CC BY
11Probl. Anal. Issues Anal. Vol. 13 (31), No 1, 2024, pp. 71-81
DOI: 10.15393/j3.art.2024.14050
71
UDC 517.587, 517.521, 517.538.3
J. Souissi
CHARACTERIZATION OF POLYNOMIALS VIA A RAISING OPERATOR
Abstract. This paper investigates a first-order linear differential operator J^, where £ = ) p C2\(0,0), and D := ^. The
operator is defined as J^ := x(xD + I) + + (2D, with I representing the identity on the space of polynomials with complex coefficients. The focus is on exploring the J^-classical orthogonal polynomials and analyzing properties of the resulting sequences. This work contributes to the understanding of these polynomials and their characteristics.
Keywords: orthogonal polynomials, classical polynomials, second-order differential equation, raising operator
2020 Mathematical Subject Classification: Primary 33C45; Secondary: 42C05
1. Introduction. An orthogonal polynomial sequence {Pn}n^o is called classical, if {P'n}n^0 is also orthogonal. This characterization is essentially the Hahn-Sonine characterization (see [9], [14]) of the classical orthogonal polynomials.
In a more general setting, let O be a linear operator acting on the space of polynomials, which 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 {0'pn}n^0 is also orthogonal.
In this paper, we consider the raising operator, J^ := x(xD +1) + + + £2D, where £ = (^1,C2) is a nonzero free parameter and I represents the identity operator. We describe all the J^-classical orthogonal polynomial sequences.
The basic idea has been deduced by starting from the raising operator
:= x(xD + I) + £2D (see [1]). Now, to obtain a raising operator, we can add to . Then we can consider the perturbed operator, given
© Petrozavodsk State University, 2024
in the previous paragraph, J^ := + ^I, where (^1,^2) ^ (0, 0) because the orthogonality is not preserved for = (0, 0).
As a result associated to , we have that the scaled Chebyshev polynomial sequence {a~nUn(ax)}n^0 with a2 = —1 is the only -classical sequence, (for more details see [1]). In [2] the others prove that the scaled
(3)
Bessel polynomial sequence [Bn2 }n^0 is the only J^-classical orthogonal polynomial sequence for £2 = 0. For the raising operator J^, the result is completely different. More precisely, in ^ ^ 0,£2 ^ 0 the Jacobi polynomial sequence {P^1'^}^ is the only J^-classical orthogonal polynomial sequence with a = -1—^= , y2 = €2, and ^ i(2k + 1),
k e Z\{_i>o}.
The structure of the paper is the following: In Section 2, a basic background about forms, orthogonal polynomials is given. In Section 3, we find the -classical orthogonal polynomials. In Section 4, we give some properties of the sequence obtained.
2. Preliminaries. Let P be the linear space of polynomials in one variable with complex coefficients. The algebraic dual space of P will be represented by P1. We denote by (u,p) the action of u e P1 on p e P and by (u)n := (u,xn), n ^ 0, the sequence of moments of u with respect to the polynomial sequence {xn}n^0.
Let us define the following operations in P1. For linear functionals u, any polynomial g, and any (a,b) e C\{0} x C, let Du = u', gu, r—bu and hau be the linear functionals defined by duality, [11]:
(u1 ,f) :=—(u,f), (gu,f) :=(u,gf), f e V,
(haU,f) :"(U,haf) = (U,f (ax)), (T—buJy :=(U,Tbf) = (U,f (X — b)y,f e V.
A linear functional u is called normalized if it satisfies (u)0 = 1.
Lemma 1. [13], [11] For any u e P1 and any integer m ^ 1, the following statements are equivalent:
(i) (u, Pm_i) ^ 0, (u, Pn) = 0, n ^ m.
m— 1
(ii) D\k e C, 0 ^ k ^ m — 1, Xm—1 ^ 0, such that u = ^ Xkuk.
k-0
As a consequence, the dual sequence {«n1S}n^0 of {Pn1S}n^o, where M^ ._ , 1 W p1
Pn J(^) := (n + 1) 1Pln+1(x), n ^ 0, is given by
DuM = —(n + 1)un+1, n ^ 0.
Similarly, the dual sequence {ûn}n:>o of {Pn}n^o, where Pn(x) : = a nPn(ax+ + b) with (a, b) e C\{0} x C, is given by
un = an(ha-i o T-b)un, n ^ 0.
The form u is called regular if we can associate with it a sequence {Pn}n^o, such that
(u, PnPm) = rn8n,m, n, m ^ 0, rn * 0, n ^ 0.
The sequence {Pn}n^o is then called a monic orthogonal polynomial sequence (MOPS) with respect to u. Note that u = (u)ouo, with (u)o * 0. When u is regular, let F be a polynomial, such that Fu = 0. Then F = 0 [11].
Proposition 1. [11]. Let {Pn}n^o be a MOPS with deg Pn = n, n ^ 0, and let {un}n^o be its dual sequence. The following statements are equivalent:
(i) {Pn}n^o is orthogonal with respect to uo;
(ii) un = (uo, PlYlPnuo, n ^ 0;
(iii) {Pn}n^0 satisfies the three-term recurrence relation
{Po(x) = 1, P\(x) = x - fio, (1)
Pn+2 (x) = (X - fin+l)Pn+l(x) - Jn+\Pn (x), n ^ 0,
where
= (uo,xP%)(uo, Ply1, n ^ 0, ln+i = (uo,P2+1)(uo,P2)-1 * 0, n ^ 0.
If {Pn}n^o is a MOPS with respect to the regular form uo, then {Pn}n^o is a MOPS with respect to the regular form ûo = (ha-i o t-b)uo, and satisfies [13]
{Po(x) = 1, A(x) = X - fio,
(JPn+2(^) = (X - fin+i )Pn+1 (x) - %+lPnPx), n ^ 0,
where ¡3n = a~ 1(ffin - b) and 7„+i = a~2ryn+i.
An orthogonal polynomial sequence {Pn}n>0 is called D-classical, if {Pn^}n>0 is also orthogonal (Hermite, Laguerre, Bessel or Jacobi ), [7], [9]. A second characterization of these polynomials, which will play the leading role in the sequel, is that they are the only polynomial solutions of the Second-Order Differential Equation (Bochner [5])
(SODE): ^{x)Pn+l{x) — ip(x)Fn+i(x) = XnPn+i{x), n > 0, (2)
where (j),^ are polynomials, fi monic, deg fi = t ^ 2, degip = 1, and Xn = (n + 1) (2 fi{0)n — ip'(0)) ^ 0, n > 0.
If {Pn}n>0 is a classical sequence satisfying (2), then {Pn}n>0 is also classical and satisfies (see [11])
(SODE): ¡P(x)P,^+l(x) — ^(x)P,^+l(x) = XnPn+i(x), n > 0, (3)
where 4>(x) = a^tfi(ax + b) and rtp(x) = al^tip(ax + b).
Now let us provide a summary of some basic characteristics of classical orthogonal polynomials. We focus on two families: the Bessel orthogonal polynomials (C1) and the Jacobi orthogonal polynomials (C2).
Bessel Orthogonal Polynomials (C1): For n > 0 and a ^ — n,
the Bessel orthogonal polynomials are denoted by Pn(x) = B^^(x), with Uo = The coefficients are given by:
1 1 — a
A =--, Pn = —■-7w—:-v, n > 0
a (n + a — 1)(n + a)
n(n + 2a — 2)
y =---- n > 1
'n (2n + 2a — 3)(n + a — 1)2(2n + 2a — 1)' '
The polynomials fi and ^ are x2 and —2(ax + 1), respectively, and Xn are (n + 1)(n + 2a) for n > 0.
Jacobi Orthogonal Polynomials (C2): For n > 0 and (a, A ^ —n,
a + A ^ —n — 1,n > 1), the Jacobi orthogonal polynomials are denoted by Pn(x) = J^Px), with u0 = jt"'^. The coefficients are given by:
a — A p a2 — A2
P0 — --—0 , 0 , An —
a + A + 2 (2n + a + A )(2n + a + A + 2)'
_ 4n(n + a + A )(n + a)(n + A ) > 1
' " (2n + a + A — 1)(2n + a + A)2(2n + a + A + 1), U > '
The polynomials 0 and ^ are x2 — 1 and — (a + ft + 2)x + a — ft, respectively, and \n are (n + 1)(n + a + ft + 2) for n ^ 0.
3. The Jf-classical orthogonal polynomials. Recall the operator
Jf : P —> P
f — Jf (f) = (x2 + ' + (x + 6) f.
Definition 1. We call a sequence {Pn}n^0 of orthogonal polynomials Jf-classical if there exist a sequence {Qn}n^o of orthogonal polynomials, such that JfPn = Qn+i, n ^ 0.
For any MPS {Pn}n^0, we define the MPS {Qn}n:>0, given by Qn+i(x) := J^l1X), n ^ 0, or, equivalently,
(n + 1)Qn+i(x) := (x2 + &)Pn(x) + (x + Ci)Pn(x), n ^ 0, (4)
with the initial value Q0(x) = 1.
Our next goal is to describe all the Jf-classical polynomial sequences. Note that we need £ ^ 0 to ensure that {Qn}n^0 is an orthogonal sequence. Indeed, if we suppose that £ = (£2) = 0, the relation (4) becomes, for x = 0, Qn+1(0) = 0, n ^ 0, which contradicts the orthogonality of
{Qn}n^0.
The operator Jf raises the degree of any polynomial. Such operator is called raising operator [6,10,15]. By transposition of the operator Jf, we get
lJi = —Jf + 2&I (5)
Denote by {un}n^0 and {vn}n^0 the dual basis in P1 corresponding to {Pn}n^o and {Qn}n^o, respectively. Then, according to Lemma 1 and (5), the relation
(x2 + €2)v'n+i + (x — Ci)Vn+1 =—(n + 1)u,n, n ^ 0, (6)
holds. Assume that { Pn}n^0 and {Qn}n:>0 are MOPS satisfying
(
Po(x) = 1,Pi (x)=x — fta,
Pn+2(x) = (x — ftn+i)Pn+i(x) — ln+iPn(x), ln+i ^ 0,n ^ 0,
\ Qo(x) = 1, Qi(x) = x — Po,
1 Qn+2 (x) = (x — Pn+i)Qn+i(x) — Qn+iQn(x), Qn+i ^ 0, n ^ 0.
Next, the first result will be deduced as a consequence of the relations (4), (7), and (8).
Proposition 2. The sequences {Pn}n>0 and {Qn}n>0 satisfy the following finite-type relation:
(X2 + &) Pn(x) = Qn+2(X) + dnQn+l (X) + WnQn(x), U > 0,
where
dn = (n + 1)(An — Pn+l) , n > 0, ™n = n'n — (n + 1)Qn+l, n > 0,
with the convention y0 = 0.
Proof. By differentiating (7), we obtain
K+2(x) = (x — An+i)Pn+i(x) — 'n+iP'n(x) + Pn+i(x), n > 0.
Multiplying the last equation by x2 + ^2 and the relation (7) by x + take the sum of the two resulting equations, and substitute (4). Then we get
(n + 3)Qn+3(x) = (n + 2)(x — An+i)Qn+2(x) — (n + 1) yn+iQn+i(x) +
+ (x2 + & Pn+i(x), n > 0.
Using the three-term recurrence relation (8), we get
(x2 + £2)Pn+i(x) = Qn+:i(x) + (n + 2){An+i — pn+2 )Qn+2(x) +
+ ((n + 1)'n+i — (n + 2) Qn+2)Qn+i(x), n > 0.
In fact, this result is valid for n + 1 replaced by n. More precisely, we have, for all n > 0,
(X2 + &)Pn(x) =
= Qn+2(x) + (n + 1)(An — Pn+i)Qn+i(x) + (njn — (n + 1) Qn+i)Qn(x),
with the convention y0 = 0. Hence the desired result. □ Note that, for n = 0, the Proposition 2 gives
Q2(x) + (Ao — pi)Qi(x) =x2 + & + Qi,
and using the fact that Qi(x) = x + we obtain
Q2(x) = x2 + (— pi)x — pi£i — Qi. (9)
By comparing (9) and (8) for n = 0, we obtain pi = 13(0+fl and q1 = — fl+f2.
Now we establish, in the next lemma, an algebraic relation between the forms uo and o.
Lemma 2. The forms u0 and v0 satisfy the following relation:
(x2 + £2) Vo = — QiUo. Proof. According to Proposition 2, we obtain
:(x2 + 6)Vo,Pn} = 0, n > 1. (10)
On the other hand, by (9), we have (x2 + £2) = Q2 + ((30 — pi)Qi — Qi, and then
@(x2 + 6)Vo, PoD = (Vo,Q2 + Wo — Pi)Qi^ — Qi(Vo)o = — Qi, (11)
since {Qn}n^o is orthogonal with respect to the form vo, where vo is supposed normalized. According to Lemma 1 and using (10) and (11), we obtain the desired result. □
It is clear that the formula (4) is a first-order differential equation satisfied by { Pn}n^o. Based on the last lemma, we obtain a first-order differential equation satisfied by {Qn}n:>o.
Proposition 3. The following fundamental relation holds:
Q'n+1(x) = (n + 1)Pn(x), n ^ 0. (12)
Proof. According to Proposition 1 (ii), the relation (6) can be written as follows:
(x2 + &)[Q'n+ i(x)Vo + Qn+ i(x)vo] + (x — Ci)Qn+ iVo = \nPn(x)uo, n ^ 0,
(13)
where Xn := —(n + 1)<Vo, Qn+i)<uo, p.l2) i, n ^ 0.
Making n = 0 in (13), we get (x2 + £2)vo = (^ — x)vo. (Xo = — q1). Substituting this relation in (13), we obtain
(^n Pn — 6iQ'n+ i)uo = 0.
Using the Lemma 2 and the fact that Ao = — Q1 and taking into account regularity of u0, we finally obtain A0Q'n+1(x) = AnPn(x), n ^ 0. Comparing the degrees in the last equation, we get An = (n + 1)A0, n ^ 0, and, then, Q'n+1(x) = (n + 1)Pn(x), n ^ 0. □
According to Proposition 3, and using the Bochner characterization, we get the J^-classical orthogonal sequence. Now, we will describe all of the J^-classical polynomial sequences.
Theorem 1. The J^-classical polynomial sequences are, up to a suitable affine transformation in the variable, one of the following D-classical polynomial sequences:
(a) if & = 0, Pn(x) = a-nUn(ax), n ^ 0, with a2 = — .
(b) if ^ = 0, Pn(x) = B{a'i](x), with & = 2.
(c) if ^ * 0 and & = —1, Pn(x) = Pn 2 ' 2 '(x), with & * 2k + 1, k p Z\{—1,0}.
(d) if (Ci, 6) 6 , Pn(x) = Pna,l3\x), with a = ^,0 = ^,
A{(o,o)}'
12,
with ^i(2k + 1), ke Z\{-1, 0}.
or ß2 = 6
Proof. Assume that {Pn}n^0 is a monic J^-classical orthogonal sequence. Then there exists a monic orthogonal sequence {Qn}n^0 satisfying (4), which gives after differentiating and inserting (12), the following SODE:
(x2 + C2)P^+i{x) + {3x + £i)PUi(x) = (n + 1)(n + 3) Pn+\(x), n > 0. (14)
(a) if & = 0, Pn(x) = a-'nU,n(ax), n ^ 0, with a2 = — &1. (see [1])
(b) if 6 = 0,
x2P'n+1(x) — (—3x — C1)PL+1(x) = (n + 1)(n + 3)Pn+1(x), n > 0.
According to Table C1, {Pn}n^0 is the Bessel sequence of parameter a if —2(ax + 1) = —3x — in this case a = | and ^ = 2. (c) if ^ * 0 and ^ = —1,
(x2 — 1)P'n+l(x) + (3x + i1)P'n+l(x) = (n + 1)(n + 3) Pn+1(x), n ^ 0.
According to Table C2, {Pn}n^0 is the Jacobi sequence of parameter (a, 0) if —(a + 0 + 2)x + a — 0 = —3x — ¡;1; in this case a = and 0 = ^, with & * 2k + 1, k 6 Z\{ — 1, 0}.
(d) if (C2\{(0,0)},
(x2 + & K+ !(x) + (:ix + 6 )P'n+ ! (x) = (n + 1)(n + 3)Pn+ !(x), n > 0.
According to Table C2, {Pn}n^o is the Jacobi sequence by a suitable affine transformation, Pn(x) = (-nP^a'l3^((x), with (2 = —£,21, a = , / = ^^, or ¡J2 = 6, with ^ x(2k + 1),
ke Z\{-1,0}.
□
4. Some properties of the sequence obtained. In the polynomial function space P , we can introduce the linear operator, denoted here by L:
L := D.
Using (12), we obtain
L(Qn+1) = (n + 1)Pn, n > 0. (15)
The operator L decreases the degree of a polynomial but preserves the orthogonality of the sequence {Pn}n^o. We have the following result:
Theorem 2. There exists a differential linear operator of order two C, for which the polynomial Pn(x), n ^ 0, is an eigenfunction. More precisely, we have:
C(Pn) = enPn, n ^ 0. (16)
with 9n = (n + 1)2 as the corresponding eigenvalues, and where C := a1(x)D2 + a2(x)D + a3(x)I,
where
a1(x) = x2 + £2, a2(x) = 3x + ^, a3(x) = 1. Proof. Applying the Jf operator, and according to (4), we get
D o Jf (Pn) = (n + 1)2Pn, n > 0.
This gives, after a simple calculation, the desired result. □
Note that, by applying the C operator to the Xn, n ^ 0, we obtain
C(xn) = dnXn + n^X"-1 + n(n - 1)&Xn-2, n ^ 0.
So, the matrix of the endomorphism C in the canonical basis {Xn}n^0 of P is given by
(Bo 6 26 0 ••• 0\
0 91 2£ 1 ... ... .
82 ... n(n- 1)^ 0
... n 1 ..
Mr =
On
0
.
Using the relation (16), we can write the matrix M£ in the bases { Pn}n^0 as follows:
id0 0 ...... 0\
L
0 01
0
0n 0 0 ..7
Acknowledgment. The author is very grateful to the referees for their constructive comments. Their suggestions and remarks have contributed to improve substantially the presentation of the manuscript.
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Received September 18, 2023. Accepted November 12, 2023. Published online December 10, 2023.
Jihad Souissi
Faculty of Sciences of Gabes
Department of Mathematics, Gabes University
Street Erriadh 6072 Gabes, Tunisia
E-mail: [email protected] & [email protected]
