A CHARACTERIZATION OF MEIXNER ORTHOGONAL POLYNOMIALS VIA A CERTAIN TRANSFERT OPERATOR Текст научной статьи по специальности «Математика»

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

[email protected]

Lotfi Kheriji

Institut Préparatoire aux Etudes d'Ingenieur El Manar, Universiteé de Tunis El Manar, Rommana 1068, Tunisia

[email protected]

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