ON A NEW COMBINATION OF ORTHOGONAL POLYNOMIALS SEQUENCES Текст научной статьи по специальности «Математика»

Vladikavkaz Mathematical Journal 2022, Volume 24, Issue 3, P. 5-20

УДК 512.62

DOI 10.46698/a8091-7203-8279-c

ON A NEW COMBINATION OF ORTHOGONAL POLYNOMIALS SEQUENCES

K. Ali Khelil1, A. Belkebir1 and M. C. Bouras1

1 Badji Mokhtar University, Mathematical Department, B. P. 12, Annaba 23000, Algeria E-mail: kalikhelil@gmail.com, belkebir.amel23@gmail.com, bourascdz@yahoo.fr

Abstract. In this paper, we are interested in the following inverse problem. We assume that {Pn }n^o is a monic orthogonal polynomials sequence with respect to a quasi-definite linear functional u and we analyze the existence of a sequence of orthogonal polynomials {Qn }„^0 such that we have a following decomposition Qn(x) + r„Q„-i(x) = Pn(x) + snPn-i(x) + tnPn-2 (x) + VnPn-3(x), n > 0, when vnrn = 0, for every n ^ 4. Moreover, we show that the orthogonality of the sequence {Qn}n^0 can be also characterized by the existence of sequences depending on the parameters rn, sn, tn, vn and the recurrence coefficients which remain constants. Furthermore, we show that the relation between the corresponding linear functionals is k(x — c)u = (x3 + ax2 + bx + d)v, where c,a,b,d € C and k € C \ {0}. We also study some subcases in which the parameters rn, sn, tn and vn can be computed more easily. We end by giving an illustration for a special example of the above type relation.

Key words: orthogonal polynomials, linear functionals, inverse problem, Chebyshev polynomials. AMS Subject Classification: 33C45, 42C05.

For citation: Ali Khelil, K., Belkebir, A. and Bouras, M. C. On a New Combination of Orthogonal Polynomials Sequences, Vladikavkaz Math. J., 2022, vol. 24, no. 3, pp. 5-20. DOI: 10.46698/a8091-7203-8279-c.

1. Introduction

Let P be the linear space of polynomials in one variable with complex coefficients and let P' be its algebraic dual. We denote by (u, f) the action of u in P' on f in P and by (u)n := (u,xn), n ^ 0, the moments of u with respect to the monomial sequence {xn When (u)0 = 1, the linear functional u is said to be normalized. For our work we need to recall some operations in P' (see [1, 2]). For any u in P', any q in P and any complex numbers a, b, c with a = 0, let Du = u', qu, hau, rbu and au be respectively the derivative, the left multiplication, the translation, the homothetic and the pair part of the linear functionals defined by duality:

(u,f) := (u, f'), (qu, f) := (u,qf), (rb u, f) := (u,T-bf) = {u, f (x + b)), (ha u, f) := (u,haf) = (u,f (ax)), (au, f) := (u,af) = {u,f(x2)), f € P.

The linear functional u is called regular (quasi-definite) if the leading principal submatrices Hn of the Hankel matrix H =(ui+j-)ij>0 related to the moments (u)n = (u,xn), n ^ 0, are nonsingular, for each n ^ 0 [1].

© 2022 Ali Khelil, K., Belkebir, A. and Bouras, M. C.

Definition 1.1 [1]. A sequence of monic polynomials {Pn}n^0 is called orthogonal with respect to the linear functional u if the following orthogonality conditions hold

(u, P„ (x)Pm(x)) = 0, n = m, (u,P,2(x)) = 0, n ^ 0,

where deg Pn = n, for every n ^ 0.

In this way, {Pn}n^0 satisfies the following two order recurrence relation:

Pn+i(x) = (x - ^n)Pn(x) - 7„P„-i(x), n ^ 1, Po(x) = 1, Pi(x) = x - ft,

where Yn = 0, for each n ^ 1.

Let u and v be two regular linear functionals and let {Pn}n^0 and {Qn}n^0 be the corresponding sequences of monic orthogonal polynomials. Assume that there exist nonnegative integer numbers M and N, and sequences of complex numbers {ri;n}n^0 and {sk,n}n^0 such that the structure relation

M N

Qn(x) + ^ ^ ri,ra i(x) — Pn(x) + ^ ^ si,ra Pn-i(x)

i=1 i=1

holds for n ^ 0. Further, assume that rM,M+N = 0 and sN,M+N = 0, det [aj-jM^ = 0, where the entries aj of the matrix are defined on the basis of {ri;n}n^0 and {sk;n}n^0. Then there exist two polynomials $ and ^ with deg $ = M and deg ^ = N such that

$(z)u = ^(z)v.

These polynomials $ and ^ can be constructed in an explicit way [3]. On the other hand, the converse result is also analyzed. A characterization theorem for the sequence {Qn}n^0 to be orthogonal assuming {Pn}n^0 is orthogonal is obtained when M = 0 and N = 1, M = 1 and N = 1, M = 0 and N = 2, M = 1 and N = 2, M = 0 and N = 3, M = 0 and N = k [4-8].

In this contribution, the main purpose is to analyze the inverse problem corresponding to the case M = 1 and N = 3, i. e.,

Qn(x) + rnQn-l(x) = Pn(x) + SnPn-l(x) + tnPn-2 (x) + V„P„-3(x), n ^ 0, (1.1)

with the initial conditions Q0(x) = P0(x) = 1 and Q-1(x) = P-m(x) = 0, for m ^ 1, and where {rn}n^0, {sn}n^0, {tn}n^0 and {vn}n^0 are sequences of complex numbers with the initial conditions r0 = s0 = t0 = t1 = v0 = v1 = v2 = 0 and rnvn = 0 when n ^ 4.

We provide necessary and sufficient conditions for the orthogonality of the monic polynomials sequence {Qn}n^0 assuming the orthogonality of the sequence of monic polynomials {Pn}n^0. In addition, we establish a relation between the linear functionals u and v, respectively, corresponding to MOPS's {Pn}n^0 and {Qn}n^0 as k(x-c)u = (x3+ax2+bx+d)v with a, b, c, d € C and k € C \ {0}.

This paper is organized as follows. In Section 2, we develop some basic results and lemmas. Section 3, is devoted to find the characterizations of the orthogonality of the monic polynomials sequence {Qn}n^0. Finally, we illustrate a special case of the above type relation.

2. 2-4 Type Relation

Let {Pn}n^0 and {Qn}n^0 be two sequences of monic orthogonal polynomials with respect to thej~egular functionals u and v respectively, where (u, 1) = (v, 1) = 1, let {ft}n^0, {Yn}n^1 and {ft}n^0 , {Yn}n^1 be the corresponding sequences of recurrence coefficients characterizing {Pn}n^o and {Qn}n^0 respectively. Suppose that these sequences are related by relation (1.1).

The initial conditions v3 = r3(t2 — r2(s1 — r1)) and v4r4 = 0 yield a relation between the linear functionals u and v such as

0u = ^v,

where 0 and ^ are polynomials of degree 1 and 3, respectively.

Firstly, if v3 = r3(t2 — r2(s1 — r1)) and r4 = 0, then there exists a complex number c such that

((x — c)u, Q4(x)) = 0.

Moreover, ((x — c)u, Qn(x)) =0, n ^ 4. Indeed

((x — c)u, Q0 (x)) = ft — c, ((x — c)u, Q1 (x)) = Y1 + (S1 — n)(ft — c), ((x — c)u, Q2(x)) = (S2 — r2)Y1 + (¿2 — r2(S1 — n))(ft — c), (2.1)

((x — c)u, Q3(x)) = (¿3 — r3(S2 — r2))Y1 + (v3 — ^3 (¿2 — r2(S1 — ^)))(ft — c), ((x — c)u,Q4 (x)) = (v4 — r4(t3 — r3(S2 — ^)))Y1 —^3 — ^(¿2 — ^(¿1 — n)))(ft — c).

Then there exists c such that

((x — c)u, Q4(x)) = 0.

This implies

C •= ft _ H ^4-r4(t3-r3(s2-r2)) 0 r4 v3 - r3(i2 - r2(si - ri))'

Thus,

((x — c)u, Qn(x)) = —rn((x — c)u, Qn-1(x)), n ^ 5. On the other hand [2]

, ^ ((x — c)u,Qi(x))

{x-c)u=^0^QmrQt{x)v-

Therefore, if v3 = r3(t2 — r2(s1 — r1)) and v4r4 = 0, we see that the relation between u and v is

(x — c)u = q(x)v,

where q is a polynomial of exact degree 3.

Lemma 2.1. Let {Pn}n^0 and {Qn}n^0 be two MOPSs with respect to the regular normalized linear functionals u and v respectively, where (u, 1) = (v, 1) = 1. Assume that there exist sequences of complex numbers {rn}n^0, {sn}n^0, {tn}n^0 and {vn}n^0 with the initial conditions r0 = s0 = t0 = t1 = v0 = v1 = v2 = 0, such that the relation

Qn(x) + rnQn-1(x) = Pn(x) + SnPn-1(x) + tnPn-2(x) + vnPn-3(x), n ^ 0,

with the initial conditions Q0(x) = P0(x) = 1 and Q-1(x) = P-m(x) = 0, for m = 1, 2, 3, holds, for every n ^ 0. Then the following implications hold:

1. If V3 = r3(¿2 - T2(si - ri)) and r4 = 0, then v,n = rn(in-i - rn-i(sn-2 - rn—2)), for n ^ 3, and rn = 0, for every n ^ 4. In this case the relation (1.1) reduce to 1-4 type relation

Qn(x) = Pn(x) + anPn-i(x) + № — 2(x) + CnPn— 3(x), n ^ 0

with

(2.3)

an • — sn rn, n ^ 1, bn •— tn rn(sn—1 rn— 1)j n ^ 2

Cn := Vn - rn(tn—1 - rn—i(sn—2 - rn—2)), n ^ 3.

2. If v3 = r3(t2 - r2(s1 - r1)) and r4 = 0, then rn = 0, n ^ 4.

3. If v3 = r3(t2 - r2(s1 - r1)) and v4r4 = 0, then vnrn = 0, for n ^ 4. Thus, in this case the relation (1.1) is a non-degenerate 2-4 type relation. < We have

/

(u,Q1(x)) = s1 - r1, (u, Q2 (x)) = ¿2 - r2(s1 - r1), (u, Q3(x)) = v3 - r3(¿2 - r2(s1 - r1)), x(u,Qn(x)) = -rn(u,Qn—1), n ^ 4.

If v3 = r3(t2 - r2(s1 - r1)), then we have the following cases:

i) ¿2 = r2(s1 - r1) and s1 = n.

ii) 12 = r2(s1 - r1) and s1 = n.

iii) t2 = r2(s1 - r1) and r3 = 0.

iv) 12 = r2(s1 - r1) and r3 = 0, ¿3 = 0.

v) ¿2 = r2(s1 - r1) and rзtз = 0.

See [8], in all these cases vn = 0, for n ^ 3.

1. If v3 = r3(t2 - r2(s1 - r1)) and r4 = 0, from (2.3), we have

(u,Qi(x)) = 0, i = 1,2,3, and (u,Qn(x)) = 0, n ^ 4.

So, there exists a polynomial q of degree 3 such that u = q(x)v [9]. Therefore,

Qn(x) = Pn(x) + anPn— 1(x) + № — 2(x) + CnPn— 3(x), for each n ^ 0, with cn = 0, n ^ 3. Again, the relation (1.1) leads to

sn - an + rn, n ^ 1,

tn - bn + rnan—1, n ^ ^

vn = cn + rnbn—1, n ^ 3,

and rncn—1 = 0, n ^ 4.

So, rn = 0, n ^ 4, and vn = 0, n ^ 3. Then, this case is the degenerate 1—4 type relation.

2. If v3 = r3(t2 - r2(s1 - r1)) and r4 = 0, according to (2.3), we have

(u, Q3(x)) = 0 and (u, Q4(x)) = 0,

and if rn = 0, for each n ^ 5, we get (u, Qn(x)) =0, n ^ 5. Assuminig that exists n ^ 5 such that rn = 0, putting no := min{n € N/n ^ 5, rn = 0}, then

where an,n0—1 = 0, n ^ n0 - 1.

Taking into account (1.1), this is not possible. Thus rn = 0, n ^ 4.

3. If v3 = r3(t2 - r2(s1 - r1)) and r4v4 = 0, then there exists a constant c such that

with q a polynomial of degree 3 and by (1.1), we can write for n ^ 0

((x - c)u, Qn (x)Qn-i(x)) = ((x - c)u, (Pn (x) + S,nPn-l(x) + tnPn-2(x) + ...

+ vnPn— 3(x))Qn-4(x)) - Tn{(x - c)u,Qn—1(x)Qn—4(x))

= Vn( u, P^—3(x)) - Tn( (x - c)u, Qn—l(x)Qn—4(x)).

Consequently,

Vn{ u, Pn—3(x)) = <(x - c)u, (Qn (x) + TnQn—l(x))Qn—4(x)> = (q(x)v, (Qn(x) + TnQn— l(x))Qn—4(x)) = t^v, q(x)Q,n—i(x)Q.n—4(x)) = &iTn(v,Qn—i(x^,

where k1 is the leading coefficient of the polynomial q. Now, it is enough to apply (2) to obtain Tn = 0, n ^ 4, and from Definition 1.1, we have vnTn = 0, n ^ 4. >

In the following proposition, we show that if v3 = r3(t2 - T2(s1 - T1)) and r4v4 = 0 this equivalence to assume that the functional (x - c)u is regular.

Proposition 2.1. Let {Pn}n^0 and {Qn}n^o be two MOPS with respect to the regular normalized linear functionals u and v respectively, such that the relation (1.1) holds and the initial conditions v3 = r3(t2 - r2(s1 - r1)) and r4v4 = 0 hold. Then the following statements are equivalent:

i) The functional (x - c)u is regular.

ii) vn = Tn (tn— 1 - Tn— 1(Sn—2 - Tn—2)), n ^ 3.

< Multiplying the relation (1.1) by Pn—1 and applying u, the same way for Pn—2 and

Pn—3, we get, respectively,

(u,Qn(x)Pn—2(x)) = (tn - Tn(sn—1 - Tn—1))(u, P^— 2(x)>, n ^ 2, (u, Qn (x)Pn—3(x)) = (vn - Tn(tn—1 - Tn—1(Sn—2 - Tn— 2))) (u, P^—3^)), n ^ 3.

Thus

vn Tn(tn — 1 - Tn— 1(Sn— 2 - Tn—2)) =0 ^ (u, Qn(x)Pn—3(x)} =0, n ^ 3. It is well known that (x - c)u is regular if and only if Pn(c) = 0, for all n ^ 0 [9].

(u, Qn(x)} = 0, n ^ n0, and (u, Qn(x)} =0, 3 ^ n ^ n0 - 1.

k=1

(x - c)u = q(x)v

(u, Qn (x)Pn— 1 (x) ) = (Sn - Tn)( u,Pn—1(x)), n ^ 1,

Moreover, we need to show that (u, Qn+3(x)Pn(x)) = 0 ^ Pn(c) = 0, for each n ^ 0. Either for

Pn(x) = (x - c)k, n ^ 0,

k=0

with an0 = Pn(c) and ann = 1.

The relation between the regular functionals u and v is

(x — c)u = q(x)v,

Hence

(u, Qn+3(x)P„(x)) = ((x — c)u, (x — c)n Qn+3(x))

n— 1

+ ^ an^(x — c)u, (x — c)k—1Q„+3(x^ + Pn(c)(u, Qn+3(x)) k=i

n— 1

= (v, q(x)(x — c)n—1Qn+3(x^ + ^ a™( v, q(x)(x — c)k—1Qn+3(x)>

k=i

+ Pn(c)(u,Qn+3(x)), n ^ 0.

Then

(u,Qn+3(x)Pn(x)) = Pn(c)(u,Qn+3(x)), n ^ 0, from Lemma 2.1 and the relation (2.3), we get

(u,Qn(x)) = 0, n ^ 3. >

3. Characterization of Orthogonality

Let {Pn}n^0 be a MOPS with respect to a regular functional u and let {ft}n^0, (7n}n^1 be the corresponding sequences of recurrence coefficients, so that

Pn+1 (x) = (x — ft)Pn(x) — YnPn— 1(x), n ^ 0, (3.1)

with the initial conditions P0(x) = 1, P—1(x) = 0 and the condition Yn = 0, for each n ^ 1.

In this section, we give the characterizations of the orthogonality of a sequence {Qn}n^0 of a monic polynomials defined by a non-degenerate type relation (1.1).

From Lemma 2.1, the conditions v3 = r3(t2 — r2(s1 — r1)) and r4v4 = 0 must hold, in order to have a non-degenerate 2-4 type relation with {Pn}n^0 and {Qn}n^0 MOPS and these conditions imply vnrn = 0, for each n ^ 4.

The following if the first characterization of the orthogonality of the sequence {Qn}n^0.

Proposition 3.1. Let {Pn}n^0 be a MOPS satisfies (3.1), and let {Qn}n^0 be a sequence of polynomials given by the structure relation (1.1) with v3 = r3(t2 — r2(s1 — r1)) and vnrn = 0 for n ^ 4. Then, {Qn}n^0 is a MOPS with recurrence coefficients {ft }n^0 and {7n}n^1; where

ft := ft — Sn+1 + Sn + rn+1 — rn, n ^ 0, (3.2)

ftn •— Tn + tn ^n+1 + Sn(sn+1 sn ^n + ^n—1)

(3.3)

— rn(rn+1 — rn — ft + ft—1), n ^ 1,

if and only if ft1ft2ft3 = 0, and the following relations hold:

anSn—1 = SnYn— 1 + tn(sn+1 — Sn — ft + ft—2) — Vn+1 + Vn, n ^ 2, (3.4)

antn—1 = tnYn—2 + Vn(sn+1 — Sn — ft + ft—3), n ^ 3, (3.5)

anvn— 1 = VnYn—3, n ^ 4, (3.6)

anrn—1 = rnftn—1, n ^ 2, (3.7)

where

an := Yn + tn — tn+1 + Sn(Sn+1 — Sn — ft + ft—1), n ^ 1. (3.8) < Substituting (3.1) in (1.1), for all n ^ 0, we have Qn+1(x) = xPn(x) —(^n — Sn+1)Pn(x) —(Yn — tn+1)Pn—1(x)+Vn+1Pn—2(x)—rn+1Qn(x). (3.9)

Applying (1.1) to xPn(x), and substituting the recurrence relation (3.1) into (3.9) for xPn— 1(x), xPn— 2 (x) and xPn—3 (x), we obtain for n ^ 0

Qn+1(x) = (x — rn+1)Qn(x) + rn(xQn— 1(x)) — (ft — Sn+1 + Sn )Pn(x) — (Yn + tn — tn+1 + Snft— 1)Pn— 1(x) (snYn—1 + tn^n — 2 + Vn — Vn+1)Pn— 2(x)

3 + tnYn—2)Pn—3(x) VnYn—3Pn— 4(x).

Using relation (1.1) for Pn, and with convention P—n(x) = 0, n ^ 1, the above relation becomes, for n ^ 0

Qn+1(x) = (x — ft)Qn(x) — rn(Qn — xQn—1) — rn(ft + Sn — Sn+1)Qn—1(x)

[Yn + tn tn+1 Sn(/n ^n—1 + Sn Sn+1)] Pn—1(x) — [SnYn—1 — tn(ft — ft—2 + Sn — Sn+1) + Vn — Vn+1] Pn—2 (x) [tnYn—2 — Vn(/n — ft— 3 + Sn — Sn+1)]Pn—3(x) — VnYn— 3Pn—4 (x),

where ft is given by (3.2). Then using again (1.1) for Pn—1, we get for n ^ 0

Qn+1(x) = (x — ft)Qn(x) — ftnQn— 1(x)

— rn [Qn(x) — (x — ft—1)Qn—1(x) + ftn—1Q

n— 2 (x) (anrn—1 rnftn—1 n— 2(x) — [SnYn—1 — tn(/n — ft—2 + Sn — Sn+1) + Vn — Vn+1 — «nSn— 1] Pn— 2(x) (3.10)

— [tnYn—2 — Vn(/n — /n—3 + Sn — Sn+1) — antn—1] Pn—3(x) — (VnYn—3 — anVn— 1)Pn—4(x),

where ftn and an are given by (3.3) and (3.8).

Hence from (3.10), {Qn}n^0 be a MOPS if and only if ftn = 0, for n ^ 1, and the conditions (3.4)-( 3.7) hold.

Suppose that {Qn}n^0 is a MOPS, then the relation (3.10) is equivalent to

(fn — anrn— 1)Qn—2(x) = (bn

anSn— 1)Pn—

2(x) + (Cn — antn— 1)Pn— 3(x) + (e n anVn—1 )Pn—4(x), n ^ 2,

where

bn := SnYn—1 + tn(Sn+1 — Sn —/n + ft—2) —Vn+1 + Vn, n ^ 2, (3.12)

cn := tnYn—2 + vn(Sn+1 - Sn - Pn + Pn—3), n ^ 3, (3.13)

en := vnYn—3, n ^ 4, (3.14)

fn := Tn7n—1, n ^ 2. (3.15) Moreover, since v3 = t3 - t3(s2 - r2) and rn = 0, for n ^ 4, then by (2.3), we deduce

(u, Qn} = 0, n ^ 3.

Applying u to both sides of (3.11), we get

(fn - Tn—1an)(u,Qn—2} = 0, n ^ 5.

This leads to

fn = T,n—1(an, n ^ 5. Multiplying (3.11) by Pn—2, Pn—3 and Pn—4 and applying u, we obtain for all n ^ 5

bn - anS'n—1l cn - antn—1, en - anvn—1, fn - anTn—1

comparing coefficients in both sides of (3.11), for n = 2, 3 and 4, we obtain

b2 - f2 = a2(S1 - T1), (3.16)

b3 - f3 = a3(s2 - T2), (3.17)

c3 - 63 (S1 - T1) = a3(t2 - S2(S1 - T1)), (3.18)

64 - f4 = a4(s3 - T3), (3.19)

c4 - 64(S2 - T2) = a4(t3 - S3(S2 - T2)), (3.20)

e4 - 64(t2 - T2(S1 - T1)) = a4[v3 - S3(t2 - T2(S1 - T1))]. (3.21) Conversely, if (3.11) is satisfied and pn = 0, for every n ^ 1, we have

Qn+1 (x)-(x-Pn)Qn(x) + 7nQn— 1(x) = -Tn [Qn(x)-(x-j3n—1)Qn—1(x)+ 7n—1Qn—2(x) , n ^ 1. Moreover, from (1.1), we obtain

Q1(x) = P1(x) + S1 - T1 = x - Po + S1 - T1 = x - /3o, we deduce recursively

Qn+1(x) = (x - Pn)Qn(x) - 7nQn— 1(x), n ^ 1.

Thus, {Qn}n^0 is a MOPS with recurrence coefficients {Pn}n^0 and {pn}n^1. >

Now, we show that the orthogonality of the sequence {Qn}n^0 can be also characterized by the fact that there are four sequences depending on the parameters rn, sn, tn, vn and the recurrence coefficients which remain constants.

Theorem 3.1. Let {Pn }n^0 be a MOPS and let {Qn}n^0 be a sequence of polynomials given by (1.1). Then the following statements are equivalent:

(i) {Qn}n^o is a MOPS with recurrence coefficients {Pn}n^0 and {pn}n^1 given by (3.2) and (3.3) .

(ii) It holds p1p2p3 = 0 together with initial conditions (3.16)-(3.21) and

v5Y2 = v4(Y5 + t5 - to + S5(S6 - S5 - P5 + P4)) (3.22) and there exist four complex numbers A, B, C and D such that, for each n ^ 4

A = ]3n — rn+\ — —, (3.23)

Tn

B = — 7„_2 + sn+1 - P„ - fin-i - Pn-2, (3.24)

vn

C =— 7ra-l7ra-2 + (— 7n-2 + Sn+1 - Pn-2) (Sn+1 - Pn - Pn-l)

vn ^vn ' (3.25)

- Sn+1(sn+2 - Pn+1) + pn—1pn - Yn+1 - Yn - Yn—1 + tn+2,

(3.26)

n YnYn—1Yn—2 . sn , a x

-D =--1--7n-l7n-2(Sn+l - Pn)

vn vn

+ — 7ra-2 [sra+l(sra+i — Sn+2 + Pn+l — Pn — Pn-l) + (3n(3n-i — — 7„ + ¿„+2]

vn

+ (sn+2 - Sn+1 - pn+1 + Pn— 1 + pn—2)(an+1 - tn+1) + (Sn+1 - Pn)(Pn — 1 Pn—2 - Yn— 1) - (Sn+1 - Pn—2)Yn + vn+2.

< Observe that the conditions (3.4)-(3.7) in the Proposition 3.1 may be written as, for each n ^ 5

vn

sn-1 —^ 7n-3 = sn7„_i + i„(sra+i - sra - P„ + pra_2) - vn+i + vn, (3.27)

vn—1

vn

tn-1-7n-3 = tnjn-2 + ^n(sn+i — sra — P„ + Pri-3), (3.28)

vn—1

vn

Jn-3 = In + tn - tn+l + sn(sn+i - sn - Pn + Pn-l), (3.29)

Vn-1

rn

7n-i = In + rn(rn+1 - ^n - Pn + Pn-l), (3.30)

Tn—1

moreover {Qn}n^0 is a MOPS if and only if the conditions p1p2p3 = 0, the initial conditions (3.16)-(3.21) and the above equations ( 3.27), (3.28), (3.29) and (3.30) hold. Firstly, we show that (3.27)-(3.30) ^ (3.22)-(3.26). For n = 5 in (3.29), yields (3.22).

From (3.30), dividing the left and the right hand sides by rn, we get

Pn - rn+i - — = Pn-l - rn - '-, (3.31)

Tn Tn— 1

Hence (3.23) holds.

Now, we will deduce (3.24).

Using (3.28), dividing the left and the right hand sides by vn, we obtain — 7n-2 + Sn+1 "Pn "Pn-l "Pn-2 = !"7n-3 +Sn ~ Pn-l ~ Pn-2 ~ Pn-3, U ^ 5. (3.32)

vn vn—1

Hence (3.24) holds.

Next, we will deduce (3.25).

From (3.27), multiplying the left and the right hand sides by we obtain

sn-1 I / о о \ -7„-27n-3 H--7n-2(Sn - ft-l - ft-2)

s -1 t Vn v (3'33)

= — 7n-l7ra-2 + — 7n-2(sra+l - ft - ft-l) + (l ~ jYra-2,

v v v

taking into account (3.29) for n + 1 instead of n, for each n ^ 5, we have

'- 7n-27n-3 + — 7n-2(sra - ft-l - ft-2) = — Yra-lYra-2 + — 7n-2(sra+l - ft - ft-l) v -1 v v v

+ Yn-2 — Yn+1 — ^+1 + ^+2 — sn+1(s n+2 — sn+1 — ft+1 + ft)) using (3.24), in the above expression, for each n ^ 5, we get

— Yn-lYn-2 + (— 7n-2 + Sn+1 - ft_i - ft_2) (sra+i - ft - ft-l)

vn vn

— Yn+1 — Yn — Yn-1 + tn+2 — sn+1 (sn+2 — ft+1 — ft-0 — ftUl

= ^ Yn-2Yn-3 + f^—- Yra-3 + Sn - ft-2 - ft-з) (sra - ft-l - ft-2) Vn-1 VVn-1 /

— Yn — Yn-1 — Yn-2 + tn+1 — sn(sn+1 — ft — ft-2) — en-2.

Lastly, we will deduce (3.26).

From (3.29), multiplying the left and the right hand sides by ; we obtain

vn

Yn-1Yn-2Yn-3 . sn /ал

--1--Yn-l7n-2(Sn - ft-l)

vn- 1 vn

Yn Yn — 1 Yn—2 sn f q \ f tn tn+1 \

(3.34)

+ — Yn-l7n-2(sra+l - ft) + (— - W-lYra-2 vn vn vn

taking into account (3.28) for n + 1 instead of n, for each n ^ 5, we have

Yra-lYn-2Yra-3 . S„ YnYn-lYn-2 , S„ / ^ x

--1--Yn-l7n-2(Sn - ft-l) =--1--Yn-l7n-2(Sn+l - ft)

vn-1 vn vn vn

tn

+ — 7»г—2 (Тга—1 ~~ 7n+l — in+l + ¿ra+2 — Sra+l(sra+2 — Sn+\ — ft+1 + ft)) (3.35)

vn

+ (sn+2 — sn+1 — ft+1 + ft-2) (Yn+1 + tn+1 -tn+2 + sn+1 (sn+2 — sn+1 — ft+1 + ft)) , using (3.33) and (3.1), we obtain

7ra7ra-l7ra-2 /£„ vn vn

+ Yn-lYn-2 - Yn-l) (Sra+l - ft)

vn

+ — 7га-г( — "in — 7ra+l + ira+2 ~ Sra+l(sra+2 — Sra+i — ft+1 + ft + ft-l) + ftft-l)

vn

+ (sn+2 — sn+1 — ft+1 + ft-2 — (sn — ft-0) (Yn+1 + tn+1 — tn+2 + sn+1(sn+2 — sn+1 -ft+l+ft)) = 7n.-l7ra-27n.-3 + iSn-l 7ra_27ra_3 _ 7ri_2VSra_^ri_1)_(Sri_^ri_3)7ri_:|

Vn-1 Vvn-1 '

H--Yra-3 ( — Yra-1 — Yra + ¿ra+1 sn(sn+1 sn ^n + ^n- 1 + en-2) + ft-1ft-2)

vn-1

+ (— sn+1 + s n + ^n ^n- 3) [— Yn + tn+1 sn(sn+1 sn ^n + ^n- 1 + en-2)+ ft-1 ft-2] ;

taking into account (3.8) and (3.12) in the above expressions and by straightforward computation, for each n ^ 5, we get

YnYn—1Yn—2 Sn , a , --1--7n-l7n-2(Sn+l - Pn)

Vn Vn

+ — Yra-2 [Sn+l(sra+i — Sn+2 + /3n+l ~ Pn ~ Pn-1) + PraPra-1 ~ Yra+1 ~ In + tn+2]

Vn

+ (sn+2 — Sn+1 — Pra+1 + Pn—1 + Pra—2)(an+1 — tn+1) + (sn+1 — Pn)(Pn—1pn—2 - Yn—1)

- (Sn+1 - Pn-2hn + Vn+2 = 7"~l7"'~27"'~3 + '" Yn-2Yn-3(Sn ~ Pra-l) (3'36)

°n yn-

Vn-1 Vn-1

t'n—1

+ —-Yra-3 [sra(sra — Sn+1 + Pn — Pra-1 — Pra-2) + Pra-lPra-2 ~ In ~ Yra-1 + ¿ra+l]

Vn-1

+ (Sn+1 - sn Pn + Pn— 2 + Pn—3)(an - tn) + (Sn - Pn—l)(Pn—2Pn—3 - Yn—2)

- (Sn - Pn—3)Yn— 1 + Vn+1.

Secondly, we show that (3.23)-( 3.26) ^ (3.27)-(3.30).

Notice that, the relations (3.27)-(3.30) are equivalent to (3.4)-(3.7) and the relations (3.23)-(3.26) are equivalent to (1.1)-(3.36) then, it is enough to show (1.1)-(3.36) ^ (3.27)-(3.30).

From (1.1), multiplying the left and the right hand sides by rn, yields (3.30). From (3.1), multiplying the left and the right hand sides by vn, yields (3.28). From (3.35), we have

Yra-1Yra-2 Vn

Vn

TVi—3 ^ïn tn + ¿n+1 + sn(sn - Sn+1 - Pn—1 + Pn)

vn—1

— ^n [ — tn(Xn+1 + ¿ra+lYra-1 + Vn+\{sn+2 ~ S-n+l ~ Pn+l + Pn-2)]

vn

+ (Sn+2 — Sn+1 — Pn+l + Pn-2) C--Yra—2 + Yra+1 + tn+1 — tn+2

\ vn

+ Sn+1(sn+2 - Sn+1 - P'n+1 + P'n^ )

using (3.5), for n + 1 instead of n, we get

' y'—3 yn tn + ¿n+1 + sn(sn - Sn+1 - Pn—1 + Pn)

■ Yn—2 - Yn+1 - ¿n+1 + ¿n+2

Yn—1Yn—2 vn

vn—1

vn+1

vn

= (sn+1 - Sn+2 - Pn—2 + Pn+1)

Vn

+ Sn+1(sn+1 — Sn+2 — Pn + Pn+1)

according to (3.22) and as v4 not equal to 0, we have V5

— Y2 - Y5 - h + ¿6 + «5(55 - S6 - PA + P5) = 0,

v4

then

Vn

—^ Yra-3 - Yra - tn + Wl + Sra(«ra ~ Sn+1 - Pra-1 + Pra) = 0, U ^ 5. Vn—1

Hence (3.29) holds.

Taking into account the expression of ^Yra-3, obtained from (3.1), the relation (3.34) rewrites as

— Yra-l7ra-2 + — Yra-2(Sra+l ~ ft ~ ft-l) ~ 7ra+l ~ In ~ Yra-1 + W2

vn vn

— sn+1(sn+2 — sn+1 — ft+1 + ^ra) + (ft + ft-1 — sn+1)(ft-1 + ft^ — ^-1

= '" 7n-27n-3 + (— 7n-2 + Sn+1 - Sn - ft + ft-3) (Sn ~ ft-1 ~ ft-2) Vn-1 /

— 7n — 7n-1 — 7n-2 + ¿n+1 — sn(sn+1 — sn — ft + ft-1)

+ (ft-1 + ft-2 - Sn)(ft-2 + ft-3) - /¡-2,

hence

Sn— 1 Sn I tn / Q . Q \

-Yn-2Yra-3 = -7n-l7ra-2 H--7ra-2(Sra+l - Sn - ft + ft_2)

vn-1 vn vn

+ Yn-2 — (Yn+1 + tn+1 — in+2 + sn+1(sn+2 — sn+1 — ft+1 + ft^

using (3.29) for n + 1 instead of n and simplifying, yield (3.27). >

In the following theorem, we observe that there is a relation between regular linear functionals when {Qn}n^0 is a MOPS with respect to a regular linear functional v.

Theorem 3.2. Let {Pn}n^0 be a MOPS with respect to a regular linear functional u and the sequence of monic polynomials {Qn}n^0 be given by the relation (1.1). If {Qn}n^0 is a MOPS with respect to a regular linear functional v, then

k(x — c)u = (x3 + ax2 + bx + d)v (3.37)

with c, a, b, d € C and k € C \ {0} and the normalizations for these linear functionals <u, 1) = <v, 1) = 1.

< Applying the regular linear functional u corresponding to the MOPS {Pn}n^0 in (1.1), we obtain, for each n ^ 4

<(x — c)u, Qn(x)) = 0.

Then, according to [2] taking into account the relation (1.1), we expand the linear functional u in terms of the dual basis { li>o MOPS {Qn}n^o as

, ^ V^ <(x — c)u,Qi)

i=0

Since {Qn}n^0 is a MOPS with respect to v, the recurrence coefficients {ft}n^0 and {Yn}n^1 are given by (3.2) and (3.3). Moreover

Indeed, making both sides of (3.37) acting on the polynomials Q0, Q1, Q2 and Q3, and taking into account (2.1), we get

k(ft — c) = /0 + (2/50 + /31)71 + (71 + /3(2) a + ftb + d, (3.39)

k [71 + (/0 — c)(s1 — n)] = (/502 + /2 + /50/51 + 71 + 72)71 + a(ft + /1)71 + 71b, (3.40)

fc{7l(«2 - + - c)[t2 - r2(si - ri)]} = (ft + ft + ft + ^7172, (3.41) &{7l[t3 - r3(S2 - r2)] + (ft - c)[v3 - r3(¿2 - r2(Si - ri))]} = 7i7273, (3.42)

where 71, 72, 73 are given by (3.2) and (3.3).

Using the relations (3.39)-(3.42) and taking into account (2.2), thus, the values of c, a, b, d and k are given as follows

k _ r4 717273 c-p0 7i V4 ~ r4(t3 ~ r3(s2 - r2)) v4 7i ' r4v3- r3{t2 ~ r2(si - n)) '

73 r4V3(S2 - r2) + (V4 - ^3)^2 - ^(Si - ri)]

a = -ft - /5i - /52 +

b = ftft+ftft+ftft - 7i - 72 -+

d = -ftftft + /072 + /27i - /0 +

V4 V3 - r3(¿2 - r2(Si - ri))

(ft + ft)73 r4v3{s2 - r2) + {v4 - r4t3)[t2 - r2{si - n)] v4 v3 - r3{t2 - r2{si - n))

7273 ^4(^3 - ^¿2) + (v4 - r4{t3 - r3s2)){si - n) v4 v3 - r3{t2 - r2{si - n))

7273 r4(V3 - r3Î2) + (V4 - ^(¿3 - r3«2))(si - ri)

V4 V3 - r3(¿2 - r2(Si - ri))

(ftft - 7i)73 r4V3(S2 - r2) + (V4 - ^3)^2 - r2(si - ri)]

V4 V3 — r3(t2 — r2(S1 — n))

717273 ~ r4(i3 ~ r3{,S2 ~ T2)) ^ ^4 V3 -T3{t2 -r2{si -ri))'

Remark 3.1. The constants A, B, C and D appearing in the Theorem 3.2 are, respectively, the coefficients c, a, b and d of the polynomial which relate the two regular linear functionals.

4. A Particular Case

In this section, we will discuss a special case of relation (1.1).

Let us consider the symmetric MOPS {Pnthis means that ft = 0, for each n ^ 0. From Proposition 3.1, the equations (3.4), (3.5), (3.6) and (3.7) become, for each n ^ 5

I ¿n- i ¿n / A 1\

Sn+l=SnH--7n-3--7n-2, (4.1)

vn-i vn

V

tn+1 =tn + Jn--— 7ra-3 + Sn(sn+1 - sra), (4.2)

vn-i

Vn+i = Vn + SnYn-i + ¿n(Sn+i - Sn) - Sn-i [7n + ¿n - ¿n+i + Sn(Sn+i - Sn)] , (4.3)

= + 4. (4.4)

Vn— i rn vn rn+i

The equations (3.2) and (3.3) become

ft = Sn - Sn+i + rn+i - rn, n ^ 0,

7n - 7n + ¿n ¿n+i + Sn(Sn+i Sn) rn(Sn+i Sn + 7n— i ) , n ^ 1

for each n ^ 5, we have

à tn tn—1 Vn Yn-3 Vn+1 Yn—2 Pn = — 7n-2--7n-3 +

n

vn Vn— 1 Vn— 1 rn vn rn+1

Yn-2 f _ Vn+A _ 7»-3 f _Vn I tn I I tn—i

Vn V rn+i) Vn—i\ rn

(4.5)

~ Vn ( tn-1 tn \ 7 , .

7n = -7n-3 - rn -7ra-3--7n-2 - rnpn-i. (4.6)

Vn-1 \Vn-l Vn J

In this case, we treat the following three subcases.

i) If sn = si and rn_i = ri, for each n ^ 5, from (4.1) and (4.4), we obtain

tn tn— i ¿3

— 7n-2 =-7n-3 = ... = — 71,

Vn Vn—i V3

Vn+i Vn V4

■ 7n-2 =-7n-3 = • • • = —71,

the relation (4.6) yields

Vn Vn—1 V3

7n = -7i, 5. (4.7)

V3

We conclude that /3n = 0 and 7n are constants, for each n ^ 5. From (4.2), we have

V4

Wi = i«+7n--7i- (4-8)

V3

ii) If sn = si, rn—i = ri and tn = t2, for each n ^ 5, from (4.8) and (4.7), we get

7n = Yn, n ^ 5.

The coefficients Yn are constants, for each n ^ 5, then (Pn}n^0 is the sequence of anti-associated polynomials of order 5 for the Chebyshev polynomials of the second kind [10].

iii) If rn—i = ri, sn = si, tn = t2 and Vn = V3, for each n ^ 5, from (4.3), we have

Vn+i = Vn + snYn—i - sn— iYn, n ^ 5,

hence

Vn+i = Vn + si(Yn—i - Yn), n ^ 6, it is clear that si(Yn—i — Yn) = 0, for all n ^ 6.

Remark 4.1. If rn— i = ri, sn = si and tn = t2 or rn—i = ri, sn = si, tn = t2 and Vn = V3, for each n ^ 5, then

7n = 0, n ^ 5,

7n = Yn = Y5, n ^ 5.

Example 4.1. Let (Pn}n^0 be the sequence of monic Chebyshev polynomials of the second kind orthogonal with respect to the weight function W(x) = (1 — x2)i/2 on (—1,1). Then /3n = 0, n ^ 0, Yn = n ^ 1, and the relations (4.1), (4.2), (4.3) and (4.4), for each n ^ 5, become

- , 1 (tn-1 tn\ _ 1 / _ _Vn_\ , _ .

Sn-\-1 — Sn ~r I I , tra+1 — n A \ / ^rav^ri+1 Sn),

4 \ Vn— i / 4 \ Vn— i /

I

Wn+1 — + — Sn ¿riv^ra+l 1

_ 1 f Vn-i

1"n - fn—1 I

1

^ "I" tn tn-\- \ Sriv^ra+1 Sn)

4 Vrn—lVn—2 rnVn—1

V

n

Assume that rn-i = ri, Sn = Si and ¿n = ¿2, for each n ^ 5, we obtain

1 .

vn+i = vn + - (sn - sn-i), n ^ 5,

in particular,

Vn+i = Vn, n ^ 6.

In this situation, we deduce constant connection coefficients, for n ^ 6.

Acknowledgments. The authors like to thank the referee for their careful revision of the manuscript. Their comments and suggestions have substantially improved the presentation of the manuscript.

References

1. Chihara, T. S. An Introduction to Orthogonal Polynomials, New York, Gordon and Breach, 1978.

2. Maroni, P. Une Theorie Algébrique des Polynômes Orthogonaux. Application aux Polynômes Orthogonaux Semi-Classiques, Orthogonal Polynomials and their Applications, IMACS Annals on Computing and Applied Mathematics, eds. C. Brezinski et al., vol. 9, Basel, Baltzer, 1991, pp. 95-130.

3. Petronilho, J. On the Linear Functionals Associated to Linearly Related Sequences of Orthogonal Polynomials, Journal of Mathematical Analysis and Applications, 2006, vol. 315, no. 2, pp. 379-393. DOI: 10.1016/j.jmaa.2005.05.018.

4. Alfaro, M., Marcellán, F., Pena, A. and Rezola, M. L. On Linearly Related Orthogonal Polynomials and their Functionals, Journal of Mathematical Analysis and Applications, 2003, vol. 287, no. 1, pp. 307-319. DOI: 10.1016/S0022-247X(03)00565-1.

5. Alfaro, M, Marcellán, F., Peña, A. and Rezola, M. L. On Rational Transformations of Linear Functionals: Direct Problem, Journal of Mathematical Analysis and Applications, 2004, vol. 298, no. 1, pp. 171-183. DOI: 10.1016/j.jmaa.2004.04.065.

6. Alfaro, M, Marcellán, F., Peña, A. and Rezola, M. L. When Do Linear Combinations of Orthogonal Polynomials Yield New Sequences of Orthogonal Polynomials, Journal of Computational and Applied Mathematics, 2010, vol. 233, no. 6, pp. 1446-1452. DOI: 10.1016/j.cam.2009.02.060.

7. Alfaro, M., Peña, A., Rezola, M. L. and Marcellan, F. Orthogonal Polynomials Associated with an Inverse Quadratic Spectral Transform, Computers and Mathematics with Application, 2011, vol. 61, no. 4, pp. 888-900. DOI: 10.1016/j.camwa.2010.12.037.

8. Alfaro, M., Penña, A., Petronilho, J. and Rezola, M. L. Orthogonal Polynomials Generated by a Linear Structure Relation: Inverse Problem, Journal of Mathematical Analysis and Applications, 2013, vol. 401, no. 1, pp. 182-197. DOI: 10.1016/j.jmaa.2012.12.004.

9. Kwon, K. H., Lee, D. W., Marcellan, F. and Park, S. B. On Kernel Polynomials and Self-Perturbation of Orthogonal Polynomials, Annali di Matematica Pura ed Applicata, 2001, vol. 180, no. 2, pp. 127-146. DOI: 10.1007/s10231-001-8200-7.

10. Ronveaux, A. and Van Assche, W. Upward Extension of the Jacobi Matrix for Orthogonal Polynomials, Journal of Approximation Theory, 1996, vol. 86, no. 3, pp. 335-357.

Received March 31, 2021 Karima Ali Khelil

Badji Mokhtar University, Mathematical Department,

B.P. 12, Annaba 23000, Algeria,

Lecturer

E-mail: kalikhelil@gmail.com

https://orcid.org/0000-0002-1032-3837

Amel Belkebir

Badji Mokhtar University, Mathematical Department, B. P. 12, Annaba 23000, Algeria, PhD Student

E-mail: belkebir. amel23@gmail. com https://orcid.org/0000-0002-7614-0187

Mghamed Cherif Bouras

Badji Mokhtar University, Mathematical Department,

B.P. 12, Annaba 23000, Algeria,

Professor

E-mail: bourascdz@yahoo.fr

https://orcid.org/0000-0001-7302-8053

Владикавказский математический журнал 2022, Том 24, Выпуск 3, С. 5-20

О НОВОЙ КОМБИНАЦИИ ПОСЛЕДОВАТЕЛЬНОСТИ ОРТОГОНАЛЬНЫХ ПОЛИНОМОВ

Али Хелил К., Белькебир А., Бурас М. Ш.1 1 Университет Баджи Мохтар, Алжир, 23000, Аннаба E-mail: kalikhelil@gmail.com, belkebir.amel23@gmail.com, bourascdz@yahoo.fr

Аннотация. В настоящая статья посвящена следующей обратной задаче. Для последовательность полиномов от одной переменной {Pn }n^o, ортогональных относительно квазиопределенного линейного функционала u, выяснить условия существования последовательности ортогональных полиномов {Qn }n^o, для которых имеет место разложение Qn(x) + rnQn-i(x) = Pn(x) + snPn-i (x) + tnPn-2 (x) + vnPn-3 (x), n ^ 0, где vnrn = 0, для всех n ^ 4. Показано, что ортогональность последовательности {Qn }n^0 характеризуется существованием последовательностей, зависящих от параметров rn, sn, tn, vn и постоянных рекуррентных коэффициентов. Кроме того, установлено, что соотношение между соответствующими линейными функционалами имеет вид k(x — c)u = (x3 + ax2 + bx + d)v, где c,a,b,d € C and k € C \ {0}. Рассмотрены также подклассы для которых параметры rn, sn, tn и vn легко вычисляются. В конце приводятся иллюстрирующие примеры.

Ключевые слова: ортогональный полином, линейный функционал, обратная задача, полиномы Чебышева.

AMS Subject Classification: 33C45, 42C05.

Образец цитирования: Ali Khelil, K., Belkebir, A. and Bouras, M. C. On a New Combination of Orthogonal Polynomials Sequences // Владикавк. мат. журн.—2022.—Т. 24, № 3.—C. 5-20 (in English). DOI: 10.46698/a8091-7203-8279-c.