CC BY
0Probl. Anal. Issues Anal. Vol. 13 (31), No 2, 2024, pp. 49-62
DOI: 10.15393/j3.art.2024.15830
49
UDC 517.58
S. Jbeli
A NEW CHARACTERIZATION OF g-CHEBYSHEV POLYNOMIALS OF THE SECOND KIND
Abstract. In this work, we introduce the notion of U^q-classical orthogonal polynomials, where U^q is the degree raising shift operator defined by '■= x(xHq + q^11-p) + ^Hq, where ^
is a nonzero free parameter, I-p represents the identity operator on the space of polynomials V, and Hq is the ^-derivative one. We show that the scaled g-Chebychev polynomials of the second kind Un(x,q),n > 0, are the only Upq>tiq-classical orthogonal polynomials.
Key words: orthogonal q-polynomials, q-derivative operator, q-Chebyshev polynomials, raising operator
2020 Mathematical Subject Classification: Primary 33C45; Secondary 42C05
1. Introduction. Chebyshev polynomials and their ^-analogues are used in many fields in the mathematics as well as in the physical sciences. Note that several contributions have been devoted to the ^-extension of the Chebyshev polynomials and their properties [1], [8], [12], [18]. Our objective in this paper is to characterize the scaled g-Chebyshev polynomials of the second kind [18] via a raising operator.
Let O be a linear operator, acting on the space of polynomials, that sends polynomials of degree n to polynomials of degree n + n0, where n0 is a fixed integer (n > 0 if n0 > 0 and n > n0 if n0 < 0). We call a sequence {Pn}n>0 of orthogonal polynomials O-classical if {OPn}n>0 is also orthogonal. An orthogonal polynomial sequence {Pn}n>0 is called classical if {Pn}n>o is also orthogonal. This is the Hahn property (see [10]) for the classical orthogonal polynomials. In [11], Hahn gave similar characterization theorems for orthogonal polynomials Pn, such that the polynomials
© Petrozavodsk State University, 2024
D^Pn or HqPn(n ^ 1) are again orthogonal; here Du is the divided difference operator and Hq is the ^-derivative operator given, respectively, by Duj f (x) = fJs+zhm, w * 0 and Hq f (x) = , q * 1.
In this paper, we consider the raising operator
Upq^) : = x(xHq + q~lI-p) + ¡1Hq,
where ^ is a nonzero free parameter and I-p represents the identity operator. We show that the scaled g-Chebyshev polynomial sequence of the second kind [18], \b-nUn(bx)\ , where b2 = — (q^)-1, is the only
Mpq,fi)-classical orthogonal polynomial sequence.
Several authors have been interested in the study of the orthogonal polynomials using the lowering, transfer, and raising operators [2], [3] [5], [4], [6], [14], [17].
The structure of the paper is as follows. In Section 2 , we give some useful results . In Section 3, we solve the problem. In Section 4, a property of the scaled g-Chebychev polynomials of the second kind is given.
2. Preliminaries. We denote by V the vector space of the polynomials with coefficients in C and by V' its dual space. The action of u e V on f e V is denoted as (u, f). In particular, we denote by (u)n : = (u,xn) ,n ^ 0, the moments of u. For instance, for any form u, any polynomial g, and any (a, c) e (C\{0}) x C, we let Hqu, gu, hau, Du, (x — c)-1u, and 5C be the forms defined as usually ( [15] and [13]) for the images related to the operator Hq
(Hq^ f) := —(U, Hqf) , (gu, f) := (U, gf) , (haU, f) := (U, haf) ,
(Du, f) :=—(u,f), ((x — c)-1 u,f) :=(u,dcf), (6c, f) : = f (c), where for all f e V and q e C := e C, z * 0,zn * 1,n ^ l|, [13]
i
Hq(f)(x) = * 0,
Hq (f )(o) = / ' (o),
(haf )(x) = f (ax) , (dcf )(X) = f(x) f(c).
x — c
In particular, this yields
(Hqu)n = —[n]q(u)n-i ,n ^ 0 ,
where (u)_1 = 0 and
qn - 1
q := q - 1 ' n > 0.
Let {Pn}n>0 be a sequence of monic polynomials with deg Pn = n, n > 0, (MPS for short) and let {un}n>0 be its dual sequence, un e V' defined by (un,Pm) := 5n>m, n,m > 0 [7], [15]. The form u is called regular if we can associate with it a MPS {Pn}n>0, such that ( [7], [15]) (u,PnPm) = Tn5n,m, n,m > 0; rn * 0, n > 0. The sequence {Pn}n>0 is then said to be orthogonal with respect to u (MOPS for short) and is characterized by the following three-term recurrence relation (Favard's theorem) (TTRR for short) [7]:
Po(x) = 1, Pi(x)= x - Po,
(1)
Pn+2(x) = (X - Pn+l)Pn+l(x) - ln+lPn(x), n > 0,
w-e A. = tire = W P C^z{0}'« > 0
The shifted MOPS {Pn := a~n(haPn)}n>0 is then orthogonal with respect to « = ha-iu and satisfies (1) with [15]
p fin ^ 7n+1 ^ n
fin = -, Pn+l = -+-, n > 0.
a a2
Moreover, the form u is said to be normalized if (u)0 = 1. In this paper, we suppose that any regular form are normalized. In addition, {Pn}n>0 is a symmetric MOPS if and only if fin = 0, n > 0 or, equivalently, (u)2n+1 = 0, n > 0 [7], [15]. When u is regular, let $ be a polynomial, such that $u = 0, then $ = 0 [15].
Lemma 1. [15], [17] Let {Pn}n>0 be a MPS and let {un}n>0 be its dual sequence. For any u e V' and any integer m > 1, the following statements are equivalent:
(i) (u, Pm_ 1 ) * 0, (u, Pn) = 0, n > m;
m— 1
(ii) e C, 0 ^ v ^ m — 1, \m-1 * 0 such that u = ^ Xvuv.
As a consequence, when the MPS {Pn}n>0 is orthogonal with respect to u, necessarily, u = u0.
Proposition 1. [15] Let {Pn}n>0 be a MPS with deg Pn = n, n > 0, and let {un}n>0 be its dual sequence. The following statements are equivalent:
(i) {Pn}n>0 is orthogonal with respect to u0;
(ii) Un = (uo, P;2)-1 PnUo, n > 0;
(iii) {Pn}n>0 satisfies the three-term recurrence relation (1). Let us recall some results in the field of g-theory.
Lemma 2. [9], [13]
Hq (fg) (x) = (hq f) (x) (Hq g) (X) + g(X) (Hq f) (X) , f,9 P V , (2)
Hq (fu) (x) = fHq U + (Hq-1 f) hq U, f P V ,U P V', (3)
ha (fg) (x) = (haf) (x) (hag) (x), f,9 P V ,0, P C — {0}, (4)
ha(gu) = (ha-1 g) (hau) ,g e V,u e V',a e C — {0}, (5)
Hq o ha = aha o Hq in V, (6)
hq-i o Hq = Hq-i in V. (7) Now, consider a MPS {Pn}n>0 as above and let [13]
p;i](x; q) := (Hqp;+i) (x), n > 0.
[n + 1jq
Denote by {«niS(g)}n>0 the dual sequence of {P^G; q)}n>0. The following equality holds [13]:
Hq {uln](q)) = —[n + 1jq Un+1, n > 0.
Definition 1. [13] The form u0 is said to be Hq-classical if it is regular and there exist two polynomials, $ monic, deg$ ^ 2, and deg$ = 1, such as:
Hq ($(x)u0) + $(x)u0 = 0,
where the pair ($,$) is admissible, i.e., (0) — 1 (0)[nJq * 0, n > 1. The corresponding MOPS {Pn},n>0 is said to be Hq-classical.
Lemma 3. [13] When u0 satisfies the equation Hq ($u0) + = 0, then P0 = ha-iu0 fulfils the equation
Hq ($+ $P0 = 0,
where $ (x) = a~ deg#$(ax), ^ (x) = a1~deg^^(ax).
Proposition 2. [13] For any orthogonal sequence {Pn}n^0, the successive assertions are equivalent:
(i) The sequence {Pn}n>o is Hq-classical.
(ii) The sequence \Pn^ is orthogonal.
n>0
(iii) There exist two polynomials, $ monic, deg$ ^ 2, deg^ = 1, and a sequence {An}n>0 , Xn ^ 0, n > 0, such that
$(x)(HqoHq-iPn+i){x)-^{x){Hq-iPn+i){x)+\nPn+i{x) = 0,n > 0.
Let us recall the g-Chebyshev MOPS of the first kind: {Tn(.,q)}n>0 orthogonal with respect to Tq and the g-Chebyshev MOPS of the second kind {Un(.,q)}n>Q orthogonal with respect to Uq. We have [18]:
ry'l - ry'l - _ml_ n > 1
11 q+l , in+l (qn + 1)(çn + l + 1) , n > 1
Hq ((x2 - 1) %)- q-1 x% = 0,
and
u,
r.n+2
ln+1 (q™ + 1 + 1)(q™+2 + 1) , n > 0,
[Hq ((x2 - q-1) Uq) + ^xUq = 0.
Denote by {Un(., q)} 0 the MOPS with respect to Uq have [18]:
Ün(x, q) = q-^ Ûn ^(px, q^ , n > 0,
:iq)
h _ 1 Uq. We
q 2 q
[x2 - 1) % "
1
1
h _iU„ =
q+1 i 2 q q + 1
U
and
Hq [Tn+1
( x, )
-n+1
1
Un(x, q), n > 0.
ad
:i2)
:i3)
q- 1
Finally, denote by {Tn}n>0, {Un}n>0 and {Un}n>0 respectively, the sequences {T^^ n>0, {Un(., n>0 and {U^^ n>0. 3. Main results. Let us introduce the operator
Up«,») : V h
>v
Uiq M f) = (x2 + ß)Hq (f) + q-1xf.
:i4)
Definition 2. The MOPS {Pn}n>0 is said to be U(q>ß)-classical if {U(q,ß)Pn}n>Q is also orthogonal.
For any MPS {Pn}n>0, the MPS {Qn}n>0 is defined by
Qn+i(x) := ,n > 0, (15)
q 1[n + 1J q
or, equivalently,
q-1 [n + 1jqQn+i(x) := (x2 + (Ji) Hq (Pn) (x) + q-1xPn(x),n > 0, (16) with Q0(x) = 1.
It is clear that the operator U(q^) raises the degree of any polynomial. Such operator is called raising operator [14]. By transposition of the operator U(q^), we have:
tU(q,,\ = —Uiq,,\. (17)
Denote by {un}n>0 and {vn}n>0 the dual basis in V corresponding to {Pn}n>0 and {Qn}n>0, respectively. Then, according to Lemma 1 and (17), we get the relation
(x2 + ^ Hq (v,n+i) + q-1xVn+1 =—q-1[n + 1jq Un, n > 0. (18)
Assume that {Pn},n>0 and {Qn},n>0 are MOPSs satisfying
I
Po (x) = 1, Pl(x)= x — fo,
Pn+2(x) = (X — f3n+\) Pn+\(x) — r)n+\Pn(x), Jn+l * 0,n > 0,
:i9)
(20)
\Qo(x) = 1, Qi(x)=x — xo, Qn+2 (%) = (X — Xn+\)Qn+l(x) — 6n+\Qn(x), On+1 * 0,n > 0.
Our goal is to describe all the -classical orthogonal polynomial sequences. Note that it is necessary that y * 0 to ensure the orthogonality of the sequence {Qn}n>0. In fact, if we suppose that y = 0, the relation (16) becomes, for x = 0, Qn+l(0) = 0, n > 0, which contradicts the orthogonality of {Qn}n>0. Indeed, from (20) we have Q-\_(x) = x and Q2(x) = (x — Xi)x — Ql. For x = 0, we obtain 9l = 0, which is impossible.
We are going to establish the connection between the two sequences {pn}n>o and {Qn}n>o.
Proposition 3. The sequences {Pn}n>0 and {Qn}n>0 satisfy the following relation:
(x2 + y) hqPn(x) = qnQn+2(x) + XnQn+l(x) + VnQn(x), U > 0,
where
\n = q^1[n + l]q {ffn - Xn+i) ,n ^ 0, = q^1 {[n]qln -[n + l]qdn+i) ,n ^ 0,
with ^o := 0.
Proof. By applying the operator Hq to (19) and using (2), we get
Hq(Pn+2)(x) = {qx-f3n+i)Hq(Pn+i)(x)-<yn+iHq(Pn)(x) + Pn+i(x), n ^ 0.
(21)
Multiply equation (21) by x2 + y and relation (20) by x. Then take the sum of these two resulting equations. Next, substituting (16), get
q^1[n + 3]qQn+3{x) =
= q~1[n + 2]q {x - fn+i) Qn+2{x) - q~1[n + l]q Jn+iQn+i{x) +
+ (x2 + y) hqPn+1{x), n ^ 0.
On account of the recurrence relation (20), we get
(:x2 + y) hqPn+i {x) =
= qn+1Qn+3{x) + q1 [n + 2]q {ffn+i - Xn+2) Qn+2{x) +
+ q1 {[n + l]qryn+1 -[n + 2]gen+2)Qn+i{x), n ^ 0.
Now, replacing n + l by n, we have for all n ^ l:
(x2 + y) hqPn{x) = qnQn+2{x) + q~1[n + l]q {ffn - Xn+i) Qn+i{x) +
+ (fl {[n]qln -[n + l]q 9n+i) Qn{x),
with the constraint ^f0 := 0.
For n = 0, the Proposition 3 gives
Q2{x) + q1 {ffo - Xi) Qi{x) =x2 + y + q l0i, and using the fact that Q1{x) = x, we obtain
Q2{x) = x2 - x + ^. (22)
By comparing (20) and (22) for n = 0, we obtain x1 " and q _ q^ i 9
In the following lemma, we establish an algebraic relation between the forms u0 and v0.
Lemma 4. The forms u0 and v0 satisfy the relation
hq ((x2 + j)vo) = -^-uo.
vv ' ' q + 1
Proof. By virtue of Proposition 3, we get
<(x2 + j) v0, hqPnD = 0, n > 1. (23)
Moreover, by (22) we have x2 + j = Q2 + -^Qi + . Since {Qn}n>0 is orthogonal with respect to the form v0, and v0 is supposed to be normalized, we obtain:
<(x2 + j) V0, P0D " (v0, Q2 + QiD + JT = . (24)
xv y 7 x q + 1 7 q + 1 q + 1
On account of Lemma 1, (23), and (24), the desired result holds. □
Using the last lemma, we are going to establish a first-order g-difference equation satisfied by {Qn}n>0.
Proposition 4. The following relation holds:
Hq PQn+i) (x) = [n + 1], q n(hqPn)(x), n > 0. (25)
Proof. Based on Proposition 1, we may write the relation (18) as
(x2 + j)Hq-1 (Qn+i)(x) hqV0 + q1 xQn+i(x) V0 + (x2 + j)Qn+i(x)Hq(V0) =
= XnPn(x)u0, n > 0, (26)
where Xn := -q^1[n + 1}q{v0,Q2n+i){u0,P^)'1, n > 0. Making n = 0 in (26) and using (3), we get:
(x2 + j xHq (v0) = — (x2 + j hqv0 — q~lx2v0 + X0u0.
Substituting this relation in (26), for n > 0 we obtain:
(xHq-1 (Qn+i)(x) — Qn+i(x))(x2 + j)hq V0 = (XnxPn(x) — X0Qn+i(x))u0.
By virtue of Lemma 4, the fact that X0 = — 9i = , and taking into account the regularity of u0, we finally get
j Hq-1 (Qn+i) (x) = (q + 1)XnPn(x), n > 0.
The comparison of the degrees in the last equation gives (q + 1 )Ara = = [n + 1 n > 0. Therefore,
Hg-1 (Qn+\) (x) = [n + 1]g-iPn(x), n > 0,
which is equivalent to
Hg (Qn+i) (x) = [n + 1], q-nhqPn(x), n > 0.
□
Now we will show that the scaled g-Chebyshev polynomial sequence {b~nUn(bx^n>Q, where b2 = — (q¡)1, is the only -classical orthogonal sequence. In particular, {Un(x)} is W(ç,_i)-classical orthogonal sequence.
Theorem 1. For any nonzero complex number i and any MPS {Pn}n>o, the following statements are equivalent:
(i) { Pn}n>0 is Upq^)-classical.
(ii) There exists be C, b ^ 0, such that Pn(x) = b~nUn(bx), n > 0.
Proof. (i) ^ (ii). Assume that { Pn}n>o is Wp^q-classical. Then there exists a monic orthogonal sequence {Qn}n>o satisfying (16). By applying vo to (16), we get for n > 0:
(vo, q l[n + 1]qQn+i(x)) = (vo, (x2 + ¡)Hq (Pn) + q'lxPn) = 0.
The preceding equation can be written as
( Hq ((x2 + vo) — q-lxvo,Pn) = 0, n > 0.
Equivalently,
Hq^x2 + ¡) v0) — q~lx v0 = 0. The choice a2 = —¡~1 in Lemma 3 gives v0 = Tq. Then, from (4) and (5),
—uo q + 1
= —¡i 1 hq{[x2 + ¡) vo) = hq (ha(x2 — 1)h0-i Tq) =
= hq o ha-i {(x2 — 1)73 = hpq-1a)-1 Uq.
Consequently, uo = hpq-ia)-i Uq. Thus, for n > 0
Qn(x) = a~nTn(ax), Pn (x) = (aq ~lynÛn(aq~1 x) = (aq -i)"n(Jn(aq-x).
The desired result is found by taking b = aq 2; so, b2 = — (qj) l. (ii) ^ (i). Let b in C, with b ^ 0, and let Pn(x) = b-nUn(bx), n > 0. It is clear that {Pn}n>0 is a MOPS. The sequence {Tn}n>0 is Hq-classical; then, according to (8), (9), it satisfies the g-diffrence equation
(x2 — 1)( Hq o Hq-iTn+i)(x) — q~lx(Hq-iT,n+i)(x) = —XnT,n+i(x), n > 0.
From (7), we get
(x2 — 1) Hq (hq -1 (HqTn+l)) (x) + q — x hq-1 (Hq Tn+^j (x) =
= —\nTn+l(x), n > 0.
On account of (13), the last equation becomes
[n + 1]q (x2 — 1) Hq (hq - lU^) (x) + q- [n + 1]q x hq-lUn(x) =
=—\nTn+l(x), n > 0.
According to (11), we get
q^[n + 1]q (x2 — 1) Hq (hq -i U^j (x) + q -r~l[n + 1]qxhq=itn(x) =
=—\nTn+l(x), n > 0.
Applying h i to the previous equation and using (6), we get q 2
q(qx2 — 1)HqUn(x) + q^^x U,n(x) = -—-hiT,n+l(x), n > 0.
[n + 1]q 2
(27)
Finally, applying hb to (27) and using (6), we get qb(x2 — (q^b)'2)Hq(hbUn) (x) + bxhbU,n(x) =
n + 1
%i)T»l(x)- n >0 (28)
For j = — (q2b) 2 and multiplying (28) by b~n, we get
-A fan1)-n+l)
(x2+j)Hq (Pn) (x)+q-lxPn(x) = ^q+1]-h^bq 1 )Tn+l(x), n > 0.
Then
pUMPn)px) = —+ 1 -h^bqi^fn+1{x), n ^ 0.
Since {Tn} an orthogonal polynomials sequence, then Upq^)Pn is also an orthogonal polynomials sequence. Therefore, {Pnis -classical. □
4. A property of the scaled g-Chebyshev polynomials.
Lemma 5. There exists an endomorphism E of V into itself, such that the polynomials Pn{x), n ^ 0, are eigenfunctions. We have:
E {Pn) = XnPn, n > 0, (29)
with
^n = q-pn+1\[n + 1], )2. (30)
Moreover,
E := bi{x)Hq-i o Hq + b2{x)Hq-i + b3{x)I-p, (31)
where
bi(x) = x2 + j, b2{x) = (q-2 + q-1 + l)x, b3{x) = q-1, (32)
and I-p represents the identity operator on the space of polynomials V. Proof. By applying the operator Hq to (15) and using (24), we obtain
Hq o U^ {Pn) = q-pn+iq {[n + 1],)2 {hqPn) , n > 0. (33)
Then, applying the operator hq-i to (33) and using (7), we get
Hq-i o U{q^ {Pn) = q-pn+iq {[n + 1],)2 Pn, n ^ 0. (34)
Consequently, from (2), (7), (15), and (34), we deduce (31)-(32). In addition, we have:
E {Xn) = \nXn + jnXn-2, n ^ 0,
with
Jn = q-(n-2)[n]q[n - 1],j, n ^ 0.
Thus, the matrix of the endomorphism £ in the canonical basis {Xn}n>0 of V is given by
0 \
M, =
0 0
0 •••
A2 ••• ^n 0
0
An
0
• /
Using the relation (29), the matrix of £ in the basis {Pn},n>0 is as follows:
0
f \o 0 0 Ai
E =
An 0
□
Remark 1.
1. When q ^ 1 in Proposition 3, Lemma 4, Proposition 4 and Theorem 1, we recover the results, as well as the characterization of Chebyshev polynomials of the second kind in [3].
2. When q ^ 1 in Lemma 5, we find the property described in [19] with £i = 0 for the Chebyshev polynomials of the second kind.
0
0
Acknowledgment. I would like to extend my sincere gratitude to the reviewers for dedicating their time to reviewing the article and for their valuable suggestions, which have significantly improved the quality of the manuscript.
References
[1] Atakishiyeva M, Atakishiyev N. On discrete q-extension of Chebyshev polynomials. Commun. Math. Anal., 2013. vol. 14(2), pp. 1-12.
[2] Aloui B. Characterization of Laguerre polynomials as orthogonal polynomials connected by the Laguerre degree raising shift operator. Ramanujan J., 2018, no. 45, pp. 475-481.
DOI: https://doi.org/10.1007/s11139-017-9901-x
[3] Aloui B. Chebyshev polynomials of the second kind via raising operator preserving the orthogonality. Period Math Hung., 2018, vol. 76, pp. 126-132. DOI: https://doi.org/10.1007/s10998-017-0219-7
[4] Bouanani A, Kheriji L, Tounsi MI. Characterization of q-Dunkl Appell symmetric orthogonal q-polynomials. Expositiones Mathematicae., 2010, vol. 28(4), pp. 325-336.
DOI: https://doi.org/10.1016/j.exmath.2010.03.003
[5] Bouras B, Habbachi Y, Marcellan F. Characterizations of the Symmetric Tpg,qq-Classical Orthogonal q-Polynomials. Mediterranean Journal Of Mathematics., 2022, vol. 19(2).
DOI: https://doi.org/10.1007/s00009-022-01986-8
[6] Ben Cheikh Y, Gaied M. Characterization of the Dunkl-classical symmetric orthogonal polynomials. Appl. Math. Comput., 2007, vol. 187, pp. 105-114. DOI: https://doi .org/10.1016/j.amc.2006.08.108
[7] Chihara T. S. An Introduction to Orthogonal Polynomials. Gordon and Breach, New York, 1978.
[8] Ercan E, Cetin M, Tuglu N. Incomplete q-Chebyshev polynomials. Filomat., 2018, vol. 32(10), pp. 3599-3607.
DOI: https://doi .org/10.2298/fil1810599e
[9] Jbeli S. Description of the symmetric Hq-Laguerre-Hahn orthogonal q-polynomials of class one. Period Math Hung., 2024.
DOI: https://doi.org/10.1007/s10998-024-00574-5
[10] Hahn W. Uber die Jacobischen polynome und zwei verwandte polynomklassen. Math. Z., 1935, vol. 39, pp. 634-638.
[11] Hahn W. Uber Orthogonalpolynome, die linearen Funktionalgleichungen genügen. Dans : Lecture Notes in Mathematics., 1985, pp. 16-35.
DOI: 10.1007/bfb0076529
[12] Kizilates C, Tuglu N, Cekim B. On the (p,q)-Chebyshev polynomials and related polynomials. Mathematics., 2019, vol. 7(136), pp. 1-12.
[13] Kheriji L, Maroni P. The Hq-classical orthogonal polynomials. Acta. Appl. Math., 2002, vol. 71, pp. 49-115.
DOI: https://doi .org/10.1023/a:1014597619994
[14] Koornwinder T.H. Lowering and raising operators for some special orthogonal polynomials. In: Jack, Hall-Littlewood and Macdonald Polynomials, Contemporary Mathematics, vol. 417, 2006.
[15] Maroni P. Une théorie algébrique des polynômes orthogonaux. Application aux polynomes orthogonaux semi-classiques. In Orthogonal Polynomials and their applications. Proc. Erice, 1990, IMACS, Ann. Comput. Appl. Math., 1991, vol. 9, pp. 95-130. Math., 9, Baltzer, Basel, 1991.
[16] Maroni P. Variations around classical orthogonal polynomials. Connected problems. Journal Of Computational And Applied Mathematics., 1993, vol. 48(1-2), pp. 133-155.
DOI: https://doi.org/10.1016/0377-0427(93)90319-7
[17] Maroni P, Mejri M. The Ipq,uq-classical orthogonal polynomials. Appl. Nu-mer. Math., 2002, vol. 43(4), pp. 423-458.
DOI: https://doi.org/10.1016/s0168-9274(01)00180-5
[18] Mejri M. q-Chebyshev polynomials and their q-classical characters. Probl. Anal. Issues Anal., 2022, vol. 11(29), no 1, pp. 81-101.
DOI: https://doi .org/10.15393/j3.art. 2022.10330
[19] Souissi J. Characterization of polynomials via a raising operator. Probl. Anal. Issues Anal., 2024, vol. 13 (31), no 1, pp. 71-81.
DOI: https://doi .org/10.15393/j3.art. 2024.14050
Received March 11, 2024. In revised form, May 26 , 2024. Accepted May 28, 2024. Published online June 07, 2024.
S. Jbeli
Université de Tunis El Manar Campus Universitaire El Manar, Tunis, 2092, Tunisie. LR13ES06 E-mail: jbelisobhi@gmail.com
