URAL MATHEMATICAL JOURNAL, Vol. 10, No. 1, 2024, pp. 4-17
DOI: 10.15826/umj.2024.1.001
A CHARACTERIZATION OF MEIXNER ORTHOGONAL POLYNOMIALS VIA A CERTAIN TRANSFERT OPERATOR
Emna Abassi
Faculté des Sciences de Tunis, Université de Tunis El Manar, Rommana 1068, Tunisia
emna.abassi@fst.utm.tn
Lotfi Kheriji
Institut Préparatoire aux Etudes d'Ingenieur El Manar, Universiteé de Tunis El Manar, Rommana 1068, Tunisia
kheriji@yahoo.fr
Abstract: Here we consider a certain transfert operator = Ip — ctw, w = 0, c € R — {0, 1}, and
we prove the following statement: up to an affine transformation, the only orthogonal sequence that remains orthogonal after application of this transfert operator is the Meixner polynomials of the first kind.
Keywords: Orthogonal polynomials, Regular form, Meixner polynomials, Divided-difference operator, Transfert operator, Hahn property.
1. Introduction and preliminaries
Let O be a linear operator acting on the space of polynomials as a lowering operator (the derivative [4, 18, 19], the q-derivative [4, 12, 14, 15], the divided-difference [1], the Dunkl [6, 8, 9, 11, 13], the q-Dunkl [5, 7, 13], other [17, 21]), a transfert operator (see [20]) or a raising operator (see [2, 3, 17]). Many researchers in this vast field cited above had the concern to characterize the O-classical polynomial sequences that is those which fulfill the so-called Hahn property: the sequences {Pn}n>0 and {OPn}n>0 are orthogonal.
By the way, in [20], the authors characterized the I(q,w)-classical orthogonal polynomials where I(q,w) is a transfert operator acting on the space of polynomials P and defined by [20]
I(q,w) := Ip + uhq, u € C \{0}, q € Cw := {z € C, z = 0, zn+1 = 1, 1+ uzn = 0, n € N},
with Ip being the identity operator in P and (hqf )(x) = f (qx), f € P (homothety). Therefore, our goal is to consider the following transfert operator M(c,w) acting on P and defined by
M(c,w) = Ip - CTw, u = 0, c € R -{0,1}, (1.1)
where
(twf)(x) = f (x - u), f € P,
(translation) and to characterize all sequences of orthogonal polynomials {Pn}n>0 having the Hahn property; the resulting up an affine transformation (that is to say up a composition of a homothety and a translation; see (1.4) below), is the Meixner polynomials of the first kind (see Theorem 2
below). Indeed, in Section 2, firstly we deal with the M(cw)-character by presenting some characterizations of it (see Theorem 1), secondly, we establish the system verified by the elements of second-order recurrence relation for the sequences {Pn}n>0 and {M(cw)Pn}n>0 and thirdly we solve it to deduce the desired result (Theorem 2). Moreover, the divided-difference equation fulfilled by its canonical form and the second order linear divided-difference equation satisfied by any Meixner polynomial are highlighted.
Let P be the vector space of polynomials with coefficients in C and let P' be its dual. We denote by (u, f} the action of u € P' on f € P. In particular, we denote by
(u)n := (u,xn}, n > 0
the moments of u. The form u is called regular if we can associate with it a sequence of monic polynomials {Pn}n>0 with deg Pn = n, n > 0 ((MPS) in short) [18] such that
(u,PmPn} = TnSn,m, n,m > 0; Tn = 0, n > 0.
The sequence {Pn}n>0 is then called orthogonal with respect to u ((MOPS) in short). In this case, the (MOPS) {Pn}n>0 fulfils the standard recurrence relation ((TTRR) in short) [10, 18]
(1.2)
I Po(x) = 1, Pi(x)= x - A) ,
\ Pn+2(x) = (x - An+l)Pn+l(x) - Yn+lPn(x), n > 0, where
(u,xP^} Tn+1 Pn = -—, Jn+l = - /0, n > 0.
Tn Tn
Moreover, the regular form u will be supposed normalized that is to say (u)0 = 1.
For any form u, any polynomial g and a, w € C\{0}, b € C, we let Tbu, hau, gu, Du = u', Duu be the forms defined by duality [18] namely
(Tbu,f} = (u,T-bf}, (hau,f} = (u,haf}, (gu, f} = (u,gf}, (u', f} = -(u, f'}, (Duu, f} = -(u, D-uf}
where
(T-bf)(x) = f(x + b), (haf)(x) = f(ax), ID.ujf){x) = f{x)~f{x~u}\ f € P,
w
and due to the well known formulas [1, 18] we have
Tb(fu) = (Tbf)(Tbu), ha (fu) = (ha-l f)(hau), u € P', f €P. (1.3)
Let Sb be the Dirac mass at b defined by
(Sb,f} = f (b), b € C, f € P. In addition, let {Pn}n>0 be the (MPS) defined by
Pn(x) = a-nPn(ax + b), n > 0, a = 0, b € C. If {Pn}n>0 is a (MOPS) associated with u, then {Pn}n>0 is a (MOPS) associated with
« = (ha-i o T-bju
and fulfilling the (TTRR) in (1.2) (fin ^ (3n, y«+i ^ Sn+1, n > 0) with [18]
S fin— b ^ Yn+1 , ,
pn =-, ln+1 = —5-, n > 0. (1.4)
a a2
Let now {Pn}n>0 be a (MPS) and let {un}n>0 be its dual sequence, un € P' defined by
(«n, Pm) = dn,m, n,m > 0.
Let us recall some results [18].
Lemma 1 [18]. For any u € P' and any integer m > 1, the following statements are equivalent
(i) (u, Pm-1) = 0, (u,Pn) = 0, n > m,
(ii) EUv € C , 0 < v < m - 1, Am—1 = 0,
such that
m—1
u = ^^ Avuv.
v=0
As a consequence,
— the dual sequence {Sn}n>0 of {Pn}n>0 is given by
Sn = an (ha-i o t—^un, n > 0,
— when {Pn}n>0 be a (MOPS) then u = u0. In this case, we have
un = r—1Pnu0, n > 0
and reciprocally. Lastly, when u0 is regular and $ is a polynomial such that $u0 = 0, then $ = 0.
The monic Meixner polynomials {Mn(.; a,c)}n>0 of the first kind are given by [10, 16]
Mn(X] a, c) = (a + 1)„ (j^j n 2^1 ( 11 " I), n> 0,
they are orthogonal with respect to the discrete weight
p(x) = —-—«N x!
for a > —1, 0 < c < 1. Here, the Pochhammer symbol (z)n takes the form
n
(z)0 = 1, (z)n + k — 1), n > 1,
k=1
and 2F1 is the hypergeometric function defined by
2F1 ,
r
p,q > ^ (p)k(q)k
£
(r)k k!'
fc=0 v
By describing exhaustively the D—w-classical orthogonal polynomials in [1], the authors rediscover the (MOPS) of Meixner {Mn(.; a, c)}n>0 orthogonal with respect to the D—1 -classical Meixner form M(a, c) for a = —n — 1, n > 0, c € C — {0,1} and the positive definite case occurring for a +1 > 0, c € (0, œ) — {1}; they establish successively the (TTRR) elements, the divided-difference
equation, the modified moments, the discrete representation and the second order linear divided-difference equation (see the following),
( c 1 + c c
fin = --(a + 1) + --n, 7„+i = ---r(??, + l)(??, + a: + l), n> 0,
1 - c 1 - c (1 - c)2
D-:L ((x + a + 1)M(a, c)) - ((1 - c-1)x + a + 1)M(a, c) = 0,
M(a,c) = (1 -c)"+1 Er(p(+ +t)fc) TT'^ °<lcl<1' a^-n-l, n > 0, k>0 (a + ) ' (x + a + 1)(D-i o DiMn+i)(x; a, c) + ((1 - c-1)x + a + 1)(DiMn+i)(x; a, c)
-(n + 1)(1 - c-i)Mn+i(x; a, c) = 0, n > 0. 2. Main result 2.1. The M(cw)-classical character
First of all, let w = 0 and c € R - {0,1}. By virtue of (1.1) we have
(M(c>w)f)(x) = f (x) - cf (x - w), f € P. (2.1)
Particularly,
(M(C>W) 1)(x) = 1 - c, (M(c>w){n)(x) = (1 - c)xn + lower degree terms, n > 1. (2.2)
When c = 1, M(i w) is not a transfert operator but a lowering one since M(i w) = wD_w. From (1.1), we have
M(clW) = Ip - ctw .
The transposed iM(c w) of M(c w) is
iM(c;W) = Ip - ct_w = M(c,-W),
leaving out a light abuse of notation without consequence. Thus,
(M(c,_w)u,f} = (u, M(c>w)f}, u € P', f € P. Particularly, by virtue of (2.2) we get
n_i in\
(M(c,_w)u)0 = 1 - c, (M(c,_w)u)n = (1 - c) (u)n - c^lj (-w)n-k (u)k, n > 1.
k=0 ^ '
Lemma 2. The following formulas hold
M(ciW)(fg)(x) = f(x)(M(i>w)g)(x) + (Twg)(x)(M(c>w)f)(x), f,g € P, (2.3)
M(c,_w)(fu) = (t_wf)(M(c,_w)u) + (M(i>_w)f)u, u € P', f € P, (2.4)
ha o M(c>w) = M(c,a-lw) o ha in P, ha o M(c,_w) = M(c,_aW) ◦ ha in P', a € C - {0}, (2.5)
Tb o M(ciW) = M(ciW) o Tb in P, Tb o M(c,-W) = M(c,-W) o Tb in P', b € C. (2.6)
Proof. The proof is straightforward since definitions and duality. □
by
Now consider a (MPS) {Pn}n>0. On account of (2.2), let us define the (MPS) {Pn](.;c,w)}n>0
= w/0, c € M - {0,1}, n > 0. (2.7)
1—c
Denoting by {uJ1](c, w)}n>0 the dual sequence of {Pn1]( .; c, w)}n>0, we have the result Lemma 3. The following formula holds
M(c,-W)(un1](c, w)) = (1 — c)un, n > 0. (2.8)
P r o o f. Indeed, from the definition it follows
(un1](c,w),pm](x;c,w)) = ¿n,m, n,m > 0,
so we have
((M(C;-w)(un1](c,w)),Pm) = (1 — c)5n,m, n, m > 0,
therefore,
(M(C;-w)(un1](c,w)),Pm) = 0, m > n + 1, n > 0;
(M(C>-W) (un1] (c, w)), Pn) = 1 — c, n > 0.
By virtue of Lemma 1, we get
n
M(c_w)(un1](c,w)) = ^ An,vuv, n > 0.
v=0
But,
,[1]
(M(c,-W)(un1](c,w)),PM) = An)M, 0 < ^ < n,
with An>M = 0, 0 < ^ < n and An,n = 1 — c. The formula (2.8) is then established. □
Definition 1. The (MPS) {Pn}n>0 is called M(c,w)-classical if {Pn}n>0 and {p!1(.; c, w)}n>0 are orthogonal.
Remark 1. When the (MPS) {Pn}n>0 is orthogonal, it satisfies the (TTRR) (1.2). When the (MPS) {Pni](.; c,w)}n>0 is orthogonal, it satisfies the (TTRR) (1.2) with the notations (fin ^ Yn+i ^ yj^+hi, n > 0).
Theorem 1. For any (MOPS) {Pn}n>0, the following assertions are equivalent.
a) The sequence {Pn}n>0 is M(c,w)-classical.
b) There exist a polynomial 0 monic, deg 0 < 1 and a constant K = 0 such that
M(c>_w)(0u0) — K-1(1 — c)u0 =0, (2.9)
1 — c — K0'(0) wn = 0, n > 0. (2.10)
c) There exist a polynomial 0 monic, deg 0 < 1, a constant K = 0 and a sequence of complex numbers {An}n>0, An = 0, n > 0, such that
(Кф(х) - 1 + с) (M(C>_W) о M(c>w)Pra)(x) +(c - 1)(Кф(х) - 1)(M(C>W)P„)(x) = A„P„(x), n > 0.
(2.11)
Proof. a) ^ b), a) ^ c).
of u0 and u
M(c,_w)(Pl1](-; c,w)u01](c,w)) = Zn Pnuo, n > 0,
From (2.8) and the regularity of u0 and u0i](c, w), we have
with
, (1 c) (Уу\с,и),(р£](.-,с,и))2)
By (2.4), we get
(t_wPW(.;c,w))M(c,_w)(u0i](c,w)) + (Ma>_w)pW(.;c,w))u0i](c,w) = ZnPnu0, n > 0. In accordance with the definition of M(c -W), one may write
M(c_w)(u01](c,w)) = u01](c,w) - с(т_шu01](c,w)),
which yields
Pl1](.; c,w)u01] (c,w) - с(т_шPl1](.; c,w))(r_w41](c,w)) = Zn PnU0, n > 0. (2.12)
Taking n = 0 in (2.12) leads to
u0i](c, w) - c(t_wu0i](c, w)) = (1 - c)u0. (2.13)
Injecting (2.13) in (2.12) gives
{P^C;c,w) - (t_wPni](.;c,w))}u0i](c,w) = {ZnPn - (1 - c)(t_wP^(.;c,w))}u0, n > 0. (2.14) Now, taking n = 1 in (2.14), we obtain
u0i](c,w)= K0(x)u0, (2.15) where K be a normalization constant since 0 monic and
1 - c Jn Yi
=-- {(1 — + uj + ^/io - 4111-
w I Yi Yi 0 J
Applying the operator t_w to (2.15), we get
(t_w u0i](c,w)) = K (t_w 0)(x)(t_w u0). (2.16)
Replacing (2.16) and (2.15) in (2.13) leads to the desired result (2.9). By virtue of (2.15), the formula in (2.14) becomes
(pW(.; c,w) - (t_wPW (.; c,w))) + (1 - c)(t_wpW(.; c,w)) - Zn Pn}u0 = 0, n > 0.
(2.17)
Therefore,
K0(pJ1](.;c,w) — (t_wpJ1](.;c,w))) + (1 — c)(t_wpJ1](.;c,w)) — ZnPn = 0, n > 0,
thanks to the regularity of u0. Moreover, from (2.1) with the change w <--w, we may write
(t_wpJ1](.; c,w)) = pJ1](.; c,w) — (M(C)_w)Pni](.; c,w))), n > 0. Consequently, the last equation becomes
(K0(x) — 1 + c)(M(Ci_w) o M(CiW)Pn)(x) + (c — 1)(K0(x) — 1)(M(CiW)Pn)(x)
= c(1 — c)ZnPn(x), n > 0.
Writing into (2.17)
' 0(x) = 0'(0)x + 0(0), (M(CiW)Pn )(x) = Pn(x) — cPn(x — w),
(M(C>_W) o M(CiW)Pn)(x) = (1 + c2)Pn(x) — c(Pn(x — w) + Pn(x + w)),
n
Pn(x) = an,fcxfc, an,n = 1, n > 0, k=0
and by comparing the degrees we obtain
1 — c — K0'(0) wn = Zn = 0, n > 0.
Hence (2.10) and a) ^ b).
Finally, (2.17) is (2.11) with An = c(1 — c)Zn = 0, n > 0. We have also proved that a) ^ c).
b) ^ a) Let us suppose that there exist a polynomial 0 monic, deg 0 < 1 and a constant K = 0 such that (2.9)-(2.10) are valid. From (2.9), we have
0 = <M(C>_W) (0u0) — K _1(1 — c)u0,1) = (1 — c)((u0,0) — K _1).
Thus,
K _1 = (u0,0) = 0'(0)fi0 + 0(0) = 0(fi0).
Necessarily, 0(fi0) = 0. Let v = K0u0. We are going to prove that the (MPS) {pJ1](.; c, w)}n>0 is orthogonal with respect to v. We have successively
(v,P01] (.; c,w)) = K (u0,0) = 1, (2.18)
for all n > 1,
KK
{v,pw(.-,c,uj)} = --<0mo,M(c,w)P„> = --(M (C,_W)(<M)),-Pn)
1 — c 1 — c
K
(K-\l-c)uo,Pn) = 0,
(2.9) 1 — c
and for m > 1, n > 0,
K
ct_„(0uo)=(<£-K-1(1-c))uo 1 - c or equivalently, for m > 1, n > 0,
(v, xmP^{- c, uj)) = --(0uo,xm(Pn(x) - cPn(x ~ uj)))
1-c
K Kc
: --<«imo,xmPn(x)) - —- (<fmo, Tuj {(( + co)mPn(0) (x))
1 - c 1 - c
KK
-- --((¡)uo,xmPn(x)) - --(cr_w(</mo), (x + uj)mPn(x))
1 - c 1 - c
K
(<M), (Xm - (x + uj)m)Pn(x)) + (uo, (X + u)mPn(x)),
c k=i ^ '
g m_fc(ti0ja.fcpn(a;)) + £
c k=0 ^ ' k=0 ^ ' from which thanks to the orthogonality of {Pn}n>0 and (2.10) we get
<v,xmPn1](.; c,u)) =0, 1 < m < n - 1, n > 2,
1 - c
(v, xnP^(.; c, uj)) = (l- (p[ ' nuj) (uq, Pi2,) ± 0, n > 1
1 - c n
By the identities in (2.18)-(2.19), we see that {Pn1](.; c,u)}n>0 is orthogonal with respect to v. We then obtain the desired result.
c) ^ b) Comparing the degrees in (2.11), we can deduce (2.10). Making n = 0 into (2.11), we obtain
A0 = c(1 - c)2. (2.20)
Moreover, from definitions, (2.11) may be written as
0((M(c>w)Pn) - (T-w o M(c,w)Pn)) + K-1(1 - c)(T-w o M(c,w)Pn) = c-1K-1AnPn, n > 0,
then,
(U0,0((M(c,w)Pn) - (T-w o M(c,w)Pn)) + k-1(1 - c)(t—w o M(c,w)Pn)) = c-1K-1An(U0, Pn), n > 0. Equivalently,
<M(c,-w)(<M)) - (M(c,-w) oTw)(0uc) + K-1(1 - c)(M(c,—w) oTwU0), Pn) = c-1K-1An(U0, Pn), n > 0. By virtue of Lemma 1 and (2.20), we get
M(c,-w)(<M)) - (M(c,-w) o Tw)(0U0) + K-1(1 - c)(M(c,—w) o TwU0) - K-1(1 - c)2U0 = 0. A similar expression is
M(c,-w) (0uc) - K-1(1 - c)U0 = (M(c,-w) o Tw)(0uc) (2 21)
-K-1(1 - c)(M(c,—w) o TwU0) - K-1(1 - c)cu0. (. )
But, by (2.6) and definition of the operator (M(c _w), we have for the right side of (2.21),
(M(c,_w) o tw)(0u0) - K_i(1 - c)(M(c,_w) o Twu0) - K_i(1 - c)cu0 = Tw (M(c,_w)(0u0^ - K_i(1 - c)rw ((M(c,_w)u0) + cr_wu0) = Tw (M(c,_w)(0u0) - K_i(1 - c)u^.
Therefore, (2.21) becomes
M(i>w)(M(c>_w)(0uo) - K_i(1 - c)u0) = 0. From the fact that the operator M(i w) is injective in P' we get (2.9). □
Lemma 4. If u0 satisfies (2.9), then «0 = (ha-i o r_b)u0 fulfills the equation
M(c,_wa-i) (a_ deg*0(ax + b)u0) - a_deg*K_i(1 - c)«0 = 0. P r o o f. We need the following formulas which are easy to prove from (1.3)
g(Tbu) = Tb((r_bg)u); g(hau) = ha((hag)u), g € P, u € P '. (2.22)
Now, with u0 = (rb o (ha) «0, we have
-K_i(1 - c)u0 = (rb o (ha)(-K_i(1 - c)«0).
Further,
M(c,-w)(0Uo) = M(c_w)(0(Tb(ha«O))) = M(c>-w^7b((r-b0)(hauo)))
= (Tb O M(c_w))((T-b0)(ha«o)) ( = )(Tb O M(c_w))(ha((ha O r-60)«o)) (2.0) (2.22)
= (Tb O ha O M(c_wa-1 )) ((ha ◦ T-b0)«o) . (2-5)
Consequently, equation (2.9) becomes
Tb O ha(M(c,-Wa-1^+ 6))«o) - K-1(1 - c)«o) =0. This leads to the desired equality. □
2.2. Determination of all M(c w)-classical (MOPS)s
Lemma 5. Let {Pra}ra>0 be a M(c,w)-classical (MOPS). The following equality holds
■ pn+i(x -u) = (ßn+1 - Ailirflite c, v) + (7n+! - 7,[!!i)41](^; C, u), n > 0. (2.23)
1 - c
Proof. From the (TTRR) (1.2) we have
Pn+2(x) = (x - ^n+i)Pn+i(x) - Yn+iPn(x), n > 0. (2.24)
Applying the transfert operator to (2.24), using (2.3) and (2.7) we obtain
(1 - c)p1+2(x; cw) =(1 - c)(x - ^™+i)pl+i(x; cw) + c^pn+i(x - w) (2
—Yn+i(1 - c)Pii](x; c,w), n > 0.
But from the (TTRR) of {Pni](.;c, w)}n>0, one may write
xP^C; c,w) = Pni+2(.; c,w) + Ani+iPn+i(.; c,w) + Yni+iPni](.; c,w), n > 0. (2.26) Now, injecting (2.26) in (2.25) leads to the desired result (2.23). □
Proposition 1. The coefficients An, Yn+i, , 7«+ satisfy the following system
=UJ-^, 77, >0, (2.27)
1-c
7n+i-7!!!i = -w2^^2(77 + l), 77 >0, (2.28)
1 + c
Pn+1 - Pn = W 1-, n > 0, (2.29)
1 — c
7,l1] = ^-T7n+i, n> 1. (2.30)
P r o o f. Firstly, the higher degree test in (2.23) yields
Pn+1 " tf+i = w , n > 0. (2.31)
1-c
Secondly, n = 0 in (2.23) gives
71 " 7!11 + 411)- (2-32)
1-c
Thirdly, applying the transfert operator M(c w) to
Pi(x) = x - A0 and by virtue of (2.7) and (2.31)-(2.32) we get (2.27) and
7i-7!11 = -W2(I^. (2-33)
Thanks to (2.27), the formula in (2.23) becomes
cwPn+i(x - w) = cwPn+i(x; c,w) + (1 - c)(Yn+i - Yn+i)Prli](x; c,w), n > 0. (2.34)
Moreover, multiplication of (2.24) by cw with the change x ^ x - w yields
cwPn+2(x - w) = (x - w - An+i)cwPn+i(x - w) - Yn+icwPn(x - w), n > 0. (2.35)
Replacing (2.34) for the index n, n + 1, n + 2 in (2.35), using (2.26) for the index n, n + 1, the formula in (2.27) and the fact that {P,ii](.; c, w)}n>0 is a basis , we obtain successively
(7!!!2-7n+2)-(7,ili-7n+i)=w2^r^2, 77 >0, (2.36)
(Yn+i - Yn+i) |(1 - c)(An - An+i) + (1 + c)w} = 0, (2.37)
(Yn+i - Yn+i)Yni] = (Yni] - Yn)Yn+i, n > 1. (2.38)
Summing on (2.36) and taking into account (2.33) lead to (2.28) and (2.37) yields (2.29). Lastly, (2.30) is a direct consequence of (2.38) and (2.28).
Now, we are able to solve the system (2.27)-(2.30). Summing on (2.29) leads to
1 + c
fin =f30+uj --n, n > 0. (2.39)
1 — c
Injecting (2.39) in (2.27) yields
fiW = fi0 -co + u ii^n, n > 0. (2.40)
1 — c 1 — c
Also, injecting (2.30) in (2.28) gives
7ra+2 ^-a;2—n>0.
n + 2 n + 1 (1 — c)2 Summing the previous equality leads to
Jn+1 = (n + 1) (71 + Ce;2(1^c)2 n), n > 0. (2.41)
After replacing (2.41) in (2.30) we deduce the following
7,[!!i = (n + 1) (71 + w2 + !)) > n ^ (2-42)
Corollary 1. Let {Pra}ra>0 be a M(c,w)-classical (MOPS). The following statements hold. 1) The recurrence elements of {Pn}n>0 are
' fi0 , 1 + c , ■ n,
□
fin = U (— + --- n), n > 0,
Vw 1 — c /
+ + n > 0.
(2.43)
2) The recurrence elements of {p!1](.; c, w)}n>0 are
'fio c 1+c
--1--h 1-'
w 1 — c 1 — c
A^-F-A + ^H n > 0,
Vw 1 — c 1 — c / 7&1 = -2T-^(n + 1) (n + 1 + ^ , n > 0.
(2.44)
Proof. The formula (2.43) is a consequence of (2.39) and (2.41). Also, (2.44) is a direct result from (2.40) and (2.42).
Theorem 2. Up to an affine transformation, the only M(c1)-classical (MOPS) is the Meixner's one of the first kind,.
Proof. The classification of the canonical situations depends on the fact that fi0 = 0 or fio = 0.
= 0. For (2.43)-(2.44), put
wfi0 = (1 — c)7i
and
c w2
Then,
— = i-(a + 1).
w 1 - c
Now, for (2.43), choosing a = w, b = 0 in (1.4) and thanks to (2.5)-(2.6) this yields
— c 1 + c Pn = T-(a + l) + -!—n, n> 0,
1 - c 1 - c
c
Jn+I = 71-r?(n + l)(n + a + l), n> 0.
(1 - c)2
Therefore (see (1.5)),
Pn = Mn(.; a,c), n > 0, with a = -n - 1, n > 0. Next, for (2.44), choosing
2wc
a = u, b = — -
1 - c
in (1.4) and thanks to (2.5)-(2.6) this yields
A«1 = 7—(« + 2) + n, n> 0, 1 - c 1 - c
7n+i = (1^c)2 (n + l)(n + a + 2), 77, >0. Thus,
P™ = Mn(.; a + 1,c), n > 0,
with a = -n - 2, n > 0.
A0 = 0. In this case, (2.43)-(2.44) become successively,
1+c
Bn = uj-n, n > 0,
1-c
2
and choosing in (1.4)
we obtain
Consequently,
c w2
wc 1-c
— c 1 + c
An = t-(a + 1) + --n, n > 0,
1 - c 1 - c
c
Jn+I = 71-r?(n + l)(n + a + l), n> 0.
(1 - c)2
Pn = Mn(.; a, c), n > 0,
(2.45)
7n+i = (^(n + 1) (« + ^ . « > 0,
+ n > 0,
V 1 - c 1 - c ) 2 (2 46)
7'i+i = (1^2 + 1) (" + 1 + . " > 0-
For (2.45), putting
with a = — n — 1, n > 0. For (2.46), putting
(1 — c)2 Yi
and choosing in (1.4) we get
Equivalently,
c w2
= a + 1
w c
a = (jj, b =--(a + 3),
1—c
ffl = (a + 2) + n n > o, 1 — c 1 — c
c
7Üi
(n + 1)(n + a + 2), n > 0.
(1 — c)2
PW = Mra(.; a + 1,c), n > 0,
with a = —n — 2, n > 0.
The theorem is then proved.
□
Remark 2. On account of Theorem 1, Theorem 2 and after some easy calculations we get for the divided-difference equation (2.9) fulfilled by the Meixner form M(a, c),
M(c _d [(x - (a + 1 )^M(a, c)) + (a + 1 )M(a, c) = 0,
and also for the second order linear divided-difference equation (2.11) satisfied by any Meixner polynomial Mn(.; a, c), for all n > 0,
+ 2C) (M(C'-D ° M(c.i)m™)(z; «>c) + (! " c) ~ c) (M^dM»)^; c)
, 2 n + a + 1 ,, , . = c( 1 - c) ^ ^ A/„(■>•: a, c).
Acknowledgements
The authors thank the anonymous referees for their careful reading of the manuscript and corrections.
REFERENCES
1. Abdelkarim F., Maroni P. The Dw-classical orthogonal polynomials. Results. Math.., 1997. Vol. 32, No. 12. P. 1-28. DOI: 10.1007/BF03322520
2. Aloui B. Characterization of Laguerre polynomials as orthogonal polynomials connected by the Laguerre degree raising operator. Ramanujan J., 2018. Vol. 45, No. 2. P. 475-481. DOI: 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, No. 1. P. 126-132. DOI: 10.1007/s10998-017-0219-7
4. Aloui B., Chammam W. Classical orthogonal polynomials and some new properties for their centroids of zeroes. Anal. Math., 2020. Vol. 46, No. 1. P. 13-23. DOI: 10.1007/s10476-020-0012-3
5. Aloui B., Souissi J. Characterization of q-Dunkl-classical symmetric orthogonal q-polynomials. Ramanujan J., 2021. Vol. 57, No. 4. P. 1355-1365. DOI: 10.1007/s11139-021-00425-8
6. Ben Salah I., Ghressi A., Khériji L. A characterization of symmetric TM-classical monic orthogonal polynomials by a structure relation. Integral Transforms Spec. Funct., 2014. Vol. 25, No. 6. P. 423-432. DOI: 10.1080/10652469.2013.870339
7. Bouanani A., Kheriji L., Ihsen Tounsi M. Characterization of q-Dunkl Appell symmetric orthogonal q-polynomials. Expo. Math., 2010. Vol. 28, No. 4. P. 325-336. DOI: 10.1016/j.exmath.2010.03.003
8. Bouras B., Habbachi Y., MarceMn F. Rodrigues formula and recurrence coefficients for non-symmetric Dunkl-classical orthogonal polynomials. Ramanujan J., 2021. Vol. 56, No. 2. P. 451-466. DOI: 10.1007/s11139-021-00419-6
9. Chaggara H. Operational rules and a generalized Hermite polynomials. J. Math. Anal. Appl., 2007. Vol. 332, No. 1. P. 11-21. DOI: 10.1016/j.jmaa.2006.09.068
10. Chihara T. S. An Introduction to Orthogonal Polynomials. New York: Gordon and Breach, 1978. 249 p.
11. Ghressi A., Kheriji L. A new characterization of the generalized Hermite linear form. Bull. Belg. Math. Soc. Simon Stevin, 2008. Vol. 15, No. 3. P. 561-567. DOI: 10.36045/bbms/1222783100
12. Ghressi A., Kheriji L. Integral and discrete representations of certain Brenke type orthogonal q-polynomials. Ramanujan J., 2013. Vol. 30, No. 2. P. 163-171. DOI: 10.1007/s11139-012-9421-7
13. Habbachi Y. A new characterization of symmetric Dunkl and q-Dunkl orthogonal polynomials. Ural Math. J., 2023. Vol. 9, No. 2. P. 109-120. DOI: 10.15826/umj.2023.2.009
14. Kheriji L., Maroni P. The Hq-classical orthogonal polynomials. Acta. Appl. Math., 2002. Vol. 71, No. 1. P. 49-115. DOI: 10.1023/A:1014597619994
15. Kheriji L., Maroni P. On the natural q-analogues of the classical orthogonal polynomials. Eurasian Math. J., 2013. Vol. 4, No. 2. P. 82-103.
16. Koekoek R., Swarttouw R.F. The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Ana,logue. Rep. Fac. Tech. Math. Inf., Delft University of Technology, The Netherlands. Report 98-17. Delft: TU Delft, 1998. 168 p.
17. Koornwinder T. H. Lowering and raising operators for some special orthogonal polynomials. In: Jack, Hall-Littlewood and Macdonald Polynomials. Contemp. Math., vol. 417. Providence, RI: AMS, 2006. P. 227-238. DOI: 10.1090/conm/417/07924
18. Maroni P. Une theorie algebrique des polynômes orthogonaux. Applications aux polynômes orthogonaux semi-classiques. In: Orthogonal Polynomials and their Applications, C. Brezinski et al. (eds.), IMACS Ann. Comput. Appl. Math., vol. 9, 1991. P. 95-130. (in French)
19. Maroni P. Variations around classical orthogonal polynomials. Connected problems. J. Comput. Appl. Math., 1993, Vol. 48, No. 1-2. P. 133-155. DOI: 10.1016/0377-0427(93)90319-7
20. Maroni P., Mejri M. The I(q,w)-classical orthogonal polynomials. Appl. Numer. Math., 2002. Vol. 43, No. 4. P. 423-458. DOI: 10.1016/S0168-9274(01)00180-5
21. Srivastava H. M., Ben Cheikh Y. Orthogonality of some polynomial sets via quasi-monomiality. Appl. Math. Comput., 2003. Vol. 141, No. 2-3. P. 415-425. DOI: 10.1016/S0096-3003(02)00961-X