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386 Probl. Anal. Issues Anal. Vol. 12 (30), No 3, 2023, pp. 86-104
DOI: 10.15393/j3.art.2023.13333
UDC 517.587, 517.538.3
Y. Habbachi
SECOND STRUCTURE RELATION FOR THE DUNKL-CLASSICAL ORTHOGONAL POLYNOMIALS
Abstract. In this paper, we characterize the Dunkl-classical orthogonal polynomials by a second structure relation. Key words: orthogonal polynomials, Dunkl-classical polynomials, regular forms, second structure relation
2020 Mathematical Subject Classification: 33C45, 42C05
1. Introduction and Preliminary Results. Classical orthogonal polynomials (Hermite, Laguerre, Bessel, and Jacobi) are characterized by several properties: they satisfy Hahn's property (that the sequence of monic derivatives of the polynomial is again orthogonal (see [2], [11], [20], [22]); they are characterized as the polynomial eigenfunctions of a second-order homogeneous linear differential (or difference) hypergeometric operator with polynomial coefficients [6], [21], [22]; their corresponding linear forms satisfy a distribution equation of Pearson type (see [15], [19], [21]); they satisfy a first structure relation (the Al-Salam and Chihara property [2]) and can be characterized by the so-called Rodrigues formula (see, for instance, [11], [13]).
Another characterization was established by J. L. Geronimus in [15]; in particular, he proved that a classical sequence of monic orthogonal polynomials {Pn(x)}n^o can be characterized by the fact that Pn(x) = Qn(x) + a,nQn-i(x) + bnQn-2(z), where Qn(x) = n+iP'n+i(x). This is the so called second structure relation for classical orthogonal polynomials (see also [20], [21], [23]).
In the recent years, many authors (see [7], [8], [9], [10], [17], [24]) have started to study Dunkl-classical orthogonal polynomials, as analogues of the Hahn definition of ^-classical orthogonal polynomials [18]. Symmetric case was studied for the first time by Y. Ben Cheikh and his coworker [4]; in particular, they proved that the only symmetric Dunkl-classical orthogonal polynomials are the generalized Hermite polynomials and the
© Petrozavodsk State University, 2023
generalized Gegenbauer polynomials. Later on, M. Sghaier [24] characterized the symmetric Dunkl-classical forms by a distributional equation of the Pearson type and he showed that the corresponding polynomials can be characterized by a second-order differential-difference equation in the space of polynomials. Another characterization called the first structure relation was established by L. Kheriji et al [5].
Non-symmetric Dunkl-classical orthogonal polynomials have been studied in [7], [8], [9], [24]. In particular in [9] the authors showed that the unique non-symmetric Dunkl-classical linear form for ^ = 0 and ^ > 1 is, up to a dilation, the perturbed generalized Gegenbauer linear form
*-1+2ir+ 2«-1)-1 g — ^
where n + a = 0, 2^ + 2a + 2n + 1 = 0, n ^ 0 and Q(a,y — 1) is the generalized Gegenbauer form [1], [3].
The aim of this contribution is to give a new characterization of Dunkl-classical orthogonal polynomials.
We begin by reviewing some preliminary results needed for the sequel. Let V be the vector space of polynomials with coefficients in C and let V' be its dual. The action of u E V' on f eV is denoted by (u, f). In particular, we denote by (u)n = (u,xn), n ^ 0, the moments of u.
Let us define the following operations on V' [22]:
The left-multiplication of a linear form by a polynomial
(9uJ) = (u,gf), f,g EV ,u EV'. The dilation of a linear form
(hau, f) = (u, haf), f EV ,u EV', a E C \{0},
where
haf (x) = f (ax), f EV, a E C \{0}. The derivative of a linear form u is the linear form Du, such that
(Du,f) = — (u,f), f EV ,u EV'.
Let {Pn}n^o be a sequence of monic polynomials with deg Pn = n,n ^ 0, and let {un}n^o be its dual sequence, un eV' , defined by (un, Pm) = 5n,m, n, m ^ 0.
The form u is called regular if there exists a sequence of polynomials [Pn}n^o, such that
(u, PnPm) = rn8n,m, n,m ^ 0, rn = 0, n ^ 0.
The sequence {Pn}n^o is then called orthogonal with respect to u. In this case, we have
un = r—1PnUo, n ^ 0. (1)
Let us recall the following result [20]:
Lemma 1. Let {Pn}n^o be a monic orthogonal polynomial sequence (MOPS, in short) with respect to u and let {un}n^o be its dual sequence. If v is an element of V', then it can be expressed as
<x
v = ^2 anun,
n=0
where
an = (v, Pn), n = 0,1, 2 .... Moreover, if v satisfies (v,Pn) = 0 for n ^ m, then
m— 1
v = V^ anun.
n=0
According to the previous lemma, we have u = Xu0, where (u)0 = X = 0. In what follows, all regular linear forms u will be taken normalized, i.e., (u)0 = 1. Then u = u0.
According to Favard's theorem, a MOPS [Pn}n^o is characterized by the following three-term recurrence relation [11]:
Po (x) = 1,Pi(x)= x - p0,
P'n+2 (x) = (X - ftn+l)Pn+l(x) - ln+lPn(x), n ^ 0,
where
p™ = ~i—G C; = ~i—G C \^^ n ^ 0 (3) {Uo ,P.'2) {uo,P2)
A form u is said to be symmetric if and only if (u)2n+l = 0,n ^ 0, or, equivalently, in (2), = 0, n ^ 0.
From (2), we have
P2(x) = x2 - % + Pi)x + AA - 11. (4)
As a consequence of the orthogonality of [Pn}n^0 with respect to u0, we have
M2 = l320 + 71. (5)
Let us introduce the Dunkl operator [14]:
T,(f ) = f + 2»H-if, (H-if )(x) = 1 {x) (-x), f eVe C.
By transposition, we define the operator T^ from V' to V' as follows:
{T,u,f) = -{u,T, f), f eV ,u eV. In particular, this yields
(T^u)n = -^n(u)n-i, n ^ 0, with the convention (u)-1 = 0, where
1 — (—iy
^n = n + 2^[n], [n] =-2-, n ^ 0.
Note that T0 is the derivative operator D.
Using the previous definitions, we get the following formula [7]:
T,(fu) = fT^u + (TJ )u + 2^(H-if )(h-m — u), f EV, u EV'. (6)
In particular, if u is a symmetric linear form, then (6) becomes
T,(fu) = fT,u + (TJ )u, f EV, u EV'. (7)
Now, consider an MOPS {Pn}n^0 and let
(x,y) = ~^—(T^Pn+i)(x), ^ = —n — 1 ,n ^ 0.
^n+1 2
Denoting by {i}n}n^ 0 the dual sequence of {Pn\-, ^)}n^0, the following result is proved in [24]:
= — Vn+iUn+i, n ^ 0. (8)
Definition 1. [5], [7], [24] An MOPS {Pn}n>0 is called Dunkl-classical or Tß-classical if Pn\^ß) is also an MOPS. In this case, the form u0 is called either a Dunkl-classical or a Tß-classical form.
Any symmetric Dunkl-classical polynomial sequence {Pn}n^o can be characterized taking into account its orthogonality as well as one of the four difference equations:
• Second-order differential equation of the Bochner type [24]
$(x)(T2ßPn+i)(x) - x)(TßPn+1)(x) + XnPn+i(x) = 0, n ^ 0. (9)
• First structure relation [5]
n+t
$(x)P^](x,ß) = ^ \n,kPk(x), n ^ 0, 0 ^ t = deg$ ^ 2.
k=n
= 0, n ^ 0.
• Rodrigues-type formula [25]
Pnuo = $nT™($nuo), n > 0. (11)
• Its canonical form u0 satisfies the Pearson differential equation [24]
Tß ($mo) + ^O = 0, _ $
2
V(0) - ™ßn = 0,n > 0, (12)
where $ is a monic polynomial of degree t, 0 ^ t ^ 2, ^ is a first degree polynomial, and [\n,k}n>o,n^n+t and {idn]ri^Q are sequences of complex numbers, such that idn = 0,n ^ 0.
Remark 1. Under conditions of relations (9)—(12), the linear form u0\ corresponding to {Pi^}^, is given by:
uH1 = (1 + 2M)-1j-1K $uq, (13)
where K is a non-zero constant chosen to make $ monic, and ^ is given by
V(x) = K-1(1 + 2p)2Pi(x). (14)
On the other hand, some characterizations of non-symmetric Dunkl-classical orthogonal polynomials have been provided (see [7], [8], [10], [16], [17]).
2. Main Result. In this section, we prove the characterization theorem in both situations.
2.1. The symmetric case.
Theorem 1. For any symmetric MOPS {Pn}n^0, the following statements are equivalent
(a) The sequence {Pn}n^0 is Dunkl-classical.
(b) There exist an integer t, 0 ^ t ^ 2, and a sequence of complex numbers {\n>kt,n-t^k^n, such that
n
Pn(x)= ^ \n, kp£](x,»), n > t, (15)
k=n-t
Xn,n = 1, n ^ t, (16)
1 + ^12 - ^n = 0, n ^ 0 when \2fl = 0. (17)
A2,0
Proof. (a) =^ (b) Assume that {Pn}n^0 is Dunkl-classical; then there exist polynomials $ (monic), deg$ = t, 0 ^ t ^ 2, and deg^ = 1, such that the canonical regular form u0 satisfies (12). Moreover, since Pn is a polynomial of degree n, then there exists a sequence of complex numbers {Xn,k}n^t, 0 ^ k ^ n, such that
n
Pn(x) = ^ \n,k Plk](x,/j), n ^ t. (18)
k=0
By comparing the degrees in the previous equation, we get
K,n = 1, n ^ t.
Therefore, (18) becomes
n—l
Pn(x) = pn](x,^ + ^ Xn,kPk1](x,ri, n ^ t. (19)
k=0
Multiplying the last equation by Pm(-,^), 0 ^ m ^ n — 1, n ^ 1, and applying $u0, we get
{$Uo,Pl1](-,^)Pn) =
n—l
($«0 ,PH](-,^)Pni](-,^)) + ($U0,P£](-,riPk1](-,ri)
k=0
= Krn($U0, (P£](-,V))2), n ^ 1.
Hence,
Xnm = (U0, ',^))pn) , 0 ^ ^ — 1,n ^ 1. (20) Since deg($p£](-,^))
= m + t, the orthogonality of {Pn}n^0 leads to (uo, ($ P£](-,v))Pn) = 0, 0 ^m + t ^n — 1,n ^ 1. So, we have
Xn,m = 0, 0 ^ m ^ n — t — 1, n ^ 1. Consequently, (19) becomes
nl
Pn(x) = P[n](x^)+ K,kPk1](x,v), n ^ t. (21)
k=n—t
It remains to prove (17). Assume that A2,0 = 0. From (21), where n = 2, we have
P2(X) = P[2](X^) + \%0P0](X,»).
Therefore,
(u0\P2) = \2,0.
But from (8) and the fact that $ is monic, we have
(u[0],P2) = (1 + 2ii) — 11—1K T'2 = (1 + 2ii) — 112K.
Then
K = (1 + 2,)X2,o
2
Substitution of (22) in (14) gives
A2,0
Therefore,
^2,0
So, condition (17) becomes an immediate consequence of the second equality in (12). Thus the desired result (15)-(17).
(b) =^ (a) Assume that there exist an integer t, 0 ^ t ^ 2, and a sequence of complex numbers {Xn,k}n^t, n—t^k^n, such that (15), (16), and (17) hold.
Let {Pn}n^0 and ^)}n^0 be sequences of monic polynomials
with {un}n^0 and {u£]}n^0 be their respective dual sequences. Using (15) and (16) for n ^ t + 1, we have
n— 1
(ul1],Pn) = (u[0],Pni](-,V)) + £ ^n,k (vg],Pj?] (-,»)) =0.
k=n—t
Thus, according to Lemma 1, there exist complex numbers ai, i E {0,... ,t}, such that
u0] = ®iUi, 0 ^ t ^ 2.
i=0
Or, equivalently,
w01] = &0u0 + a1 u1 + a2u2. (23)
On account of (1), the previous equation becomes
«01] = («0 + a1r—1P1 + a2r—1P2)u0. Therefore, there exists a polynomial $, deg$ ^ 2, such that
u0] = k$U0, (24)
where
k$ = a0 + a,1r—1P1 + a2r—1P2, (25)
and the non-zero constant k is chosen to make $ monic.
Moreover, $ is an even polynomial. Indeed, since P1 (x) = P11 (x, = x, we have
0 = (41] ,Pi](-,V)) = k[a0{u0,Pi )+alr—1{u0,Pl)+a2r—1 {u0,PiP2)) = kav Hence, a1 = 0.
Thus, taking into account (25) and the fact that P2(x) = x2 — 71, we can easily see that $ is even.
On the other hand, putting n = 0 in (8), we obtain
= — (1 + 2/i)ui.
Substitution of (24) in the previous equation gives (12), with
^(x) = k—1-f—1 (1 + 2/i)Pl (x).
To complete the proof, we will show that the second equality in (12) is fulfilled. Indeed, from (23) we have
a2 = (u[0],P2). But from (15) and (16), where n = 2, we have
P2(x) = p21](x,iJ) + \20P01](x,ii).
Thus,
®2 = A2,0.
On the other hand, taking into account (23) and the fact that u0 and «0^ are normalized, we get
a0 = 1.
Therefore, (25) becomes
k$(x) = 1 + \2,0r—1P2 (x). (26)
So, we distinguish two cases: A2,0 = 0 and A2,0 = 0.
The first case: \2,0 = 0. In this case, deg$ = 0; then $''(0) = 0 and, since $ is monic, we get k = 1. Therefore,
$''(0)
*'(0)--^ Vn = 0) = l—\1 + 2fi) = 0,n ^ 0.
The second case: A20 = 0. In this case, deg$ = 2. But, since $ is $''(0)
monic, —-— = 1. Furthermore, identification of degrees in (26) gives
k = \2,0r2
l
Therefore,
¥'(0) - ™ ^ = (A±M l2 - ^ = 0, n ^ 0 (by (17) ).
A2,0
So, according to relation (12), the sequence {Pn}n^0 is Dunkl-classical. □
In the sequel, using the previous theorem, we will determine the second structure relation for the generalized Hermite polynomials and the generalized Gegenbauer polynomials.
Put $(x) = x2 + $(0) and ¥(x) = ¥'(0)x and let {Pn}n^o be a symmetric Dunkl-classical MOPS, such that its associated regular form u0 satisfies (12). So, from (15)-(16) we have:
Pn(x) = P[n\x, ¡j) + Xn,n-iPn--i(x, v) + \n,n-2Pn--2(x> V), n ^ t. (27) Since the sequences {Pn}n and {p[\-, n^0 are symmetric,
\nn-l = 0, n ^ t. (28) The coefficient Xn,n-2 is given by
¥'(0) Vn-2
with the convention X0,n-2 = 0. Indeed, from (20), we have
{U0, ($P[aU-,V))Pn)
Kn-2 = —, s 2 -In, n ^ t, (29)
^n,n —2 = -¡Ti-, n ^ t.
, ($«0, (pn-U-^))2)
($^0, \*:>n—2(',ß)2
Writing
[1] $ (0) n
$(x)P„i—2(x, y) = —-— xn + lower degree terms.
On the one hand, from the orthogonality of {Pn}n^0 with respect to u0, we have
(U0, ($Pn—2(-,v))Pn) = (u0,xnPn) = ^(U0,P2), n > t.
On the other hand, from (7) and the fact that $«0 is symmetric, we have
($«o, ( Pl'U-^))2) =--— ),Pn-1) =
ßn-1
1 (Tß( pn-U-,ß))$Uo + pn--2 (;ß)Tß(^Uo),Pn-l).
^n—l
Taking into account (12), we get ($«0, ( P[aU;V))2) = — (PHU-,^0 —T,(P[—(-,ll))$U0,Pn—l).
I^n—l
Hence, the orthogonality of {Pn}n^0 with respect to u0 gives
—'(0) _ (o)
($U0, ( pn-U-iß))2) = —-2-^nzl(Uo,Xn-1Pn-l)
ßn-1
<s>''(o)
= n0) - ' ßn-2 n
ßn-1
Consequently, from the second equality of (3) we deduce (29). Substitution of (28) and (29) in (27) gives
Pn(x) = Pn](x,ß) + 2 ß-1-^P^x^), n ^ t. (30)
-(0) - ^2°>ßn-2
Corollary.
1) The generalized Hermite polynomial {H^^n^ is characterized by the following second structure relation:
H(n»)(x) = (H(rr)f\x), n > 0. (31)
2) The generalized Gegenbauer polynomial {S^'^ 1}n^o is characterized by the following second structure relation:
S(n'"-1 )(x) = (S<n'ß-1 ))[1](x)-
— --- (sna'ï-2))[1l(x), n > 2.
(2n + 2 a + 2ß — 1)(2n + 2a + 2ß + 1)y n-2 ' yh '
(32)
Proof. 1) The sequence of generalized Hermite polynomials {H<i^")}n^o satisfies (2) with (see [11]):
ßn = 0,1n+1 = ^, n > 0, (33)
where the regularity condition is
1
H = —n — -, n ^ 0.
This sequence is Dunkl-classical and its associated form satisfies (12) with (see [7])
<£(x) = 1, V(x) = 2x. (34)
So, using (33) and (34) the proof of (31) is an immediate consequence of (30).
2) The sequence of generalized Gegenbauer polynomials {S^''1 2 satisfies (2) with (see [11]):
(n + 1 + 8n)(n + 1 + 2a + 8n)
ßn = 0, Jn+1
(2n + 2a + 2fi + l)(2n + 2a + 2/i + 3) (35) 8n = ^(1 + (-1)n), n ^ 0,
where the regularity conditions are
a = -n, [ = -n, a + [ = -n, n ^ 1.
This sequence is Dunkl-classical and its associated form Q(a,^ — 2) satisfies (12) with (see [7])
$(x) = x2 - 1, ¥(x) = —2(a + 1)x. (36)
Then, using (35) and (36), equation (32) is deduced from (30). □ Remark 2.
1) From equation (31), we can recover again the following structure relation established by T. S. Chihara [12]:
xDH^(x) = -»{1 + (-1)n)U^l(x)+
+ (n + 1 + ^(1 +(-1)n)^xH([i)(x), n ^ 0. (37)
Indeed, using the definition ofT^ and the fact {H^}^ is symmetric, equation (31) becomes
nj(P) (T)
DH^x) + ii(1 - (-1)n+l)= ^n+iH{n)(x), n > 0.
x
Therefore, multiplication of the last equation by x gives (37).
2) The relation (32) can be written of the following form:
2 )(x) = S(n+ ^— 2 V) —
^n^n—1 -stf-2V), n > 2.
(2n + 2 a + 2/i — 1)(2n + 2a + 2fi + 1) This result is deduced from (32) and the fact that iiSn+1 2 = Hn+iSn 2 (see [4]). 2.2. The non-symmetric case. In this subsection, we will present a second structure relation for non-symmetric Dunkl-classical polynomial sequences. But first, let us recall the following result.
Theorem 2. [7] Let {Pn}n^0 be a MPS orthogonal with respect to a linear form u0. For ^ = 0 and ^ = -, the following statements are equivalent:
(a) The sequence {Pn}n^0 is Dunkl-classical.
(b) There exist K G C* and three polynomials $ (monic), B and ^ with deg$ ^ 2, degB ^ 3, and deg^ = 1, such that
K$ (0) 2 K B (0)
v(°) + 2{ 1—(VM — n) + 3{1 — ¡¿2)mm — n) = 0, (38)
and
/ \ 1 _ 4u2
T^$U0 — 2nh—i($U0)) + KP ^«0 = 0, (39)
with
x$(x)u0 = h—1(B(x)u0). (40)
The authors [9] used Theorem 2 to classify all Dunkl-classical linear forms. In particular, they proved that the unique non-symmetric Dunkl-classical
linear form for ^ = 0 and ^ = - is, up a dilation, the perturbed generalized Gegenbauer linear form u0, satisfying
L^(x2 — 1)u^ — 1 + 2fi (x — @0)U0 = 0
TJ (x2- 1W--(x-8())un = 0, (41)
(x — 1)u0 = h—1((x — 1)u0), (42)
with the regularity conditions:
¡30 G {0,1}, 1 + 2/1 + 00(n — 2/i[n]) = 0, n ^ 0. (43)
Remark 3. If {Pn}n^0 is a non-symmetric Dunkl-classical MOPS, then {Pn^ (■, is orthogonal with respect to [10]
= 7T^(x2 - 1)uo, Po / {0,1}. (44)
Po — 1
Theorem 3. Let {Pn(x)}n>o be a MOPS fulfilling (2) with (43). Assume that its corresponding linear form u0 satisfies (42). The following statements are equivalent:
(a) The sequence {Pn}n^0 is Dunkl-classical.
(b) {Pn}n^0 satisfies the second structure relation
Pn(x) = P^](x,ß) + \n,n-iP}1_l(x, ß) + \n,n-2Pn-2(x,ß), n ^ 0,
(45)
where
. = juo, (x2 - l)P^[l(-)ß)Pn)
/^n,n— 1 ril ,
, U, (x2 - 1)(P^i(;ß))2) . juo, (x2 - 1)Pl_2(-,ß)Pn) . 0
K,n_2 = -_-, n ^ 0.
, juo, (x2 - 1)(P^ß))2)
(46)
Proof. (a) (b) Suppose that {Pn}n^0 is Dunkl-classical; then its canonical regular form u0 satisfies (41)-(42). Moreover, since Pn is a polynomial of degree n, there exists a sequence of complex numbers {\n,k}n^0, 0 ^ k ^ n, such that
Pn(x) = Y,X^Pl1](x,ß), n > 0. (47)
j [1]/
An,kP k=0
By comparing the degrees in the previous equation, we get
An,n = 1, n ^ 0.
Therefore, (47) becomes
n— 1
Pn(x) = P[2\x,ß) + Y,KkP[k\x,ß), n > 0. (48)
k=0
It is clear that (45) holds for n = 0, where A0,-1 = A0,_2 = 0. For n ^ 1, multiplying the previous equation by Pm(■,y), 0 ^ m ^ n — 1, n ^ 1, and applying (x2 — 1)u0, we get
((x2 — 1)u0,Pj1](^,^)Pn) =
--((x2 — 1)u0,p£](;v)pni](;v)) + ((X2 — 1)u0,PH](;^)Pk1](;^))
k=0
= K,m((x2 — 1)U0, (PH](^,^))2), n > 1.
n— l
[l]
Hence,
(U0, ((x2 — 1)Pi1](;^))Pn) 0 < < _ > _ (49) Anm =-¡-j-, 0 ^m ^n — 1, n ^ 1. (49)
, {(x2 — 1)U0, (Pi\;v))2) Since deg((x2 — 1)P^^,»))
= m + 2, the orthogonality of {Pn}n^0 leads to (U0, ((x2 — 1)P^t,p))Pn) = 0, 0 ^m + 2 ^n — 1. So, we have
An,m = 0, 0 ^ m ^ n — 3, n ^ 3. Consequently, (48) becomes
Pn(x) = P[](x+ An,n—lPjn—i(x,^) + \n,n—2Pn-2(X,v), n ^ 0
with the equalities in (46) are obtained by (49). Therefore, (45) holds.
(b) (a) Let {Pn}n^0 and , ^)}n^0 be sequences of monic
polynomials with {un}n^0 and {uinbe their respective dual sequences. Suppose that {Pn}n^0 satisfies (45). From (45) for n ^ 3, we have
(u[0],Pn) =
= (v0],pW(;ri) + \nn—i(ul0\PnU^,V) + Kn—2(ul0\Pn—2(^)) = 0.
Thus, according to Lemma 1, there exist complex numbers ak, 0 ^ k ^ 2, such that
«0^ = a0u0 + a1u1 + a2u2. (50)
On account of (1), the previous equation becomes
«01 = (a0 + air—lPi + a2r—1P2)u0. (51)
Taking into account (50) and the fact that u0 and «0^ are normalized, we get
a0 = 1. (52)
According to (50), we have
«1 = (u[0],Pi}.
Making n = 1 in (45), we get
P1(x) = P11](x,y) + A1[P01](x,y).
Therefore,
a1 = A1,o.
Using the first equality in (46) for n = 1, we have
A Po + P1
A1,0 =7-s-7 r1■
( u ) 2 — 1
Then
Po +P1
«1 = —r-- r-1. (53)
(Uo)2 — 1
From (50), we have
a2 = (uU[l\P2). But from (45), where n = 2, we have
P2(x) = P21](x,y) + A2AP11](x,y) + A2,oP0?](x,ij).
Thus,
a2 = A2,o■
On the other hand, from the second equality in (46) for n = 2, we deduce
2
a2 = A2,o = 7—:-7. (54)
(uo)2 — 1
Substitution of (52), (53) and (54) in (51) gives
[1] A , po + p1 P , 1 p\ uo + P1+ M1—T P2)uo■
Using (4)-(5) and the fact that P1(x) = x — P0, the last equation becomes
u01] = p2 ^ — 1 (x2 — 1)uo. (55)
But from (42), it is easy to see that
(Uq)2 = (Uo)l.
Since (uo)1 = ß0, from (5) we have
H = Po - fa. (56)
Therefore, equation (55) becomes
ul0] = (x2 - IK (57)
For n = 0 in (8), we obtain
T^ = -(1 + 2/i)^1(x - Po)uo. (58)
Substitution of (56) and (57) in (58) gives (41).
So, according to Theorem 2, the sequence [Pn}n^o is Dunkl-classical. □
Remark 4. Theorems 1 and 3 are the main results of this paper. From them, we carry out the complete study of the Dunkl-classical orthogonal polynomials.
Acknowledgements. The author would like to thank the anonymous referees for their useful comments and many valuable suggestions for this paper.
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Received January 31, 2023. In revised form, June 30, 2023. Accepted July 2, 2023. Published online October 12, 2023.
Y. Habbachi
Laboratory of Mathematics and Applications, Faculty of Sciences, Gabes University, Omar Ibn Khattab Street 6029 Gabes, Tunisia E-mail: E-mail: Yahiahabbachi@gmail.com
