Connection formulas and representations of Laguerre polynomials in terms of the action of linear differential operators Текст научной статьи по специальности «Математика»

24

Probl. Anal. Issues Anal. Vol. 8(26), No3, 2019, pp. 24-37

DOI: 10.15393/j3.art.2019.6290

UDC 517.587, 517.521.1

B. Aloui, L. Kheriji

CONNECTION FORMULAS AND REPRESENTATIONS OF LAGUERRE POLYNOMIALS IN TERMS OF THE ACTION OF LINEAR DIFFERENTIAL OPERATORS

Abstract. In this paper, we introduce the notion of O£-classical orthogonal polynomials, where O£ := I + eD (e = 0). It is shown that the scaled Laguerre polynomial sequence {a-n L^ (ax)}n^.o, where a = —e-1, is actually the only O£-classical sequence. As an illustration, we deal with some representations of Laguerre polynomials Lno)(x) in terms of the action of linear differential operators on the Laguerre polynomials Lnm) (x). The inverse connection problem of expanding Laguerre polynomials L^^x) in terms of L^ (x) is also considered. Furthermore, some connection formulas between the monomial basis {xn}n^0 and the shifted Laguerre basis {Lnm)(x + 1)}n^0 are deduced.

Keywords: Classical polynomials, Laguerre polynomials, lowering and raising operators, structure relations, higher order differential operators, connection formulas

2010 Mathematical Subject Classification: 33C45, 42C05

1. Introduction. Let O be a linear operator that acts on the space P of polynomials in one variable and maps polynomials of degree n to polynomials of degree n + n0 (n0 is a fixed integer). We call a sequence {pn}n^o of orthogonal polynomials O-classical if there exist a sequence {qn}n>0 of orthogonal polynomials such that Opn = qn+n0, where n ^ 0 if n0 ^ 0 and n ^ n0 if n0 < 0. (This is Hahn's property [1-4], [6-8], [12], [13], [15], [18], [19], [23]).

It is known that the monic Laguerre polynomial sequence {Ln, )}n^0, where a = —n, n ^ 1, is classical and satisfies the relation (see [10], [16])

DLna) = nLn_+1), n ^ 1. (1)

© Petrozavodsk State University, 2019

In [2], the first author introduced the notion of Ra-classical orthogonal polynomials and put in evidence, for a E C\{0, -1, — 2,...}, the following relation

RL^ (x) = L^ (x), n > 0. (2)

Note that whereas the first expression involves the operator D that lowers the degree and raises the parameters, the second one involves Ra := (x — a)I — xD, which raises the degree and lowers the parameters. The two operators together are called shift operators. In [19], the authors proved that {Lia)}n^0 is the only Fa-Appell orthogonal sequence and satisfies

F«Lia)(x) = OnL^x), n ^ 0, (3)

where Fa := DxD + aD, a E C\{ —1, — 2,...}, is called lowering operator (it lowers the degree and preserves the parameter) introduced by Dattoli and Ricci (see [11]). We also see, by the second order differential equation satisfied by the Laguerre polynomials, that [9]

LL^(x) = \nLn\x), n > 0, (4)

where L := xD2 — (x — a — 1)D, is called Jacobi's operator. Note that the classical orthogonal polynomials (Hermite, Laguerre, Bessel, and Ja-cobi) are essentially the only eigenfunctions of the Bochner's operator, i.e., satisfy the same relation (4) (see [9]).

Furthermore, the present contribution is a natural continuation of a previous works. More precisely, in view of Eqs (1) —(4), it is natural to study the same problem with respect to the operator which, for example, raises the parameters and preserves the degree of the polynomial Lia)(x), n ^ 0. The operator is O£ := I + eD (e = 0). The basic idea has been deduced by starting from the so called second structure relation [20,21]

L^x) = (n + 1)-1LiQ+;(x) + LW(x), n ^ 0,

which gives, by using (1), the following relation

DLia)(x) = Ln+1)(x), n ^ 0, (5)

where O := I — D, with I as the identity operator. This means that the above family of standard orthogonal polynomials is an O-classical polynomial sequence with respect to the operator O, i.e., it is an orthogonal

polynomial sequence, whose sequence of O is also orthogonal. For a given £ = 0, let us consider Oe : P ^ P the linear operator defined in the linear space P of polynomials with complex coefficients

Oe := I + £D (O-i = O).

The aim of this paper is to put in evidence the relation (5) and characterize the Oe-classical orthogonal polynomials.

The further contents of this paper is as follows. Section 2 gives some preliminaries, while the main result is proved in Section 3. In Sections 4 and 5, we give some new properties related to the above operator and the Laguerre polynomials.

2. Preliminaries. Let P be the linear space of polynomials in one variable with complex coefficients. Let P' be the algebraic linear dual of P. We write (u,p) := u(p) (u G P', p G P). A linear functional u G P' is said to be regular or quasi-definite [10], [22] if det(u,xl+j)i,j=1,...,ra = 0 for n ^ 0. This is equivalent to the existence of a unique sequence of monic polynomials {Pn}n^0 of degree n such that (u, PnPm) = rnin,m, n, m ^ 0, with rn = 0 (n ^ 0). The sequence {Pn}n^0 is then called a monic orthogonal polynomial sequence (MOPS) with respect to u.

Theorem 1. (Favard's Theorem [10]). Let {Pn}n^0 be a monic polynomial sequence. Then {Pn}n^0 is orthogonal if and only if there exist two sequences of complex numbers {/n}n^0 and {Yn}n^0, such that Yn = 0, n ^ 1 and satisfies the three-term recurrence relation

(TTRR) j P0(x) = 1 Pi(x) = x - A), (6)

( ) \ Pn+2(x) = (X - An+l)Pn+1 (x) - Yn+lPn(x), n ^ 0. ()

When {Pn}n^0 is a MOPS, then {Pn}n^0, where Pn(x) = a-nPn(ax + + b), (a, b) G C*"x C, is also a MOPS and satisfies [20], [21]

P0(x) = 1, Pi (x) = xj- /0, ^ (7)

pPn+2(x) = (x - An+i)Pn+i (x) - pn+iPn(x), n ^ 0,

where An = a-i(/„ - b) and pn+i = a-27„+i.

An orthogonal polynomial sequence {Pn}n^0 is called classical, if {Pn}n^0 is also orthogonal (Hermite, Laguerre, Bessel, or Jacobi), [10]. This is essentially the Hahn-Sonine characterization (see [12], [24]) of the classical orthogonal polynomials.

It is well-known that any classical polynomial sequence {Pn}n^o can be characterized taking into account its orthogonality as well as the First Structure Relation (FSR), or the Second Structure Relation (SSR) [5], [20], [21]:

(FSR) 0(x)Pn+i(x) = r(x,n)P„+i(x) + sraPra(x), n ^ 0, (8)

(SSR) Pra(x) = (n + 1)-1Pl+i(x) + a„P>n(x) + &„p;_i(x), n ^ 0. (9)

Note that if Pn(x) = Lia)(x), (a = —n, n ^ 1) is the monic Laguerre polynomial, then we have a MOPS for which formulas (6), (8), and (9) were given for n ^ 0 by [16], [20], [22]

L0 (x) = 1, Ll (x) = x — a — 1, (x) = fx — (2n + a + 1)) Ln+)1(x)

— (n + 1)(n + a + 1)L"a)(x).

(TTRR) { LiQ+2(x) = (x — (2n + a + 1))L^x) — (10)

(FSR) xL^+l(x) = (n + 1)L"+)1(x) + (n + 1)(n + a + 1)L"a)(x), n ^ 0, (11) (SSR) L"a) = (n + 1)-1LÍf1 (x) + L"a)'(x) n ^ 0. (12)

Note that the monic Laguerre polynomial can be expressed by the Rodrigues formula [17]

L"a)(x) = (—1)"exx-a—(e-xx"+a), n ^ 0. It also satisfies the following explicit representation [25]:

L<")(x)=¿ (—1)"-" (n) r;n+a+;> , n » 0. ^

W r(v + a + 1)

Recall the two formulas [25], [26]: for all n ^ 0, a > —1,

L"a) (tx)=¿ (—1)n-k (n) r(n+a+1) (1—t)n-k Lka)(x),

and for all n ^ 0, a > — |:

i

L"a)(x) = n^Ua^l) /(1 — t2)a-2H2"(tvx) dt, (15) Vn(2n)!r(a + 2) J v y

:14)

where Hn(x) is the Hermite polynomial on degree n.

3. Hahn's property with respect to the operator De.

Recall that the operator De is defined by

O£ : P P

f f + ef', (e = 0).

Our purpose here is to describe all the De-classical orthogonal polynomials, i.e., the SMOP {Pn}n^o such that the monic sequence {Qn}n^o, where

Qn(x) = Pn(x) + ePn(x), n ^ 0, (16)

is also orthogonal. Suppose that {Pn}n^o and {Qn}n^o are SMOP satisfying

Po(x) = 1, Pi(x) = x - ßo, (17)

Pn+2(x) = (x - ßn+i)Pn+i(x) - Yn+iPn(x), Yn+i = 0, n ^ 0,

iQ0(x) = 1, Qi(x) = x - Xo, (18)

\Qn+2(x) = (x - Xn+i)Qn+i(x) - Öra+iQra(x), 0„+i = 0, n ^ 0.

We have the following result.

Lemma 1. The sequences {Pn}n^o and {Qn}n^o are related as follows:

Pn+i(x) = Qn+i(x) - e(n + 1)Qra(x), n ^ 0, (19)

Pn+i(x) = (n + 1)Qn(x), n ^ 0, (20)

where {ßnWo, {Xn}n^o, {YnWo and {^Wo satisfy

Xn = ßn - e, n ^ 0, (21)

0n+i = Yn+i + e2(n +1), n ^ 0. (22)

Proof. By starting (16), with n replaced by n + 2, and using (17) and (18), we obtain

(x - Xn+i)Qn+i(x) - 0n+iQn(x) =

= (x - ßn+i)Qn+i(x) - Yn+iQn(x) + ePn+i(x), n ^ 0.

Equivalently,

(ßn+i - Xn+i)Qn+i(x) + (Yn+i - Ön+i)Qn(x) = ePn+i(x), n ^ 0.

By comparing the degrees in the last equation, we obtain Xn+i = /5n+i — e, n ^ 0 and then

eQn+i(x) + (Yn+i — ^n+i)Qn(x) = ePn+i(x), n ^ 0. (23)

Making n =1 in (16), we get Xo = ^o — e; then (21) is valid. Inserting (16), with n replaced by n +1, in (23) we obtain

e2Pn+i(x) = (0n+i — Yn+i)Qn(x), n ^ 0.

After analysis of the degree, we obtain (22). Hence, (19) and (20) are valid. □

Based on Lemma 1 and the SSR of Laguerre polynomials, we can state that the scaled Laguerre polynomial sequence (ax)}n^0 where

a = — e-i, is the only O£-classical orthogonal sequence. More precisely, for all n ^ 0,

Pn(x) = (—e)nLna)(—e-ix) and Qn(x) = (—e^Li^H—e-ix).

Theorem 2. For any nonzero complex number e and any monic polynomial sequence (Pn}n^0, the following statements are equivalent.

(i) (Pn}n^0 is an 0£-classical orthogonal sequence.

(ii) There exists a G C, a = 0 such that Pn(x) = a-nLna)(ax), n ^ 0.

Proof. (i)^(ii). Assume that (Pn}n^0 is a monic O£-classical orthogonal sequence. Then, there exists a monic orthogonal sequence (Qn}n^0 that satisfies (16) and gives, after inserting in (19),

Pn(x) = —^rPn+i(x) — ePn(x), n ^ 0. (24)

n +1 +

Essentially, (24) corresponds to the scaled Laguerre polynomial sequence

{(—e)nLna)(—e-ix)}n,0,

(see (12)), i.e., Pn(x) = (—e)nz4a)(—e-ix), n ^ 0, (a = —n, n ^ 1), where, from (10) and (7), we have

= —e(2n + a + 1), n ^ 0 and Yn+i = e2(n + 1)(n + a + 1), n ^ 0.

In the same way, from (21) and (22), we obtain

Xn = —e(2n + a + 2), n ^ 0, 0n+i = e2(n + 1)(n + a + 2), n ^ 0.

Then, we also conclude that Qn(x) = (—e)nLia+1)(—e 1x), n ^ 0.

(ii)^(i). Let a in C, with a = 0 and let Pn(x) = a-nLia)(ax), n ^ 0. It is clear that {Pn}n^0 is a MOPS. By using the the (SSR) (12) satisfied by Lia)(x), n ^ 0 and the relation (1), we have

Lia+1) (x) = Lia) (x) — Lia)' (x), n ^ 0. (25)

Besides, from (25), where x is replaced by ax, it comes that

Lna+1)(ax) = L^(ax) — a-1 (L^ax))', n ^ 0,

or, equivalently, a-nLia+1)(ax) = ^I — a-1Dja-nLna)(ax), n ^ 0, i.e.,

O£P„(x) = a-nLna+1)(ax), n ^ 0,

where e = —a-1. Hence, (i) holds, since (ax)}„^o is a MOPS. □

4. Higher-order differential relations. As a consequence of Section 3, we have

O£Lna)(—e-1x) = Lia+1)(—e-1x), n ^ 0. If we take e = —1 and O-1 := O, we have the canonical situation

OLia)(x) = Lia+1)(x), n ^ 0, which gives, by induction on m E N,

OmLna)(x) = Lna+m)(x), n ^ 0, (O0 = I). (26)

Note that by using (1), the polynomial Li"+m)(x) can be written as follows

Lr+m)(x) = ( + 1)( +1-DmLiQ+m(x), n ^ 0, m ^ 0,

(n + 1)(n + 2) ••• (n + m) +

n' nmr(a) / i M D Ln+m

( n + m)

DmLn++m(x), n ^ 0, m ^ 0,

and, then, we get the following relation between Om and Dm

OmL(a)(x) = 7-—T DmLia+m(x), n ^ 0, m ^ 0,

n w (n + m)! n+mA '

with the convention O0 = D0 = I.

By (26), with a = 0, and using the fact that Om = (I — D)m and the binomial formula, we can state the following result.

Lemma 2. The monic Laguerre polynomials Lnm)(x), m ^ 0, are represented in terms of the action of linear differential operators on the Laguerre polynomials Ln0)(x), as follows:

m ,

4m)(x)=£(—1n m W0) (x), n ^ 0.

v=0 ^ '

Having Lemma 2, it is natural to study if the reciprocal is true. Firstly, we need the following relation, obtained from the explicit expression for the Laguerre polynomials

xL^' (x) + ¿ni)(x) = (n + 1)Ln0)(x), n ^ 1. (27)

Theorem 3. The representation of the Laguerre polynomials Ln0)(x) in terms of action of linear differential operators on the Laguerre polynomials Lnm)(x), is given by

Ln°)(x) (n + m)! ^

v=0

V! xm-vDm-VLnm)(x), n ^ 0. (28)

Proof. We prove this by induction on m G N. For m = 0 this is obvious. Now suppose (28) holds and prove the same for m+1 instead of m. Indeed, by differentiating both sides of (28) and using (1), with a = 0, we get, for all n ^ 1,

m

n!

(n + m)!

v ' v=0

v! xm-vDm-V+

+ (m — v)xm-v-iDm-v-i) Ln-+i)(x) = Ln-i(x).

Multiplying both sides of the previous equation by x, applying the operator D, and using the identity (27), we obtain for all n ^ 1

(n — 1)! [ ^ (m!)2 xm+i-vDm+i-v +

(n + m)! 1 ^ [(m — v)!]2v!

2

2

+

m

£

v=0

(2m — 2v + 1)(m!)2

[(m — v )!]2v!

xm-v Dm-v+

+

(m!)2

m- 1

V=0 [(m — 1 — v)!]2v!

x

m-1-vDm-1-v Km+^x) = Ln0)1(x)

By replacing v by v — 1 (resp. v — 2) in the second (resp. third) sum, we obtain for all n ^ 1

(n — 1)!

(n + m)! ^ | [(m — v)!]2v! ' [(m +1 — v)!]2(v — 1)!

£

(m!)2

(2m — 2v + 3)(m!)2

+ r / . \no/ -i\i +

+

(m!)2

[(m + 1 — v )!]2(v — 2)!

>xm+1-v Dm+1-v Lnm++1)(x)+

(n — 1)!

+7-^ fxm+1Dm+1+(m+1)2xmDm+(m+1)! i)L^^x) = L^^x).

( n + m)! - -

After some calculations, with n replaced by n+ 1, we finally obtain for all n > 0

n!

m+1

Ln0)(x) = 7+ +1)!^

n (n + m + 1)!

v ' v=0

£

m

v! xm+1-vDm+1-v Lnm+1) (x).

Hence the desired result is proved. □

4. Integral formulas. Consider the integral operator [14]:

Sc(P)(x)= / te-iP(t(x — o) + o) dt, o E C, P eP . (29)

In particular, for P(x) = (x — c)n, we have

Sc((x — c)n) = (n + 1)!(x — o)n, n ^ 0.

(30)

By (30) and (13), it is easily seen that for every integer m E N \ {0}

60 (xm-1Lnm)(x)) = (n + m)!xm-1(x — 1)n, n ^ 0.

Equivalently,

1

x

( n + m)!

tme-tLnm) (t(x + 1)) dt, n ^ 0.

(31)

2

v

Now, as an application of (31), some connection formulas between the monomial basis {x"}n^0 and the shifted Laguerre basis {L"m)(x + 1)}n^0 are deduced.

Theorem 4.

(i) For every integer m ^ 1, the following formulas hold for all n ^ 0

+ ^0

■n - £ © Lkm)(x + 1) / tm+k(1 - t)"-ke_i dt, (32)

k=0 0

x" = ££ Of"-1 i)! Lkm)(x +1). (33)

. k J \ i ) (m + k)! k=0 i=0 4 7 v 7 ^ 1

(ii) For m = 0, we have for all n ^ 0

k! n + 1

x" = £ ^n+T / 'e-'Li0)(t(x +1)) dt.

k=0 k! n + 1

Proof. By inserting (14), with a replaced by m and x by x + 1 , in (31), we obtain

£ (") (m+9 Lkm)(x + 1) / tm+k(1 - t)"-ke-t dt, n ^ 0.

x"

k=0 0

k) (m + k)! k Then (32) follows.

Substitute the binomial formula for (1 — t)n-k, in this last equality, obtaining

x" = t 0 (m+9 Lkm)(x + D t(—— *) (m + k + i)! =

fc=0 v 7 ^ ' i=0 v 7

=t t (n )(n—k)(—1);mm+kk!+i)! ^+«•

Hence we obtain (33).

For (ii), using (30) and (13) with a = m = 0, the operator S0 satisfies

60(Ln0)(x))(x) = n! £(—1)n-v (v) (v + 1)

v=0 V /

(x))(x ) = n! 2_J— 1) I I ( v +1)

v=0

nn

n!

B—^(n — vW — 1)! xV + £(—1

n!

£(—1)n-v-1(" ^ xv + (x — 1)n

v=0 ^ '

= n!(x — 1)n-1[(n + 1)x — 1], n ^ 1. (34)

Using (34), with x replaced by x + 1, we obtain the following integral relation

(n + 1)xn + nxn-1 = 1 j te-tLn0) (t(x + 1)) dt, n ^ 1.

0

This gives, by summation, the following result for all n ^ 1

+oo

n r 1 f n ( —1)fc

£ [(k + 1)(—x)k — k(—x)fc-1] = te-t £ Lk0)(t(x + 1)) dt. fc=1 0 k=1 !

Taking a telescopic sum, we get

0

(n + 1)(—x)n — 1 = / te-t £ Lk0) (t(x + 1)) dt, n ^ 1.

fc=1 !

Thus, the basic {xn}n^0 satisfies

n

xn = T / te-i £ TLk0) (t(x + 1)) dt, n ^ 0.

0 k=0

Hence, the desired result is proved. □

Remark 1. By substituting (15) into (31), with a replaced by m and x by x +1, we obtain for all n ^ 0

1

m

ii'iii"

n

n!m!4m I I , _ ^m-h+m„-t-

xn = (2n)!(2m)!J J (1 — y2)m-2^ ^ni ^t(x + 1) ) dt -1 0

which gives, for n = 0,

m!4m f i f

^cmn/(1 _ y2)m-2 dy/re-'di

-1 0

1

\2 Am r

(m!)24 ,, m-

(2m)! n

(1 _ y2)m-2 dy. (35)

1

Then, if we pose y = sin 9 in (35), we recover the Wallis integral

2

, 2m „ (2m)!n

sin2mQ dQ = ^ ' ,NO, m ^ 0.

J 22m+1(m!)2'

0

Acknowledgment. The authors are very grateful to the referees for the constructive comments and for making us pay attention to a certain reference.

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Received May 14, 2019. In revised form, September 23, 2019. Accepted October 01, 2019. Published online October 09, 2019.

B. Aloui

Université de Gabés, Institut Supérieur des Systémes Industriels de Gabés, Rue Salah Eddine Elayoubi 6033 Gabés, Tunisia. E-mail: [email protected]

L. Khériji

Université de Tunis El Manar, Institut Préparatoire aux Etudes d'Ingénieur El Manar, Campus Universitaire El Manar, B.P. 244, 2092 Tunis, Tunisia. LR13ES06.

E-mail: [email protected] or [email protected]