A Mathematical Journal
Vol. 7, No 2, (39 - 55). August 2005.

On the classical 2−orthogonal polynomials
sequences of Sheffer-Meixner type

Boukhemis Ammar 1

Department of Mathematics, Faculty of Sciences,
University of Annaba, B.P.12 Annaba 23000, Algeria

aboukhemis@yahoo.com

ABSTRACT
The polynomial sequences of Sheffer-Meixner type designed by {Sn}n≥0, are

defined by the generating function

G(x, t) = A(t)e
xH(t)

=
�

n≥0
Sn(x)

tn

n!

We are interested, in this work, in studying the sequences when they are
2−orthogonal. We will give the general properties of these sequences, and we
study in details those which are classical.

RESUMEN

Las sucesiones polinomiales del tipo Sheffer-Meixner denotadas por {Sn}n≥0
son definidas por la función generatriz

G(x, t) = A(t)e
xH(t)

=
�

n≥0
Sn(x)

tn

n!

En este trabajo estamos interesados en estudiar aquellas sucesiones que son
2− ortogonales. Mostraremos sus propiedades generales y estudiaremos en de-
talle aquellas que son clásicas.

1The author was partially supported by : L’Agence Nationale pour le Développement de la
Recherche Universitaire -ANDRU-.



40 Boukhemis Ammar
7, 2(2005)

Key words and phrases: d−Orthogonal polynomials, Relations of recurrence,
Sheffer-Meixner’s polynomials, Generating function,
Classical polynomials, Operator of Hahn.

Math. Subj. Class.: 42C05, 33C45

1 Introduction.

Let P be the vector space of polynomials with coefficients in C and P′ its algebraic
dual. Let us given d scalar linear forms Γ1, Γ2, · · · , Γd defined from P into C.

A monic sequence {Pn}n≥0 ( i.e. Pn(x) = xn + · · · , n ≥ 0) is said d−orthogonal
with respect to Γ = ( Γ1, Γ2, · · · , Γd )T when it satisfies [7, 10, 12, 18, 19, 23]

{
〈Γα,xmPn(x)〉 = 0, n ≥ md + α, m ≥ 0
〈Γα,xmPmd+α−1(x)〉 �= 0, m ≥ 0, (1.1)

for every 1 ≤ α ≤ d, and where 〈 , 〉 is the dual bracket between P and P′.
Among the d−orthogonal sequences, we will be interested here by a particular

class, but nevertheless important. Indeed, theses sequences have many applications
and have extensively investigated.

This class consists of sequences of polynomials {Sn}n≥0 defined by the generating
function

G(x,t) = A(t)exH(t) =
∑
n≥0

Sn(x)
tn

n!
(1.2)

where
A(t) =

∑
n≥0

ant
n and H(t) =

∑
n≥1

hnt
n

with
A(0) = 1, H(0) = 0 and H′(0) = 1

These sequences are said to be of Sheffer-Meixner type.
The case d = 1 has been first studied by Meixner [20] and Sheffer [22] and then,

completed by other authors [2, 13, 21].
Meixner has shown that this class consists of 5 sequences, namely, Hermite polyno-

mials, Laguerre polynomials, Charlier polynomials, Meixner polynomials and Meixner-
Pollaczek polynomials.

In the case d = 2 [5], we have shown that the functions H and A satisfy, respec-
tively the equations

⎧⎪⎪⎪⎨
⎪⎪⎪⎩

H′(t) =
1

(1 −αt)(1 −βt)(1 −γt), α,β,γ ∈ C

A′(t)
A(t)

=
σ0 + σ1t + σ2t2

(1 −αt)(1 −βt)(1 −γt), σ0,σ1,σ2 ∈ C; σ2 �= 0.
(1.3)



7, 2(2005)
On the classical 2−orthogonal polynomials sequences of ... 41

If we note by J the inverse function of H ( i.e. J(H(t) = t ), and by D =
d

dx
, then

we have [1, 20]
J(D)Sn+1(x) = (n + 1)Sn(x), n ≥ 0. (1.4)

Moreover, the polynomials Sn (n ≥ 0) are characterized by the four-term relation
[5]

Sn+3(x) = [(x−σ0) + (n + 2)(α + β + γ)] Sn+2(x)
−(n + 2) [σ1 + (n + 1) (αβ + αγ + βγ)] Sn+1(x)
−(n + 1) (n + 2) (σ2 −nαβγ) Sn(x), n ≥ 0

(1.5)

We also have proved that this class is composed of 9 sequences, namely
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(a) α = β = γ = 0,
(b) α = β �= 0 and γ = 0,
(c) α �= 0 and β = γ = 0,
(d1) α �= β �= 0 and γ = 0, α , β ∈ R,
(d2) α �= β �= 0 and γ = 0, α , β ∈ C,
(e) α = β = γ �= 0,
(f) α = β �= γ �= 0,
(g1) α �= β �= γ �= 0 α,β , γ ∈ R,
(g2) α �= β �= γ �= 0 α,β ∈ R and γ ∈ C.

(1.6)

We recall in paragraph 2, the principal properties of the d−orthogonal sequences
[ 6, 11, 18, 19]. The paragraph 3 is denoted to the characterization of those which
are in addition classical [3, 4, 12].

In paragraph 4, we show that the sequences (a), (b), (c) and (d) are classical
sequences and we give certain of their properties. Whereas in paragraph 5, we exhibit
an integral representation of the forms with respect to which these sequences are
2−orthogonal in cases (a) and (b).

2 Properties of d−orthogonal sequences.
Definition 2.1 Let {Pn}n≥0 be a sequence monic polynomials. We call dual sequence
of the sequence {Pn}n≥0, the sequence of linear forms {Fn}n≥0 defined by

〈Fn,Pn(x)〉 = δn,m, n,m ≥ 0 (2.1)

Proposition 2.2 [11] If we denote by D the operator of derivation i.e. D =
d

dx
and by {F̃n}n≥0 the dual sequence associated to the monic sequence {Qn}n≥0 of the
derivatives of {Pn}n≥0, and defined by

Qn(x) =
DPn+1(x)
n + 1

, n ≥ 0,

then
DF̃n = −(n + 1)Fn, (2.2)



42 Boukhemis Ammar
7, 2(2005)

with 〈
DF̃n,r(x)

〉
= −

〈
F̃n,Dr(x)

〉
, ∀r ∈P.

Proposition 2.3 [18, 19] Let L∈P′ be and q an integer, in order that L satisfies
〈L,Pn(x)〉 = 0 n ≥ q
〈L,Pq−1(x)〉 �= 0 (2.3)

it is necessary and sufficient that there exists λν ∈ C, 0 ≤ ν ≤ q − 1, λq−1 �= 0, such
that

L =
q−1∑
ν=0

λνFν. (2.4)

Corollary 2.4 According to the preceding lemma, we have

Γα =
α−1∑
ν=0

λανFν, λαα−1 �= 0, 1 ≤ α ≤ d,

and in a equivalent manner

Fν =
ν+1∑
α=0

τνα Γ
α; τνν+1 �= 0, 0 ≤ ν ≤ d− 1.

Consequently, every d−orthogonal sequence {Pn}n≥0 with respect to Γ =
( Γ1, Γ2, · · · , Γd )T is also d−orthogonal with respect to F = (F0,F1, · · · ,Fd−1)T .
Theorem 2.5 [18, 23] With the same notations as previously we have the following
equivalences

(a) The sequence {Pn}n≥0 is d−orthogonal with respect to F = (F0,F1, · · · ,Fd−1)T .
(b) The sequence {Pn}n≥0 satisfies a recurrence of order d + 1( d ≥ 1 )

Pm+d+1(x) = (x−βm+d)Pm+d(x) −
d−1∑
ν=0

γd−1−νm+d−νPm+d−ν−1(x), m ≥ 0 (2.5)

with the initial data{
P0(x) = 1, P1(x) = x−β0, and if d ≥ 2
Pn(x) = (x−βn−1)Pn−1(x) −

∑n−2
ν=0 γ

d−1−ν
n−1−νPn−2−ν (x), 2 ≤ n ≤ d

(2.6)
where γ0m+1 �= 0, m ≥ 0. ( Regularity conditions ).

(c) For every (n,v), n ≥ 0, 0 ≤ ν ≤ d− 1, there exists d polynomials V μ(n,ν),
(0 ≤ μ ≤ d− 1) such that

Fnd+ν =
d−1∑
μ=0

V μ(n,ν)Fμ, n ≥ 0, 0 ≤ ν ≤ d− 1, (2.7)



7, 2(2005)
On the classical 2−orthogonal polynomials sequences of ... 43

where ⎧⎨
⎩

deg V μ(n,μ) = n, 0 ≤ μ ≤ d− 1,
deg V μ(n,ν) ≤ n, 0 ≤ μ ≤ ν − 1, if 1 ≤ ν ≤ d− 1,
deg V μ(n,ν) ≤ n− 1, ν + 1 ≤ μ ≤ d− 1, if 0 ≤ ν ≤ d− 2.

(2.8)

Theorem 2.6 [18] For every sequence {Pn}n≥0 d−orthogonal with respect to F =
(F0,F1, · · · ,Fd−1)T , the following statements are equivalent

(a) It exists L∈P′ and an integer s ≥ 1 such that
{
〈L, Pn(x)〉 = 0, n ≥ s,
〈L, Ps−1(x)〉 �= 0. (2.9)

(b) It exists L∈P′ and d polynomials φα, 0 ≤ α ≤ d− 1 such that

L =
d−1∑
α=0

φαFα,

with the following properties
if s− 1 = qd + r, 0 ≤ r ≤ d− 1, we have⎧⎨

⎩
deg φr = q, 0 ≤ r ≤ d− 1, if d ≥ 2,
deg φα ≤ q, 0 ≤ α ≤ r− 1, if 1 ≤ r ≤ d− 1,
deg φα ≤ q − 1, r + 1 ≤ α ≤ d− 1, if 0 ≤ r ≤ d− 2.

(2.10)

3 The d−orthogonal sequences and the finite
differences operators Δωand ∇ω.

Let us consider the progressive finite differences operators Δω ( Hahn’s operator) and
regressive operator ∇ω, defined respectively by

Δωf(x) =
f(x + ω) −f(x)

ω
, and

∇ωf(x) =
f(x) −f(x−ω)

ω
= Δ−ωf(x)

These operators enjoy the following properties

Proposition 3.1 Let F ∈P′ then we have
〈F, Δωf(x)〉 = −〈∇ωF, f(x)〉 , ∀f ∈ C∞. (3.1)

Proof. We know that

Δωf(x) =
eωD − 1

ω
f(x),

and that by definition we have

〈DF, f(x)〉 = −〈F,Df(x)〉 ,



44 Boukhemis Ammar
7, 2(2005)

therefore

〈F, Δωf(x)〉 =
〈
F,
∑
k≥0

ωk

(k + 1)!
Dk+1f(x)

〉
=

〈∑
k≥0

(−1)k+1ωk
(k + 1)!

Dk+1F, f(x)
〉

=
〈
e−ωD − 1

ω
F, f(x)

〉
= −〈∇ωF, f(x)〉 .

Proposition 3.2 Let {Qωn}n≥0 be the sequence of the monic polynomials defined by

Qωn(x) =
ΔωPn+1(x)

n + 1
=
Pn+1(x + ω) −Pn+1(x)

(n + 1)ω
, n ≥ 0 (3.2)

and {F̃n}n≥0 the dual sequence associated to the sequence {Qωn}n≥0, then we have
∇ωF̃n=Δ−ωF̃n = −(n + 1)Fn+1; n ≥ 0. (3.3)

Proof. Indeed, we have

δn,m =
〈
F̃n,Qm(x)

〉
=

1
m + 1

〈
F̃n, ΔωPm+1(x)

〉
= − 1

m + 1

〈
Δ−ωF̃n,Pm+1(x)

〉
,

i.e.
−
〈
Δ−ωF̃n,Pn+1(x)

〉
= (m + 1)δn,m

but from the lemma (2.1), ∃ λν ∈ C, 0 ≤ ν ≤ n + 1, such that

Δ−ωF̃n =
n+1∑
ν=0

λnνFν,

with λnν = 0, 0 ≤ ν ≤ n and λnn+1 = n + 1.
Lemma 3.3 We have the following properties

Δω [(x−ω)mPn(x)] = xΔω
[
(x−ω)m−1Pn(x)

]
+ (x−ω)m−1Pn(x), m ≥ 0 (3.4)

and

xmΔωPn(x) = Δω [(x−ω)mPn(x)] − [mxm−1 −
m(m− 1)

2
ωxm−2 + Rωm−3(x)]Pn(x),

m ≥ 0, where Rωm−3(x) is a polynomial of degree (m− 3) in x.
(3.5)

Proof. Clearly

Δω [(x−ω)mPn(x)] =
xmPn(x + ω) − (x−ω)mPn(x)

ω

=
x[xm−1Pn(x + ω) − (x−ω)m−1Pn(x)] + ω(x−ω)m−1Pn(x)

ω
= xΔω

[
(x−ω)m−1Pn(x)

]
+ (x−ω)m−1Pn(x), m ≥ 0.



7, 2(2005)
On the classical 2−orthogonal polynomials sequences of ... 45

Repeating m times the expression (3.4) we get

Δω [(x−ω)mPn(x)] = xmΔωPn(x) +
[

m−1∑
k=0

xm−k(x−ω)k
]
Pn(x),

as ∑m−1
k=0 x

m−k(x−ω)k = ∑m−1k=0 ∑kj=0 (kj
)
(−1)jωjxm−1−j

=
∑m−1

j=0

∑m−1
k=j

(
k
j

)
(−1)jωjxm−1−j

= mxm−1 − m(m− 1)
2

ωxm−2 + Rωm−3(x).

from which we obtain (3.5).

Definition 3.4 [4, 11, 14, 15 ] A sequence of polynomials {Pn}n≥0 d−orthogonal
(d ≥ 1) with respect to F = (F0,F1, · · · ,Fd−1)T ,those the monic sequence of finite
differences {Qωn}n≥0 defined by

Qωn(x) =
ΔωPn+1(x)

n + 1
, n ≥ 0

is also d−orthogonal (d ≥ 1) with respect to F̃ =
(
F̃0,F̃1, · · · ,F̃d−1

)T
is said to be

classical.

Remark 3.5 In the case ω = 0, the operator Δω becomes D =
d

dx
.

Theorem 3.6 With the above hypothesis we have the following equivalence
(a) The sequence {Pn}n≥0 is classical d−orthogonal.
(b) The functional F satisfies the vectorial functional equation

∇ω(ΦF) + ΨF = 0, (3.6)
where Ψ and Φ are 2 matrices d × d of polynomials

Ψ(x) =

⎛
⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 1 0 . . . 0
0 0 2 . . . 0
. . . .
. . . .
. . . .
0 0 0 . . . d− 1

ψ(x) ξ1 ξ2 . . . ξd−1

⎞
⎟⎟⎟⎟⎟⎟⎟⎟⎠

(3.7)

and ψ is a polynomial of degree 1 and ξμ, 1 ≤ μ ≤ d− 1 are constants,

Φ(x) =

⎛
⎜⎜⎜⎜⎜⎜⎝

φ00(x) φ
1
0(x) . . . φ

d−1
0 (x)

. . .

. . .

. . .

φ0d−2(x) φ
1
d−2(x) . . . φ

d−1
d−2(x)

φ0d−1(x) φ
1
d−1(x) . . . φ

d−1
d−1(x)

⎞
⎟⎟⎟⎟⎟⎟⎠

(3.8)



46 Boukhemis Ammar
7, 2(2005)

where φνα, 0 ≤ α,ν ≤ d− 1 are polynomials such that⎧⎨
⎩

deg φνα ≤ 1, 0 ≤ ν ≤ α + 1 if 0 ≤ α ≤ d− 2
deg φνα = 0, α + 2 ≤ ν ≤ d− 1 if 0 ≤ α ≤ d− 3
deg φ0d−1 ≤ 2 and deg φνd−1 ≤ 1, 1 ≤ ν ≤ d− 1

(3.9)

In addition, if we write
⎧⎨
⎩

ψ(x) = e1x + e0, φ0d−1(x) = c2x
2 + c1x + c0

φα+1α (x) = kαx + lα, 0 ≤ α ≤ d− 2,
then ⎧⎪⎪⎨

⎪⎪⎩
c2 �=

e1
m + 1

, m ≥ 0, e1 �= 0,

kα �=
α + 1
m + 1

, m ≥ 0, for 0 ≤ α ≤ d− 2.
(3.10)

Remark 3.7 a) It is easy to show that :
{
F̃ = ΦF
∇ωF̃ = −ΨF

(3.11)

(b) When ω = 0 the functional equation (3.6) may be written [11]

ΨF + D(ΦF) = 0 (3.12)

and the conditions (3.7), (3.8), (3.9) and (3.10) remain unchanged.
c) The proof of this theorem is the same as in the case ω = 0 [11], if we take into

account the relation (3.4).

4 Classification of the sequences 2−orthogonal of
Sheffer-Meixner type.

Let us consider now the sequences of polynomials {Sn}n≥0, Sheffer-Meixner type
defined by the relation (1.5).

We noted by {mωn}n≥0 and {Mωn}n≥0 the sequences of monic polynomials defined
respectively by

mn(x) =
DSn+1(x)
n + 1

; n ≥ 0, (4.1)
and

Mωn (x) =
ΔωSn+1(x)
n + 1

; n ≥ 0. (4.2)
Then we have



7, 2(2005)
On the classical 2−orthogonal polynomials sequences of ... 47

Lemma 4.1 In the case (b) ( the case (a) if α = 0), the sequence of derivatives of
monic polynomials defined by the relation (4.1) satisfies the following recurrence⎧⎨
⎩

mn+3(x) = [(x−σ0) + (2n + 5) α] mn+2(x)
−(n + 2)

[
σ1 + (n + 2)α2

]
mn+1(x) − (n + 1)(n + 2)σ2mn(x); n ≥ 0

m0(x) = 1; m1(x) = x−σ0 + α; m2(x) = (x−σ0 + 3α) m1(x) −
(
σ1 + α2

)
(4.3)

Proof. Indeed, in the case (b) J is such that [5]

J(D) =
D

1 + αD
,

then by the relation (1.4) we have

DSn+1(x) = (n + 1) [Sn(x) + αDSn(x)]

consequently

Sn+1(x) = mn+1(x) − (n + 1)αmn(x); n ≥ 0.
Differentiating the recurrence(1.5) and replacing Sn+1 by {mν}n+1ν=n−1,we obtain

the relation ( 4.3 ).

Lemma 4.2 In the case (d) ( the case (c) if β = 0), the sequence of finite
differences of monic polynomials defined by the relation (4.2) satisfies the following
recurrence⎧⎪⎪⎨
⎪⎪⎩

M
α−β
n+3 (x) = [(x + α−σ0) + (n + 2) (α + β)] Mα−βn+2 (x)

−(n + 2) [σ1 + (n + 2)αβ] Mα−βn+1 (x) − (n + 1)(n + 2)σ2Mα−βn (x); n ≥ 0
M

α−β
0 (x) = 1; M

α−β
1 (x) = x−σ0 + α; Mα−β2 (x) = (x−σ0 + 2α + β) Mα−β1 (x)

−σ1 −αβ
(4.4)

Proof. Indeed, in the case (d) the function J is such that [5]

J(D) =
Δα−β

1 + αΔα−β
,

i.e. by the relation (1.4).

Δα−βSn+1(x) = (n + 1) [αΔα−βSn(x) + Sn(x)] ,

consequently

Sn+1(x) = M
α−β
n+1 (x) − (n + 1)αMα−βn (x); n ≥ 0.

By acting the operator Δα−β on the recurrence (1.5) and replacing Sn+1 by
{Mα−βn+1 }n+1ν=n−1, we obtain the relation (4.4).

Thus, we have the following classification.

Theorem 4.3 The sequences (a), (b), (c) and (d) are classical sequences and
the 2− orthogonal polynomials sequences {mn}n≥0 and

{
Mα−βn

}
n≥0 are “2−Kernel”

polynomial [8, 9, 17] for the 2− orthogonal polynomials sequences {Sn}n≥0 .



48 Boukhemis Ammar
7, 2(2005)

5 Integral representation of the functional F0 and
F1.

In this paragraph, we will be interested by the integral representation problem of
the linear functional F0 and F1 in the cases (a) and (b).

5.1 Properties of the functional F0 and F1.
Lemma 5.1 In the case (d) ( a fortiori the cases (a), (b) and (c)) we have

Φ(x) =

[
1 −α
− α
σ2

(x−σ0) 1 + α
σ1
σ2

]
and Ψ(x) =

⎡
⎣ 0 11
σ2

(x−σ0) −
σ1
σ2

⎤
⎦

Proof. With the same notations as in theorem (3.1), we have

deg φ00(x) ≤ 1, deg φ10(x) ≤ 1, deg φ01(x) ≤ 2, deg φ11(x) ≤ 1, and deg ψ(x) ≤ 1.
Putting ⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

φ00(x) = d0 + d1x
φ10(x) = e0 + e1x
φ01(x) = a0 + a1x + a2x

2

φ11(x) = b0 + b1x
ψ(x) = c0 + c1x

the relations (3.11) may be written respectively
{
F̃0 = (d0 + d1x)F0 + (e0 + e1x)F1
F̃1 =

(
a0 + a1x + a2x2

)
F0 + (b0 + b1x)F1

(R5.0)

and {
∇α−βF̃0 = −F1
∇α−βF̃1 = −(c0 + c1x)F0 − ξ1F1

(R5.1)

By letting, firstly, the functional F̃0 and F̃1 act successively on S0(x),S1x),S2(x)
S3(x), and S0(x),S1(x), · · · ,S4(x), respectively we determine the coefficients of the
polynomials Φji (x), (i,j = 0, 1), secondly, we let ∇α−βF̃1 act on S0(x),S1x) and S2(x)
to determine the coefficients c0,c1 and ξ1.

Proposition 5.2 For α �= 0, the functional F0 is solution of the equation
∇α−β

{
∇α−β [

(
α2x−σ2 −ασ1 −α2σ0

)
F0] − (2αx−σ1 − 2ασ0)F0

}
+ (x−σ0)F0 = 0, (5.1)

Proof. From the relation (3.6) we see that
⎧⎨
⎩
∇α−βF0 −α∇α−βF1 = −F1
− α
σ2
∇α−β [(x−σ0)F0] +

(
1 + α

σ1
σ2

)
∇α−βF1 = −

1
σ2

(x−σ0)F0 +
σ1
σ2
F1



7, 2(2005)
On the classical 2−orthogonal polynomials sequences of ... 49

and by substitution we obtain the relation

F1 = ∇α−β
{[

α2

σ2
(x−σ0) −

(
1 + α

σ1
σ2

)]
F0
}
− α
σ2

(x−σ0)F0 (5.2)

Therefore, letting ∇α−β act on this last one and replacing ∇α−βF1 and F1 by
there respective values with respect to ∇2α−βF0, ∇α−βF0 and F0 in the first relation,
we find the expected result.

Remark 5.3 In the case (b) ( the case (a) if α = 0 ), the relations (5.1) and (5.2)
may be written respectively

D
{
D[
(
α2x−σ2 −ασ1 −α2σ0

)
F0] − (2αx−σ1 − 2ασ0)F0

}
+ (x−σ0)F0 = 0 (5.3)

and
F1 =

α

σ2
D
{[
α(x−σ0) −

(σ2
α

+ σ1
)]
F0
}
− α
σ2

(x−σ0)F0 (5.4)

5.2 Determination of weight functions in the cases (a) and (b).

The problem consists now in representing the functional F0 and F1 as an integral
by putting ⎧⎨

⎩
〈F0,p(x)〉 =

∫
C
F0(x)p(x)dx, and

〈F1,p(x)〉 =
∫

C
F1(x)p(x)dx, ∀p ∈P

(5.5)

where the weight functions F0(x) and F1(x) are supposed “booth regular ” and C is
a contour to be determined.

Proposition 5.4 If F0 is a weight function representing the functional F0 and C the
contour of this representation, then F0 and C must satisfy, respectively in the case
(b) (the case (a) if α = 0)

Θ(x)
d2F0(x,α)

dx2
+
[
Ω(x) + 2α2

] dF0(x,α)
dx

+ [Π(x) − 2α] F0(x,α) = 0 (5.6)

and

[Θ(x)F0(x,α)p
′(x) −{(Θ(x)F0(x,α))′ + Ω(x)F0(x,α)}p(x)]C = 0, ∀p ∈P (5.7)

where ⎧⎨
⎩

Θ(x) = α2x−
(
σ2 + ασ1 + α2σ0

)
Ω(x) = −2αx +

(
2α2 + σ1 + 2ασ0

)
Π(x) = x−σ0.

Proof. A solution of the equation (5.3) must satisfy

〈D{D[Θ(x)F0] + Ω(x)F0} + Π(x)F,p(x)〉 = 0, ∀p ∈P ,



50 Boukhemis Ammar
7, 2(2005)

i.e.
〈F0, Θ(x)p′′(x)〉−〈F0, Ω(x)p′(x)) + 〈F, Π(x)p(x)〉 = 0,

as∫
C

Θ(x)F0(x,α)p
′′(x)dx−

∫
C

Ω(x)F0(x,α)p
′(x)dx +

∫
C

Π(x)F0(x,α)p(x)dx = 0,

by an integration by parts we obtain

[Θ(x)F0(x,α)p′(x) −{(Θ(x)F0(x,α))′ + Ω(x)F0(x,α)}p(x)]C
+
∫

C

{
[Θ(x)F0(x,α)]

′′ + [Ω(x)F0(x,α)]
′ + Π(x)F0(x,α)

}
p(x)dx = 0,

in particular if we take

[Θ(x)F0(x,α)]
′′ + [Ω(x)F0(x,α)]

′ + Π(x)F0(x,α) = 0

and

[Θ(x)F0(x,α)p
′(x) −{(Θ(x)F0(x,α))′ + Ω(x)F0(x,α)}p(x)]C = 0, ∀p ∈P.

Remark 5.5 In the case (a), the weight function F0 and the contour C must satisfy
respectively

−σ2
d2F0(x)
dx2

+ σ1
dF0(x)
dx

+ (x−σ0)F0(x) = 0 (5.8)
and [

−σ2 {F0(x)p(x)}′ + σ1F0(x)p(x)
]
C

= 0, ∀p ∈P (5.9)

Theorem 5.6 When σ2 < 0, the differential equation (5.6) has a general solution

F0(x,α) = (x + k)
λ

2 e
x

α

⎧⎨
⎩c1Jλ

⎡
⎣q(x + k)

1
2

⎤
⎦ + c2Yλ

⎡
⎣q(x + k)

1
2

⎤
⎦
⎫⎬
⎭ (5.10)

where

λ =
∣∣∣∣α

3 −ασ1 − 2σ2
α2

∣∣∣∣ , k = −α
2σ0 + ασ1 + σ2

α2
and q =

√−σ2
α2

and Jλ and Yλ are the Bessel functions of first and second kind respectively.

Proof. The equation (5.6) can be written

(x + k)
d2F0(x,α)

dx2
− 2

[x
α
−
(σ0
α

+
σ1
α2

+ 2
)] dF0(x,α)

dx

+
1
α2

(x−σ0 − 2α) F0(x,α) = 0
(5.11)



7, 2(2005)
On the classical 2−orthogonal polynomials sequences of ... 51

Let us denote by

r(x) =

x

α
−
(σ0
α

+
σ1
α2

+ 2
)

x + k
and put

F0(x) = W(x) exp
[∫

r(x)dx
]
,

then equation (5.11) may be written

(x + k)2
d2W(x)
dx2

− σ2
α2

[
x−α−σ0 −

σ1
2σ2

(
α2 − σ1

2

)]
W(x) = 0.

This last equation admits as a general solution

W(x) = (x + k)
1
2

⎧⎨
⎩c1Jλ

⎡
⎣2q(x + k)

1
2

⎤
⎦ + c2Yλ

⎡
⎣2q(x + k)

1
2

⎤
⎦
⎫⎬
⎭ .

as ∫
r(x)dx = (x + k)

λ− 1
2 exp(

x

α
),

we find (5.10).

Theorem 5.7 In the case (b), choosing, as a contour, the interval C =] − k,∞[ ,
then the function

F b0 (x,α) = Const.(x + k)
λ

2 e
x

αJλ

⎡
⎣2
√−σ2
α2

(x + k)
1
2

⎤
⎦ ,α < 0 and σ2 < 0 (5.12)

is an integral representation of the functional F0, i.e.

〈F0,p(x)〉 =
∫

C

F b0 (x,α)p(x)dx, ∀p ∈P.

Proof. We have
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩

lim
x→−k+

F b0 (x,α) = lim
x→∞

F b0 (x,α) = 0

lim
x→−k+

(x + k)F b0 (x,α) = lim
x→−k+

(x + k)
dF b0 (x,α)

dx
= 0

lim
x→∞

(x + k)F b0 (x,α) = lim
x→∞

dF b0 (x,α)
dx

= 0

consequently the condition (5.7) is satisfied.
As F b0 (x,α) is a solution of the equation (5.6), with the choice of the interval

C =] −k,∞[ as a contour, F b0 (x,α) may be an integral representation of F0.



52 Boukhemis Ammar
7, 2(2005)

Corollary 5.8 In the case (b), the function F b1 (x,α) defined by

F b1 (x,α) =
α

σ2

{[
α (x−σ0) −σ1 −

σ2
α

] dF b0 (x,α)
dx

− (x−σ0 −α)F b0 (x,α)
}

(5.13)

is an integral representation of F1.

Proof. From the relation (5.4) we have

〈F1,p(x)〉 =
α

σ2
〈D

[
α (x−σ0) −σ1 −

σ2
α

]
F0 − (x−σ0)F0,p(x)〉

=
α

σ2
〈D

[
α (x−σ0) −σ1 −

σ2
α

]
F0,p(x)〉−

α

σ2
〈(x−σ0)F0,p(x)〉

= − α
σ2

∫
C

[
α (x−σ0) −σ1 −

σ2
α

]
F b0 (x,α)p

′(x)dx

− α
σ2

∫
C

(x−σ0) F b0 (x,α)p(x)dx, ∀p ∈P.

Integrating by parts the first term in the right hand side we find

∫
C
F b1 (x,α)p(x)dx =

α

σ2

∫
C

[
α (x−σ0) −σ1 −

σ2
α

] dF b0 (x,α)
dx

p(x)dx

− α
σ2

∫
C

(x−σ0 −α)F b0 (x,α)p(x)dx
− α
σ2

[{
α (x−σ0) −σ1 −

σ2
α

}
F b0 (x,α)p(x)

]
C

As the last term is zero we obtain the relation (5.13).

Theorem 5.9 When α = 0 ( the case (a) ), the equation (5.8) admits as general
solution

F0(x) =
(
x−σ0 +

σ21
4σ2

)1
2

exp(
σ1
2σ2

x)

⎧⎪⎨
⎪⎩k1J1

3

⎡
⎢⎣ 2

3
√−σ2

(
x−σ0 +

σ21
4σ2

)3
2

⎤
⎥⎦

+ k2Y1
3

⎡
⎢⎣ 2

3
√−σ2

(
x−σ0 +

σ21
4σ2

)3
2

⎤
⎥⎦
⎫⎪⎬
⎪⎭

(5.14)

Proof. The equation (5.8) may also be written as

d2F0(x)
dx2

− σ1
σ2

dF0(x)
dx

− 1
σ2

(x−σ0)F0(x) = 0. (5.15)

Let us put
F0(x) = V (x) exp(

σ1
2σ2

),



7, 2(2005)
On the classical 2−orthogonal polynomials sequences of ... 53

by substitution, V must then satisfy

d2V (x)
dx2

− 1
σ2

(x−σ0 +
σ21
4σ2

)V (x) = 0.

This equation is of the type
d2V (X)
dX2

− 1
σ2
XV (X), where X = x−σ0 +

σ21
4σ2

, the

general solution of which is

V (X) = X
1
2

⎧⎨
⎩k1J1

3

⎛
⎝ 2

3
√−σ2

X

3
2

⎞
⎠ + k2Y1

3

⎛
⎝ 2

3
√−σ2

X

3
2

⎞
⎠
⎫⎬
⎭ .

Going back to the initial variable x and the function F0, we find (5.13).

Theorem 5.10 Choosing as a contour the interval C =]σ0 −
σ21
4σ2

,∞[, the function

F a0 (x) = Const.
(
x−σ0 +

(σ1)2

4σ2

)1
2

exp (
σ1
2σ2

x)J1
3

[
2

3
√−σ2

(
x−σ0 +

(σ1)2

4σ2

)3
2
]
,

σ2 < 0 is an an integral representation of functional F0 in the case (a).
Proof. As F a0 is a solution of (5.8) and⎧⎪⎨

⎪⎩
lim

x→a+
F a0 (x) = lim

x→∞
F a0 (x) = 0, and

lim
x→a+

dF b0 (x)
dx

= 0 = lim
x→∞

dF a0 (x)
dx

= 0, where a =
σ21
4σ2
−σ0,

the conditions of the proposition (5.2) are then satisfied and F a0 is an integral
representation of the functional F0.
Corollary 5.11 In the case (a), the function F a1 (x) defined by

F a1 (x) = −
dF a0 (x)
dx

(5.16)

is an integral representation of the functional F1.
Proof. It suffices to note, according to (5.4), that F1 = −DF0.
Remark 5.12 We just proved that the class of 2−orthogonal polynomials of Sheffer-
Meixner type consists of 9 sequences. 5 of which are classical and 2 of them have
continuous weight functions. The investigation of the last 3 sequences (c), (d1) and
(d2) will be the subject of another talk.

Received: April 2003. Revised: November 2003.



54 Boukhemis Ammar
7, 2(2005)

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