CC BY
20Probl. Anal. Issues Anal. Vol. 11 (29), No 2, 2022, pp. 29-41
DOI: 10.15393/j3.art.2022.11310
29
UDC 517.587, 517.538.3
Y. Habbachi, B. Bouras
A NOTE FOR THE DUNKL-CLASSICAL POLYNOMIALS
Abstract. In this paper, we give a new characterization for the Dunkl-classical orthogonal polynomials. The previous characterization has been illustrated by some examples.
Keywords: orthogonal polynomials, Dunkl operator, Dunkl-classical polynomials
2020 Mathematical Subject Classification: 33C45, 42C05
1. Introduction and preliminary results. Let V be the vector space of polynomials with coefficients in C. An orthogonal polynomial set (OPS for short) {Pn}n^o in V is called classical (resp. A-classical, Hq-classical) if {DPn}n^l (resp. {AP„}n^[, {HqPn}n^l) is also an OPS, where D (resp. A, Hq) denotes the derivative operator D = (resp. A the difference operator, Hq the Hahn operator given, respectively, by Af (x) = f (x + 1) - f (x) and Hqf (x) = ,q = 1J eV).
In [10], the authors characterized the so-called classical orthogonal polynomials (Hermite, Laguerre, Jacobi, and Bessel) by a new characterization. In particular, they showed that a MOPS {Pn}n^0 is classical if and only if there exists a polynomial an of degree n ^ 0, and a polynomial $ (monic) of degree less or equal to 2, such that Pn+lu = D(an$u), n ^ 0, where u is the corresponding form to {Pn}n^o. Later on, this characterization has been extended for the classical discrete and ^-classical (discrete) polynomials (see [2]).
A natural question arises: Is there a similar characterization for Dunkl-classical orthogonal polynomials?
The aim of this paper is to answer this question. Namely, we prove the Theorem 2 (see section 2).
We begin by reviewing some preliminary results needed in the sequel. Let V be the dual of V. We denote by (u,f) the action of u e V on
© Petrozavodsk State University, 2022
f EV. In particular, we denote by (u)n = (u,xn), n ^ 0, the moments of the form u (linear functional).
Let us introduce some useful operations in V'. For any form u, any polynomial p and any (a, c) E C \ {0} x C, let fu, hau, 5C and (x — c)-1u be the forms defined by duality:
(fu,P) = (u, fp) ; (haU,p) = (U, haP) ,
(5c,p) = p(c); ((x — c)-lu,p) = (u, Bcp),
where hap(x) = p(ax) and (9Cp)(x) = —^^.
x — c
Then, it is straightforward to prove that for c E C and u E V
(x — c)-l((x — c)u) = u — (u)05c.
Let {Pn}n^0 be a sequence of monic polynomials (MPS for short) with deg Pn = n, n ^ 0. The dual sequence for {Pn}n^0 is the sequence {un}n^0, un E V', defined by (un, Pm) = 8nm, n,m ^ 0, where bnm is the kronecker symbol.
The linear form u is called regular if there exists a MPS {Pn}n^0, such that [8]:
(u, PmPn) = r-nSn,m, n,m ^ 0, r,n = 0, n ^ 0.
The sequence {Pn}n^0 is then said to be orthogonal with respect to u. In this case, we have
Un = ((uo,P2))-1PnUo,n > 0. (1)
Moreover, u = Xu0, where (u)0 = X = 0 [13].
In what follows, all regular linear functionals u are assumed to be normalized, i.e, (u)0 = 1.
A polynomial set {Pn}n^0 is called symmetric if and only if Pn(—x) = (—1)nPn(x),n ^ 0/
According to Favard's theorem, a monic orthogonal polynomial sequence (MOPS) is characterized by the following three-term recurrence relation [8]:
{
P0(x) = 1,Pi (x) = x — p0,
Pn+2(x) = (X — Pn+l)Pn+l(x) — Jn+lPn(x), U ^ 0.
with ([3n, jn+l) e C x C \ {0}, n ^ 0. The first associated with {Pn}n^o is the MOPS {Pirl)}n>o, defined by
t
P^1)(x) = 1, P(11)(x) = x - fa
P&te) = (x - fln^PilUx) - ln+2Pil\x), n > 0. Let us introduce the Dunkl operator [9]:
T,(f) = f' + 2»H_if, (H-if )(x) = f (x) (-x), f eV e 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-l, n ^ 0,
with the convention (w)_;l = 0 where
1 - (-1)n
= n + 2^£n,£n =-2-,n ^ 0. (4)
Note that T0 is reduced to the derivative operator D.
Using the previous definitions, we get the following formula [5]:
T^(fu) = fT^u + (T^f )u + 2/j(H_if )(h_iu - u),f eV ,u eV'. (5)
Now, consider a MOPS {Pn}n^o and let
P^(x,y) =—^(TpPn+i)(x), y = -n - 1 ,n ^ 0.
^n+l 2
Denoting by {the dual sequence of {Pn\-, we have [14]
T^U1l](^) = -^n+iun+i, n ^ 0. (6)
Definition 1. [4,7,14] A monic orthogonal polynomial sequence {Pn}n^o is said to be T^-classical (or Dunkl-classical) polynomial sequence if {T^Pn}n^L is an orthogonal polynomial sequence. In this case, the form u corresponding to {Pn}n^o is called T^-classical form.
B. Bouras proved in [5] the following theorem: Theorem 1. Let {Pn}n^o be a MPS orthogonal with respect to a linear form u0. For and № = 0, the following statements are equivalent:
(a) The sequence {Pn}n^o is Dunkl-classical.
(b) There exist a non-zero complex number K and three polynomials $ (monic), $ and ^ with deg$ ^ 2, deg$ ^ 3 and deg^ = 1, such that
K $"(0^0 K i>" '(0)
*'(0> + {4,j2'" - n) + W-W) n - "Î = 0 P)
and
T^$uo - 2fih-i($Uo))+1—4t~^uo = 0, (8)
with
x$ (x)u0 = h-1(&(x)u0). (9)
Remark 1. Symmetric Dunkl-classical forms are well described in [4]. In particular, two canonical forms appear: the generalized Hermite and the generalized Gegenbauer forms; however, for the non-symmetric case one canonical case appears: it is the regular perturbed generalized Gegenbauer form [6]
Ç(a,» - 2) = A(x - 1)-1G(a, ^ - 1) + 5U (10)
where
A =____, (11)
2a + 2/i + 1' y J
and G(a,^ - 2) is the generalized Gegenbauer form [1], [3].
The MOPS corresponding to Q(a, 2), which we denote {S^'^ 2^ }n^0, satisfies the three-term recurrence relation (2) with [8]
¡3n = 0 and Jn+1 = 7-+ 2a- , n > 0, (12)
' In+1 (2n + 2a + 2/i + 1)(2n + 2a + 2/i + 3)' ' V '
where ßn+1 is given in (4).
Lemma 1. [5], [7]. If [Pn}n^o is a Dunkl-classical MOPS, then u0\ß) satisfies
( Pni](;ß))2) = + - n) +
+ K&''(0) A (uo,P2+i) (13)
+ ^-V(^ -n)J-+~. (13)
3(1 - 4n2) J nn+i
2. Main Result. The main result of this section is as follows: Theorem 2. Let {Pn}n^0 be a MPS orthogonal with respect to a linear form u0. For ^ = 0, y, the following statements are equivalent.
(a) The sequence {Pn}n^0 is Dunkl-classical.
(b) There exist a non-zero complex number K and three polynomials $ (monic), deg$ ^ 2, $, deg$ ^ 3 and deg^ = 1 and a polynomial Qn, deg(Qra) = n,n ^ 0, such that
K ( \
Pn+iUo = 1 - 2T^(Qn($Uo - 2fih_i($Uo))j , n > 0, (14)
K$''(0) K&''(0)
¥(0) + w-W) t>- n)+W-U) «i - n)=0 (15)
with
x<& (x)u0 = h_L( $ (x)u0). (16) Proof. (a) ^ (b) From the assumption, we have
un =((uo,P*))_lPnuo,n > 0 (17)
and
= ((u[l(v), (P}l](-,v))2))_lP£](-,v)u0;](v),n > 0. (18) Substitution of (17) and (18) in (6) gives
T„(p£\-,»)v0\ii)) = -Vn+i — Pn+iUo, n ^ 0, (19)
f n+l
where rln = (u[\y,), (Pn\-,^,))2) and rn+i = (uo,P2+l). For n = 0, equation (19) becomes
T,u[](^) = - 1+2^Pmo. (20)
li
Using formula (5), equation (19) is transformed to
PP^^T^ifi) + u[](i) + 2fih-1u[(l](fi) = -2 u0. (22)
x (h-iu[](ii) - u[1](^)) = -fin+i^—Pn+iUo, n ^ 0. (21 V / rn+i
For n =1, equation (21) becomes
Ji] 1
T2
Substitution of (20) in (22) gives
ull (i) + 2ih-i ull (i) = K $u0, (23)
where
K$ = 1+lPiPl1\^i) - 2r-±-P2, (24)
li r2
and the non-zero constant K is chosen to make $ monic. Applying the operator h-i to (23), we get
h-iuH](i) + 2iu[](i) = Kh-i($u0). (25)
Multiplying (25) by 2i and subtracting the result from (23), we get
u[\l) = *\ 2($uo - 2ih-i($u0)). (26)
K
40 1 -Substitution of (13) and (26) in (19) gives
K /
T„[Pii](;l)($uo - 2ih-i($uo))
1 -
. k $''(0) K(§>'''(0) \
= \*'(0) + 2( 1- V) (4l2^-n) + 3K1 - (¿2)l(Cn-n))Pn+iuo,n > 0.
Thus, (14) follows, where
Qn =--^^---,n > 0.
/ K$''(0) K&"'(0) \
r'^+W-W) i'(i"-n))
Now, putting n = 2 in (21), we obtain
+ 2/iН-гР21](;»)(h-iиЦ](ц) - м^Ы) = -X2P3U0. (28) Taking into account (20) and (26), we get
(V^M - (l + 2l)H-iP^,ï))h-i(*>U0) =
= (^PPM - Tp%i)+
+ 1—Ï*H-iP?(;i) -X2Ps)u0. (29)
Applying the operator h-l to the last equation and taking into account the fact that
(T,Pll\;l))(x) - (l + 2l)(H-iPil\;l))(x) =
= ( P2])'(x,l) - (H-iP2i])(x,l) = 2x
and the formulas
h-i(xv) = -xh-iv and h-i(h-iv) = v,v eV', we obtain (15), where
ф(x) = (Pi(x)P?(x,i) - 1S>(x)(T^)(x^) +
+ ï~ïФ(x)(H-iP?)(x,l) -X2Ps(x)). (30)
(b) ^ (a) Putting n = 0 in (14), we get
К / \
Piuo = l - Q0T^Фщ - 2ih-i(&u0)). (31)
Then, according to Theorem 1, the sequence {Pn}n^o is Dunkl-classical P
with Ф = - P-. □ Qo
3. Examples. In this section, we will illustrate Theorem 2 by giving some examples. For this, we need the following results.
Let {S^'1 i')}n>0 be the sequence of orthogonal polynomials with respect to the form G(o,ß — l ) (see (10)).
The sequence {Sin*'1 2^}n>0 satisfies the recurrence relation
S^-2\x) = l,$r-1 \x) = x — ßo, (_9)
,_1 ) __._(aß— 1 ) ■_(ocß—1 ) (32)
Sn+2 2 (x) = (X — ßn+l)SnTl 2 (x) — 7n+lSn'ß 2 (x),n > 0,
with [12]
ßo = —a(0O) = 1 + A, ßn+i = a^ — al°% %+i = — a^(l + a^), > 0, where can"' is given by Maroni [12]
a(o) = _sj+i-2)(1) + A(SÎa"1-2})(l)(1) n > 0
Cn = ,-----U ,-----1N . . ,'1 > 0. (33)
s{na,i- 2 ] (l) + A(s{naT1 ]Yl)(i)
The relationship between {Sl?'1 2and [sl?'1 2is (see [12])
Sl+i-2) = S-Ti-2) + a^S^-2\ n > 0. (34)
Lemma 2. The coefficient ais given by
a<+ =---,n > 0. (35)
n 2n + 2a +2n + V V J
Proof. We will prove (35) by induction on n. Using (3), (11), (12), and (33), we get
s[a'1-2\l) + \(S0a'i-2))(l)(1) _ n Vi
/i | \\ __r^i f^Zfi.']
s0+'1-2 )(1) + \(S{+r2 ))(i)(1) - = - .
Hence, (35) is true for n = 0.
Assume that (35) is true until n and let us prove it for n +1. From (33), the recurrence hypothesis, and the three-term recurrence relation
fulfilled by {S<n+'i 2we have
a
(o) _ S{nT1 )(1) + A(S(na+i-1 ))(l)(1)
n+l S((+ï-1 ) (1) + A(S(n'i-1 ])(l)(1)
_ S^-2(l) + \(S{n->r2))(1)(1)X = / Jn+i\
I ln+lS^~1 )(1) + A(Sn^'it-2 ) )(D(1)' ^
= _(+_ ^n+1 + 2a ) (by (12)) = V 2n + 2a + 2 ¡i + 3J K y y "
n + 2 + 1 _ ¡(_1)n ¡n+2
2n + 2 a + 2i + 3 2n + 2 a + 2^ + 3
This completes the proof. □
Remark 2. From (35), it is easy to see that a^ satisfies the following relation:
¡n+1 x a{n+1 = ¡n+2 x a^l1),n Z 0. (37)
Lemma 3. We have the following results:
1) The generalized Hermite polynomials 'H^ satisfy [11]
T^nlAx) = ¡n+1'H<ni)(x),n Z 0. (38)
2) The generalized Gegenbauer polynomials S^*'" 2 ) satisfy [4]
tS7i-2V) = in+1S{:l1'"-1 \x),n Z 0. (39)
3) The sequence of orthogonal polynomials S^*'" 2 ) satisfy
tS+i-1 \x) = InnSt11'"-2)(x),n Z 0. (40)
Proof. We aim at proving (40); from (34) and (39), we have:
T q( - 1 ) (\ = T S (a,"- 1 ) (\ + (a) T S (a,"-2 )(\ =
T"Snl2 (-L) = ±"°nl2 ('L)+ anl1T"Snl1 (d,) =
= S(*+1'"-2) , ) + (a) Sit*11"-2)( , =
— 1nl2Sni1 (X ) + ani1lnl1Sn (X ) —
(a+12 ), -, (a) ¡n+1 a(a
= / S (al1'"-2) +a(a) 1nl1 S (al1'"-2 )(A =
= ¡nl2\ Snl1 (X) + an+1 Sn (Xn =
v 1n+2 J
¡n+2 (Slail1'"-2\x) + a^Si*11'»-1 )(x)) by (37) :
= ¡¡ni2S{nli''1 2\x),n Z 0. Moreover, it is clear that T»s1*'" 2\x) = 1 + 2i = ¡1s0*11'" 2\x).
Hence the desired result. □
Example 1. Generalized Hermite polynomials. The sequence of generalized Hermite polynomials [H<;t1}n^o is symmetric Dunkl-classical and its associated form H(i) satisfies (7)-(9) with [5]
$(x) = 1, $(x) = -x, tf(x) = 2x, K = 1 + 2/i. (41)
Using (27), (38), and (41), we can easily prove that
Qn(x) = -2 Hnil)(x),n Z 0. The form H(i) is symmetric, i.e, H(i) = h-lH(i). Then H(i) - 2nh-iH(n) = (1 - 2n)H(n).
According to Theorem 2, the sequence {Hn1}^ satisfies the following relation:
H<IiH(ii) = - 1Ti(Hnil)H(i)), n z 0
1
Ln+i/Lw — 2 i
Example 2. Generalized Gegenbauer polynomials. The sequence
of generalized Gegenbauer polynomials {sI?'1 2) }nzo is symmetric Dunkl-classical and its associated form Q(a,i - |) satisfies (7)-(9) with [5]
$(x) = x2 - 1, $(x) = -x(x2 - 1),
, s , s (1 + 2a)(a + 1 + 2)
V(x) = (2 a + 2i + 3)x,K = ---—---^. (42)
a + 1
From (27), (39) and (42), we can easily prove that
Qn(x) = --^-—-7-—- S(:+l'i-2)(x),n Z 0.
Q() (a + i + | )(n + 2 + 2a + i(1 - (-1)n)) ( ), '
Moreover, since Q(a,i - |) is symmetric, then we have
G(a,l - ^ = h-i(<3(a,i - 2)),
Multiplying the last equation by x| - 1, we obtain
(x2 - 1)Q(a,i - 1) = h-i((x2 - 1)Q(a,i - 2)),
Therefore,
{x2-1)G {a, a-2)-2lih-1((x2-1)g {a, a-1)) = {1-2a)){x2-1)g {a, a-1).
Thus, according to Theorem 2, the sequence [S^'^ 2satisfies the following relation:
2 )n( L 1
Sn+1 2 У {a,a--) =-;-;-r^X
n+l ^ 2 n + 2 + 2a + a{1 - (-1)n)
x T^{Sia+1'^-2 V - 1)g {a,a - 1)), n Z 0.
Example 3. Non-symmetric Dunkl-classical orthogonal polynomials. The sequence [S2 is non-symmetric Dunkl-classical and its associated form G{a,a - J) satisfies (7)-(9) with [5]
Ф{х) = {x - 1){x + 1 + ), Ф{x) = x{x - 1){x - 1 - ^), y(x)= {1 + 2a + 2a)2 x _ 1 + 2a ч K = {2a - 1){l + 2g + 2a) {x) 2a V 1 + 2a + 2a J, 2a
(43)
for a = 0.
On the one hand, we use (27), (40), and (43) and we get
Qn{x) = - {2a + 2a + 1){n +1+2a + a{1 +{-1)n)) ^^ 2){x),n Z 0 On the other hand, from (10) we have
{x - 1)g{a,a - 1) = xg{a,a - 1). (44)
Since Q{a, a - 2) is symmetric, we have Q{a,a - 2) = h-l{Q{a, a - 2)), or, equivalently, in (44):
{x - 1)<5{a,a - 1) = h-i{{x - 1)Q{a,a - ^^ Multiplying the last equation by x - , we obtain
{x-1)(x- <3{a,a-1) = -h-i{{x- 1)(x + <3{a,a-1)).
(45)
Now, from the first equality in (43) and (45), we have
$(x)§(a, ß - 2) - 2ßh-i(&(x)Q(a, ß - \)) = (1 + 2ß)(x2 - 1)Ç(a, ß - 1).
„ ! (46)
Consequently, according to Theorem 2, the sequence {S^n'^ 2satisfies the following relation:
o(<*'ß-2 ) ni ^ 1 w
S.,,^ 2 y(a,ß - - ) =-------—-— x
n+1 v 2' n + 1 + 2a + ß(1 + (-1)n)
x Tß(S^-2 V - 1)g(a,ß - 2)),n z 0.
Acknowledgements. The authors would like to thank the editor and the anonymous referees for their useful comments and suggestions.
References
[1] Alaya J., Maroni P. Symmetric Laguerre-Hahn forms of class s = 1. Int. Transf. and Spc. Funct., 1996, vol. 4, pp. 301-320.
DOI: https://doi.org/10.1080/10652469608819117
[2] Alfaro M., Alvarez-Nodarse R. A characterization of the classical orthogonal discrete and q-polynomials. J. Comput. Appl. Math., 2007, vol. 201, pp. 48-54.
DOI: https://doi.org/10.1016/j.cam.2006.01.031
[3] Belmehdi S. Generalized Gegenbauer polynomials. J. Comput. Appl. Math., 2001, vol. 133, pp. 195-205.
DOI: https://doi.org/10.1016/S0377-0427(00)00643-9
[4] Ben Cheikh Y., Gaied M. Characterizations of the Dunkl-classical orthogonal polynomials. App. Math. Comput., 2007, vol. 187, pp. 105-114. DOI: http://doi. org/10.1016/j.amc.2006.08.108
[5] Bouras B. Some characterizations of Dunkl-classical orthogonal polynomials. J. Difference Equ. Appl., 20 (2014), no. 8, 1240-1257.
DOI: http://doi.org/10.1080/10236198.2014.906590
[6] Bouras B, Habbachi Y. Classification of nonsymmetric Dunkl-classical orthogonal polynomials. J. Difference Equ. Appl. 2017, vol. 23, pp. 539-556. DOI: https://doi.org/10.1080/10236198.2016.1257615
[7] Bouras B, Alaya J, Habbachi Y. A D-Pearson equation for Dunkl-classical orthogonal polynomials. Facta Univ. Ser. Math. Inform., 2016, vol. 31, pp. 55-71.
[8] Chihara T. S. An Introduction to Orthogonal Polynomials. Gordon and Breach, New York, 1978.
[9] Dunkl C. F. Integral kernels reflection group invariance. Canad. J. Math., 1991, vol. 43, pp. 1213-1227.
DOI: http://doi. org/10.4153/CJM-1991-069-8
[10] Marcellan F., Branquinho A., Petronilho J. Classical orthogonal polynomials: A functional approach, Acta. Appl. Math., 1994, vol. 34, pp. 283-303. DOI: http://doi. org/10.1007/BF00998681
[11] Ghressi A, Kheriji L. A new characterization of the generalized Hermite form. Bull Belg Math Soc Simon Stevin., 2008, vol. 15, pp. 561-567. DOI: http://doi.org/10.36045/bbms/1222783100
[12] Maroni P. Sur la suite de polynômes orthogonaux associée a la forme u = öc + \(x — c)-lL. Period. Math. Hungar., 1990, vol. 21, no. 3, pp. 223-248.
[13] Maroni P. Variation around Classical orthogonal polynomials. Connected problems, J. Comput. Appl. Math., 1993, vol. 48, pp. 133-155.
DOI: http://doi.org/10.1016/0377-0427(93)90319-7
[14] Sghaier M. A note on Dunkl-classical orthogonal polynomials. Integral. Transforms. Spec. Funct., 2012, vol. 24, pp. 753-760.
DOI: http://doi.org/10.1080/10652469.2011.631186
Received December 12, 2021. In revised form, March 21, 2022. Accepted March 22, 2022. Published online April 2, 2022.
Department of Mathematics, College of Sciences, Gabes University, Tunisia
Y. Habbachi
E-mail: Yahiahabbachi@gmail.com B. Bouras
E-mail: Belgacem.Bouras@issatgb.rnu.tn
