∞,a ? 2=0w(x)=|x| α(a ?−1 1qx;q)∞ (a−1 1x;q)∞ (a) /?Laguerre a16={0 ∞,a 2=0 a ? 1=∞,a ? 2=0 w(x)=|x| α1 (a −1 1x;q) ∞ ,(a) / ? Bessel a? 1=0,a ? 2=0 w(x)=|x| α (a 1 q/x;q)∞(b) 0−Laguerre/?Jacobi a? 16=0,a ? 2=0 w(x)=|x| α (a ?−1 1qx;q)∞(a) a1=∞,a 2=0 / ? Bessel a? 1=0,a ? 2=0 w(x)=|x| α p x logq1 x+1 (d) We have not foundany reference concerningthese functionsin the literature. In the first case, fixing ba=1applying the standard normalization (non zero factor and dilation) over the distributional equation, the only free parameter is ¯ b. Choosing it so that ¯ b=2q 2−αthen [10] ω(x)=|x| α1 (a ? 1/x;q)∞=|x|αeqa? 1 x=|x|αeq[−(1 −q)2/x], and limq→1−ω(x)=|x| αexp(−2/x)is the Bessel weight function. The q−weight functions for the 0−families are shown in table 3 (a)α=−2+Log q ¯a ¯a ?,(b)α=−3+Log q b a b a ?,(c)α=−2+Log q b a ¯a ?,(d)α=3+Log q b a ? ¯a φ= b ax2+¯ax +˙a, ψ= b bx +¯ b, b a ? =q −1 b a 4. q−integral representation of the positive definite cases. The q−integral is a Riemann sum on an infinite partition {aqn,n≥0}, Ra>0 0fd q:= P∞ n=0 f(aqn)(aqn−aqn+1)=(1−q)aP∞ n=0 f(aqn)qn, R0 a<0fd q:= P∞ n=0 f(aqn)(aqn+1 −aqn)=−(1 −q)aP∞ n=0 f(aqn)qn, ETNA Kent State University [email protected] 120 q−Classical orthogonal polynomials defined in such a way that we can apply the q−analogue of the Barrow rule, Rb aΘFd q=F(b)−F(a). This allows us to get the following integration by parts rules Rb afΘgd q=H −1 f·g| b a−qRb agΘ ? fd q,Rb afΘgd q=fg|b a−Rb aHgΘfd q.(4.1) On the other hand it can be generalized to unbounded intervals and to unbounded functions. The Riemann-Stieltjes discrete integrals related with the q−classical polynomialscan be represented as q−integrals. For example, for the 0−Jacobi case (little q−Jacobi) we have P∞ k=0 (bq;q)k (q;q)k(aq)kpm(qk)pn(qk)=KR1 0x α(qx;q)∞ (qβ+1x;q)∞pm(x)pn(x)dqx, a=q α,b=q β, w(x)=x α(qx;q)∞ (qβ+1x;q)∞(= xα[1 −qx]β,in the Hahn notation). Notice that the previous polynomials correspond to a positive definite case for −1<α.For −1<α<0the q−integral converges. The positive definite cases are deduced from the TTRR, (1.1), when βn>0,n≥1.If φ(x)=bax2+¯ax +˙a,ψ(x)=b bx +¯ b,andH n φ(x)=φ(q n x)then βn+1 =− qn[n+1][n−1] b a+ b b [2n−1] b a+ b b[2n+1] b a+ b b·Hnφ−[n]¯a+¯ b [2n] b a+ b b,n≥0.(4.2) This representation of βnin terms of the coefficients of φand ψwas obtained by N. Smaili [13] and S. H¨acker [6] using different techniques. The determination of the positive definite cases has been done case by case for any real value of q,|q|6=1,byH¨acker. A more global vision of the used procedures and, mainly, the positive definite cases which have not been considered by H¨acker can be found in [10]. In all positive definite cases it is possible to represent the orthogonality relation using the q−integral and the q−weight function. Thus, we have a self-adjoint form of the q−Sturm-Liouville equation, q−integration by parts. We only need two points a, b ∈ R ,a6=b, zeros of certain functions, so that Rb aPnPmwdq=0,n6=m, see (1.4). If n6=m,thenλ n6=λ mand (λn−λm)Zb a wPnPmdq=Zb a (wλnPn)Pmdq−Zb a (wλmPm)Pndq= =Zb a ΘH−1(φw)Θ?Pn |{z } g1 Pm |{z} f1 dq−Zb a ΘH−1(φw)Θ?Pn |{z } g2 Pn |{z} f2 dq= =H −1 (φw)Θ?Pn·Pm|b a−Zb a HH−1(φw)Θ?PnΘPmdq− −H−1(φw)Θ?Pm·Pn|b a |{z } H−1(φw)(a)=0=H−1(φw)(b) +Zb a HH−1(φw)Θ?PmΘPn |{z } φwΘPmΘPn dq=0. ETNA Kent State University [email protected] F. Marcell´an and J.C. Medem 127 SCHEME OF BASIC HYPERGEOMETRIC ORTHOGONAL POLYNOMIALS (4) q−Racah (3) q−Hahn ∅  q−Jacobi q−1Jacobi Dual q−Hahn ∅ f  q−Jacobi q−1Jacobi Big q−Jacobi (2) q−Meixner ∅  q−Jacobi q−1Laguerre Quantum q−Krawtchouk ∅ f  q−Jacobi q−1Laguerre q−Krawtchouk 0 f  q−Jacobi q−1Laguerre Affine q−Krawtchouk ∅ f  q−Laguerre q−1Jacobi Dual q−Krawtchouk (1) Alternative q−Charlier 0  q−Bessel q−1Laguerre q−Charlier 0  q−Jacobi q−1Laguerre Al-Salam Carlitz I ∅  q−Hermite q−1Jacobi Al-Salam Carlitz II ∅  q−Jacobi q−1Hermite (0) Discrete q−Hermite I ∅  particular case ASC I Discrete q−Hermite II ∅  particular case ASC II