BESSEL POLYNOMIALS AND SOME CONNECTION FORMULAS IN TERMS OF THE ACTION OF LINEAR DIFFERENTIAL OPERATORS Текст научной статьи по специальности «Математика»

URAL MATHEMATICAL JOURNAL, Vol. 8, No. 2, 2022, pp. 4-12

DOI: 10.15826/umj.2022.2.001

BESSEL POLYNOMIALS AND SOME CONNECTION FORMULAS IN TERMS OF THE ACTION OF LINEAR DIFFERENTIAL OPERATORS

Baghdadi Aloui

University of Gabes, Higher Institute of Industrial Systems of Gabes Salah Eddine Elayoubi Str., 6033 Gabes, Tunisia [email protected]

Jihad Souissi

University of Gabes, Faculty of Sciences of Gabes

Erriadh Str., 6072 Gabes, Tunisia [email protected] [email protected]

Abstract: In this paper, we introduce the concept of the Ba-classical orthogonal polynomials, where Ba is the raising operator Ba :— x2 ■ d/dx + (2(a — 1)x + l)I, with nonzero complex number a and I representing the

identity operator. We show that the Bessel polynomials B£a)(x), n > 0, where a — —m/2, m > —2, m € Z, are the only Ba-classical orthogonal polynomials. As an application, we present some new formulas for polynomial solution.

Keywords: Classical orthogonal polynomials, Linear functionals, Bessel polynomials, Raising operators, Connection formulas.

1. Introduction

Let {BÎa) }n>0 be the monic Bessel polynomial sequence. It satisfies the following explicit expression [10, 23]

for a = -m/2, m € N. To complete the definition, B(a)(0) is set equal to

R(°)ffV> r(w + 2a - 1)

B» (0)=2 r(2n + 2a-l)' (1J)

It is well known that the monic Bessel polynomial sequence is classical and satisfies the following relations [8, 10, 16, 23]:

-The Second-Order Differential Equation (SODE)

x2B(*)"(x) + 2{ax + l)B{ay(x) = n(n + 2a - 1)B{a)(x), n > 0. (1.3)

-The Lowering Relation (LR)

DB(a)(x) = nB(-11)(x), n > 1, (1.4)

where D := d/dx is the standard derivate operator.

After a simple calculation, the SODE can be written for n > 0 as follows

(x2^'(x))' + (2((a - 1)x + i)Bna)(x^' = (n + 1)(n + 2a - 2)5^(x). (1.5)

Using the LR (1.4), the equation (1.5) becomes for n > 0

(x2^'(x) + 2((a - 1)x + 1)Bia)(x^' = (n + 2a - 2)£n+-1)'(x). Using the primitive of the last equation, we get

x2B^a)' (x) + 2((a - 1)x + 1)B^a)(x) = (n + 2a - 2)Bn+"11) (x) + K, with (a = -m/2, m > -2, m € Z), and where, using (1.2), we have

K = 2B(a) (0) - (n + 2a - 2)B(+-11) (0) = 0.

Then we finally obtain the following Raising Relation (RR) satisfied by the monic Bessel polynomials

B«Bna)(x) = (n + 2a - 2)BS-11)(x), (1.6)

where Ba := x2D + 2 ((a - 1)x + 1) I is called the degree raising shift operator for the Bessel polynomials with I representing the identity operator. For more details see also the degree raising shift operator for the family of classical orthogonal polynomials [13].

In view of (1.6), we can say that {Bia)}ra>0 is an Ba-classical polynomial sequence, since it satisfies the Hahn's property with respect to the operators Ba, i.e., it is an orthogonal polynomial sequence whose sequence of Ba-derivatives is also orthogonal. Note that an orthogonal polynomial sequence {pn}ra>0 is called classical, if {pn}n>0 is also orthogonal (see [16-19]). This characterization is essentially the Hahn-Sonine characterization (see [11, 21]) of the classical orthogonal polynomials.

In the same context, a natural question arises about the characterization of Ba-classical orthogonal polynomials. The purpose of this paper is to introduce the concept of the Ba-classical polynomial sequence and to give a complete description of this family of orthogonal polynomials. Note that many researches have been devoted to these topics where lowering, transfer and raising operators have been used (see for example [1-7, 9, 11, 12, 20]).

The paper is organized as follows: Section 2 gives the basic notations and tools that will be used throughout the paper. Section 3 deals with Ba-classical orthogonal polynomial sequence. In Section 4, we put in evidence some differential relations satisfied by the polynomials solution of our problem. In Section 5, we give a conclusion.

2. Preliminaries

Let P be linear space of polynomials in one variable with complex coefficients and P' be its dual space, whose elements are linear functionals. We write (u,p) := u(p) (u € P', p € P). In particular, we denote by (u)n := (u, xn), n > 0, the moments of u. Let us define the following operations on P'. For any linear functional u, any polynomial f and any (a, b) € C\{0} x C, let Du := u', fu, hau and Tbu be the linear functionals defined by the duality [15, 16]

(fu,p) := (u, fp), (u',p) := (u, p'), (hau,p) := (u, hap) = (u,p(ax)), (r6u,p) := (u,T-6p) = (u,p(x + b)).

A linear functional u is called normalized if it satisfies (u)0 = 1. We assume that the linear functionals used in this paper are normalized.

Let {pn}n>0 be a sequence of monic polynomials with degpn = n, n > 0 (MPS in short) and let {un}n>0 be its dual sequence, un € Pdefined by (un,pm) = ¿n,m, n, m > 0. Notice that u0 is said to be the canonical functional associated with the MPS {pn}n>0 (see [16-18]).

Let us recall the following result.

Lemma 1 [16, 17]. For any u € P' and any integer m > 1, the following statements are equivalent :

(i) (u, Pm— l) =0, (u,Pn) = 0, n > m,

m— 1

^v uv •

(ii) EUV € C, 0 < v < m — 1, Am-1 = 0 such that u = ^ Avuv.

v=0

As a consequence, the dual sequence {u(1] }n>0 of {p(1] }n>0 where

Pn](x) := (n + 1)-1 Dpra+1(x), n > 0,

is given by [16, 19] as

Du(1] = — (n + 1)un+1, n > 0. Similarly, the dual sequence {un}n>0 of {pn}n>0, where

pn(x) := a_npn(ax + b)

with (a, b) € C\{0} x C, is given by [16, 19]

u„ = an(h„-i o T_b)ura, n > 0.

A linear functional u is called regular if we can associate with it a MPS {pn}n>0 such that [16, 19]

as

(u,PnPm) = r„n, m > 0, r„ = 0, n > 0.

The sequence {pn}n>0 is then called a monic orthogonal polynomial sequence (MOPS in short) with respect to u. Note that u = (u)0u0 = u0, since u is normalized.

Proposition 1. [16]. Let {pn}n>0 be a MPS and let {un}n>0 be its dual .sequence. The following statements are equivalent:

(i) {pn}n>0 is orthogonal with respect to u0,

(ii) {pn}n>0 satisfies the linear recurrence relation of order two

P0(x) = 1, P1(x) = x — ft,

Pn+2(x) = (x — ^n+1)Pn+1(x) — Yn+1Pn(x), n > 0,

where

^n = (u0,xp()(u0,p(;)_1, n > 0,

and

Yn+1 = (u0,p(+1)(u0,p(;)_1 =0, n > 0, (iii) the dual sequence { u ( } ( > 0 satisfies :

un = (u0,p(()_1pnu0, n > 0.

A MOPS }n>0 is called D-classical, if {Dpn}ra>0 is also orthogonal (Hermite, Laguerre, Bessel or Jacobi) [19]. Moreover, if {pn}n>0 is orthogonal with respect to u0, then there exists a monic polynomial 0 with deg 0 < 2 and a polynomial ^ with deg ^ = 1 such that u0 satisfies the Pearson's equation (PE) [19]

D(0uo) + ^uo = 0.

A second characterization of these polynomials is that they are the only polynomial solutions of the SODE [8, 19],

0(x)pn+i(x) - ^(x)ï4+i(x) = A„pra+i(x), n > 0,

where

A„ = (n + 1) (^0"(O)n - 0'(O)) / 0, n > 0.

Note that if pn(x) = Bia)(x), n > 0, (a = —n/2, n > 0) is the monic Bessel polynomial and we write B(a) for u0, then the regular form B(a) satisfies the following PE [16, 19]

D(x2B(a)) — 2(ax + 1)B(a) = 0, (2.1)

and Bia)(x), n > 0 satisfies the SODE (1.3).

3. The Ba-classical polynomials

Recall the operator

1« : P P,

f B«(f):= x2/' + 2((a - 1)x + 1)f,

with a = —m/2, m > -2, m € Z.

Clearly, the operator Ba raises the degree of any polynomial. Such an operator is called raising operator [14, 22].

Definition 1. We call a sequence {Pn}n>0 of orthogonal polynomials Ba-classical if {BaPn}n>0 is also orthogonal.

For any MPS {Pn}n>0 we define

Qn+i(x; a) := ——I-- BaPn(x), n > 0,

n + 2a - 2

or equivalently

(n + 2a - 2)Qra+i(x; a) := x2P^(x)+2((a - 1)x + l)Pra(x), n > 0, (3.1)

with initial value Q0(x; a) = 1.

Clearly, {Qn+1(.; a)}n>0 is a MPS and

deg Qn+i(x; a) = n + 1.

In the sequel, we write

Qn(x) := Qn(x; a), n > 0,

if there is no ambiguity. Our next goal is to describe all the Ba-classical polynomial sequences. Assume that {Pn}ra>0 and {Qn}ra>0 are MOPS satisfying

P«,+2(x) = (x - ro„+i)P„+i(x) - Yn+iPn(x), n > 0, (3.2)

with initial values P0(x) = 1, P^x) = x — ro0, and

Qn+2(x) = (x — 0n+1)Qn+1(x) — Zn+1Qn(x), n > 0, (3.3)

with initial values Q0(x) = 1, Q1(x) = x — 00.

Next, a first result will be deduced as a consequence of relations (3.1), (3.2) and (3.3).

Proposition 2. The sequences {Pn}n>0 and {Qn}n>0 satisfy the following finite type relation x2Pn(x) = Qn+2(x) + SnQn+1 (x) + tnQn(x), n > 0,

where

Sn = (n + 2a — 2) (OTn — 0n+O, n > 0, tn = (n + 2a — 3)Yn — (n + 2a — 2)Zn+1, n > 0,

with the convention y0 = 0.

Proof. Differentiating (3.2), we obtain

Pn+2(x) = (x — OTn+1)Pn+1(x) — Yn+1Pn(x) + Pn+1 (x), n > 0.

We multiply the last equation by x2 and the relation (3.2) by 2((a — 1)x + 1), take the sum of the two resulting equations, and substitute (3.1). Then, we get

(n + 2a)Qn+3(x) = (n + 2a — 1)(x — ®n+1)Qn+2(x) —(n + 2a — 2)Yn+1Qn+1(x) + x2Pn+1(x), n > 0.

Using the relation (3.3), we get

x2Pn+1(x) = Qn+3(x) + (n + 2a — 1)( ^n+1 — ^+2) Qn+2(x) + ((n + 2a — 2)Yn+1 — (n + 2a — 1)(n+^ Qn+1 (x), n > 0.

In fact, this result is valid if n + 1 is replaced by n with the convention y0 = 0. Hence we got the desired result. □

Note that, for n = 0, the Proposition 2 gives

x2 = Q2(x) + (2a — 2)(ot0 — 01)Q1(x) — (2a — 2)(1^0(x), (3.4)

and using the fact that

Qi{x) = x - 0o = x H---,

a — 1

we obtain

Q2 (x) = x2 + (2a — 2)(01 — OT0)x + (2a — 2)Z1 + 2(01 — ^0). It gives by comparing with (3.3) for n = 0

. —00 + 2(a — 1)^0 1 , 2(a — 1)

=- = -r^-r H--TOO,

1 2a — 1 (a — 1)(2a — 1) 2a — 1 0

. 0001 + 2(^0 — 01) —1 {1 = ~ ~

2a - 1 (a - 1)2'

Denote by u0 and v0 the regular forms (linear functionals) in P' corresponding to {Pn}n>0 and {Qn}n>0 respectively. Then we can state the following result.

Lemma 2. The following algebraic relation between the regular forms uo and vo holds

2 2

X Vo = --— u0.

(a - 1)

Proof. According to Proposition 2, we obtain

<x2vo,P„(x)> = 0, n > 1. (3.5)

On the other hand, by (3.4) we have

<x2vo,Po(x)> = <vo,Q2(x)> +2(a - 1)(^o - 0i)(vo,Qi(x)) - 2(a - 1)Ci(vo,Qo(x))r

2 (3.6)

since }n>o is orthogonal with respect to the normalized form vo. According to Lemma 1 and using (3.5) and (3.6), we obtain the desired result. □

Based on PE satisfied by the linear functional of B(a), we can state the following theorem.

Theorem 1. The sequence of Bessel polynomials {Bia)}ra>o, with a = -m/2, m >-2, m € Z, is the only Ba-classical orthogonal sequence. More precisely, Pn(x) = Bia)(x) and Qn(x) = Bia-1)(x), n > 0.

Proof. If we apply vo in (3.1), we get for n > 0

<vo, (n + 2a - 2)Qn+i(x)> = ^vo, x2P^(x) + 2((a - 1)x + 1)Pra(x)) = 0.

But the right hand side may be read as

^ - Dx2v^ + 2((a - 1)x + 1)vo, Pra(x)^> = 0, n > 0.

Hence we have for all polynomials P, expanding P in the basis {Pn}n>o, the following relation

^ - D(x2vo) + 2((a - 1)x + 1)vo, P(x)^> = 0.

In other words we have ( ) ( )

(x2vo)' - 2((a - 1)x + 1)vo = 0. (3.7)

This implies that vo is the Bessel functional B(a-1) according to the corresponding PE (2.1), i.e.,

Qn(x)= B^-1)(x), n > 0,

with a = -m/2, m > -2, m € Z.

Multiplying (3.7) by x2 and using Lemma 2, we obtain

(x2uo)' - 2(ax + 1)uo = 0. (3.8)

Essentially (3.8) corresponds to the PE of linear functional B(a) of the sequence of Bessel polynomials {B^}n>o. Hence, Pn(x) = Bia)(x), n > 0. □

In conclusion, we give the following relation, which is satisfied by Bessel polynomials x2Bna)'(x) + 2((a - 1)x + 1)B^a)(x) = (n + 2a - 2)Bn+-1)(x), n > 0 with a = -m/2, m > -2, m € Z.

4. Representations of Bessel polynomials in terms of the action of linear

differential operators

In this section, we prove some higher order differential relations between the Bessel polynomials (solution of our problem). First, we need the following fundamental relation

(xD + (n + a - 1)l)Bna/2)(x) = (2n + a - 1)B«a+1)/2)(x), (4.1)

which is obtained after a simple calculation from (1.1).

Theorem 2. The representation of Bessel polynomials Bn(a+m)/2)(x) in terms of action of linear differential operators on the Bessel polynomials Bna/2) (x) is given by

B{{a+m)/2), ) r(2??. + a - 1) y. /m\ r(w + a + m - 1) m-kDm-kB(a/2) ( x

T(2n + a + m-l) \k J T(n + a + m, - k - I)' n (4.2)

n > 0, m > 0.

Proof. We prove this by induction on m € N. For m = 0 this is obvious. Now, suppose (4.2) holds and prove the same for m + 1 instead of m. Indeed, by differentiating both sides of (4.2) and using (1.4), we get, for all n > 1,

B((a+m+2)/2), x _ F(2n + a - 1) y-v /m\ r(??, + a + m - 1)

"-1 W ~ r(2n + a + m - 1) ^ \ k J r(" + « + m - k ~

x [(m - k)xm-fc-1Dm-fc-1 + xm-kDm-k]Bn-+2)/2)(x), n > 1. Replacing a + 1 by a, n - 1 by n and using the identity (4.1) we obtain for all n > 0

B((a+m+i)/2)( s = r(2w + a - 1) yv /m\ r(w + a + m - 1) ; r(2n + a + m) ^ \ fc j T(n + a + m - fc - 1)

x [(m - k)xm-fc-1Dm-fc-1 + xm-kDm-k] (xD + (n + a - 1)I)Bna/2)(x), n > 0. Equivalently

'm\ r(n + a + m - 1)

R((a+m+i)/2)/_^ = r(2ra + a - 1) y. /m\ W T(2n + a + m)^\kJ

+ a + m) ' V k 7 r(n + a + m - k - 1) k=0

(m - k)(n + a + m - k - 2)xm-fc-1Dm-fc-1 +(n + a + 2m - 2k - 1)xm-kDm-k + xm+1-fcDm+1-fc] B^72 (x), n > 0.

After some calculations, we finally obtain for all n > 0

B((a+m+i)/2)( s = r(2w + a - 1) y^ /m + 1\ r(n + a + m) ' r(2n + a + m)^\ k J T(n +a + rn - k)

xxm+1-fcDm+1-kBna/2)(x), m > 0. Hence the desired result is proved. □

5. Conclusion

We have described the Ba-classical orthogonal polynomials using the Pearson's equation that the corresponding linear functionals satisfy. More precisely, we have proved that the Bessel polynomial sequence (B^ (x)}n>0, where a = —m/2, m > -2, m € Z, is the only Ba-classical sequence. As a consequence, some connection formulas between the corresponding polynomials are deduced.

Acknowledgements

The authors are 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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Appendix

Table A. Bessel polynomials.

|Bn}n>0 IS(a)

$(x) = x2, ^(x) = -2(ax + 1),

1 1 - a

ßo =--, ßn+l = 7-:-TT-:-—tt, n > 0,

a (n + a)(n + a + 1)

(n+ !)(» +2a-1) 7n+1 (2n + 2a- l)(n + a)2(2n + 2a + 1)' " ' x2Bn+i(x)+2(ax + 1)Bn+i(x) - (n + 1)(n + 2a)B„+i(x) = 0,

x2B'n+1(x) = (n + 1) (x--— ) Bn+i(x) - (2n + 2a + 1)7n+1Bn(x),

y n + a )

/+œ / t-\

x2a-2e-2W J T2ae2/«s(£)d£j f (x)dx, J(a) := 4 jf+°° t3-8ae2/t4e-t sin(t) ( £ x2a-2e-2/xdx) dt,

/0 \Jo

x < 0,

s(x)

1 -1/4 • 1/4 ^ n

sinx1/4, x > 0.