c
Applied Mathematics E-Notes, 10(2010), 210-234 Available free at mirror sites of http://www.math.nthu.edu.tw/∼amen/
ISSN 1607-2510
A Survey On D-Semiclassical Orthogonal Polynomials∗
Abdallah Ghressi†, Lotfi Khériji‡§
Received 17 June 2009
Abstract
In this paper, a survey of the most basic results on characterizations, methods
of classification and processes of construction of D-semiclassical orthogonal polynomials is presented. In particular, the standard perturbation, the symmetrization process are revisited and some examples are carefully analyzed. Symmetrization after perturbation is studied and some examples are given.
1
Introduction and Preliminaries
By D-classical monic orthogonal polynomials sequences: (MOPS), we refer to Hermite,
Laguerre, Bessel and Jacobi polynomials where D is the derivative operator. The
orthogonality considered here is related to a form (regular linear form) [6,30] not only
to an inner product. Since 1939, a natural generalization of the D-classical character is
the D-semiclassical one introduced by J. A. Shohat in [41]. From 1985, this theory has
been developed, from an algebraic aspect and a distributional one, by P. Maroni and
extensively studied by P. Maroni and coworkers in the last decade [1,12,16,30,34,36].
A form u is called D-semiclassical when it is regular and satisfies the Pearson equation
D(Φu) + Ψu = 0
(1)
where (Φ, Ψ) are two polynomials , Φ monic with deg Φ ≥ 0 and deg Ψ ≥ 1. The corresponding (MOPS) {Bn }n≥0 is called D-semiclassical. Moreover, if u is D-semiclassical,
the class of u, denoted s is defined by
s := min max(deg Φ − 2, deg Ψ − 1)
(2)
where the minimum is taken over all pairs (Φ, Ψ) satisfying (1). In particular, the class
s is greater to 0 and when s = 0 the D-classical case is recovered [34].
In 1985, M. Bachene [3, page 87] gave the system fulfilled by the coefficients of the threeterm recurrence relation of a D-semiclassical (MOPS) of class 1 using the structure
∗ Mathematics
Subject Classifications: 42C05, 33C45.
des Sciences de Gabès Route de Mednine 6029-Gabès, Tunisia.
‡ Institut Supérieur des Sciences Appliquées et de Technologie de Gabès Rue Omar Ibn El Khattab
6072-Gabès, Tunisia.
§ The second author is pleased to dedicate this work to Professor Pascal Maroni founder of the
research group in orthogonal polynomials at Gabès, Tunisia.
† Faculté
210
A. Ghressi and L. Khériji
211
relation (see (26) below). In 1992, S. Belmehdi [4,page 272] gave the same system (in
a more simple way) using the Pearson equation (1). This system is not linear and it is
very difficult to solve. As a consequence, D-semiclassical forms of class 1 are classified
by S. Belmehdi in [4] through a distributional study by taking into account (1) and by
giving an integral representation for any canonical case except the case of Bessel kind.
In 1996, J. Alaya and P. Maroni have established the linear system fulfilled by
the coefficients of the three-term recurrence relation of a symmetric D-Laguerre-Hahn
(MOPS) of class 1 [1] that is to say the (MOPS) associated with a regular form u
satisfying the functional equation
D(Φu) + Ψu + B x−1 u2 = 0
where Φ, Ψ, B are three polynomials with Φ monic and
max deg Ψ − 1, max(deg Φ, deg B) − 2 = 1.
Consequently, the authors give the classification of symmetric D-semiclassical forms of
class 1 as a particular case (B ≡ 0). There are three canonical situations:
• The generalized Hermite form H(µ) (µ 6= 0 , µ 6= −n− 12 , n ≥ 0) and its (MOPS)
satisfying
(
βn = 0 , γn+1 = 12 (n + 1 + µ(1 + (−1)n )) , n ≥ 0,
(3)
D(xH(µ)) + {2x2 − (2µ + 1)}H(µ) = 0.
• The generalized Gegenbauer G(α, β) (α 6= −n−1 , β 6= −n−1 , β 6= − 12 , α+β 6=
−n − 1 , n ≥ 0) and its (MOPS) satisfying

βn = 0 , n ≥ 0,




(n+β+1)(n+α+β+1)

 γ2n+1 = (2n+α+β+1)(2n+α+β+2)
, n ≥ 0,
(n+1)(n+α+1)

γ2n+2 = (2n+α+β+2)(2n+α+β+3)
, n ≥ 0,


n
o


 D x(x2 − 1)G(α, β) + − 2(α + β + 2)x2 + 2(β + 1) G(α, β) = 0.
(4)
For further properties of the generalized Hermite polynomials, the generalized
Gegenbauer polynomials and their orthogonality relations see [1,4,6,14,40].
• The form B[ν] of Bessel kind (ν 6= −n − 1 , n ≥ 0) and its (MOPS) satisfying

1
βn = 0 , γ1 = − 4(ν+1)
,




n+1

 γ2n+2 = 41 (2n+ν+1)(2n+ν+2)
, n ≥ 0,
(5)
1
n+1+ν
γ2n+3 = − 4 (2n+ν+2)(2n+ν+3) , n ≥ 0,



n
o


 D x3 B[ν] − 2(ν + 1)x2 + 1 B[ν] = 0.
2
212
D-Semiclassical Orthogonal Polynomials
For an integral representation of B[ν] and some additional features of the associated
(MOPS) see [13,39].
Other families of D-semiclassical orthogonal polynomials of class greater to 1 were
revealed by solving functional equations of the type
P (x)e
u = Q(x)u
(6)
where P, Q are two polynomials cunningly chosen and u, e
u two forms. In fact, the
product of a form by a polynomial is one of the old construction process of forms;
in 1858, Christoffel proved that the product of a positive definite form by a positive
polynomial is still a positive definite form [7]. This result has been generalized by J.
Dini and P. Maroni in [9] since 1989 where it was proved that, under certain regularity
conditions, the product of a regular form u
e by a polynomial P is a regular form. In
that work, the D-semiclassical character is studied. It is also interesting to consider the
inverse problem, which consists of the determination of all regular form u
e satisfying (6)
where Q(x) = −λ ∈ C − {0} and u is a given regular form. P. Maroni has considered
the case P (x) = x − τ , τ ∈ C in 1990 [31], the case P (x) = x2 in 1996 [35] and the
case P (x) = x3 in 2003 [36].
In 1992, F. Marcellan and P. Maroni have considered the case P (x) = Q(x) =
x − τ , where u is a given regular D-semiclassical form [23]. For other published papers
concerning the problem (6) see [12,15,17,21,22,43].
Symmetric D-semiclassical forms of class 2 satisfying (1) with Φ(0) = 0 are well
described in [39] by M. Sghaier and J. Alaya (in 2006) through there an original characterization by taking into account (6). In 2007, symmetric D-semiclassical forms of
class 2 are also classified by a distributional study likewise in [4], by A. M. Delgado
and F. Marcellan [8] but it seems that integral representations are given only in the
positive definite cases (see Remark 4.2. below).
For other relevant research work on the subject from other points of view and with
perhaps other operators see [2,5,20,38].
The aim of this survey is threefold. First, to present an overview about the characterizations and processes of construction of D-semiclassical orthogonal polynomials
( see section 1 and section 2). Then, to revise the standard perturbation (see (6) for
deg P = 1 and deg Q = 0) by taking into account the framework [31] and the limiting
case (q → 1) in [12] and to analyze the symmetrization process according to [2,33] by
adding some complementary results (see section 3 and section 4). Finally, to highlight
some examples of D-semiclassical orthogonal polynomials of class greater to 1 by using
[37] ( see section 2), by applying the standard perturbation ( see section 3) and by
combining the processes of symmetrization and perturbation (see section 4).
Now, we are going to introduce the material concerning orthogonal polynomials
and regular forms that find their origin in the book of T. S. Chihara (1978) [6] and
developed by P. Maroni since 1981 [24,25,29,30].
Let P be the vector space of polynomials with coefficients in C and let P 0 be its
topological dual. We denote by hu, fi the effect of u ∈ P 0 on f ∈ P. In particular,
we denote by (u)n := hu, xn i , n ≥ 0 the moments of u. Moreover, a form u is called
symmetric if (u)2n+1 = 0, n ≥ 0. For any form u, any polynomial g , let gu , be the
form defined by duality
hgu, fi := hu, gfi , f ∈ P.
(7)
213
A. Ghressi and L. Khériji
For f ∈ P and u ∈ P 0 , the product uf is the polynomial
xf (x) − ζf (ζ)
(uf) (x) := u,
.
x−ζ
(8)
The derivative u0 = Du of the form u is defined by
hu0 , fi := − hu, f 0 i , f ∈ P.
(9)
(fu)0 = f 0 u + fu0 , u ∈ P 0 , f ∈ P .
(10)
We have [30]
0
The formal Stieltjes function of u ∈ P is defined by
S (u) (z) := −
X (u)
n
.
z n+1
(11)
n≥0
Similarly, with the definitions
hha u, fi := hu, hafi = hu, f (ax)i , u ∈ P 0 , f ∈ P , a ∈ C − {0} ,
(12)
hτb u, fi := hu, τ−b fi = hu, f (x + b)i , u ∈ P 0 , f ∈ P , b ∈ C
(13)
hδc , fi := f(c) , f ∈ P , c ∈ C.
(14)
hu, PmPn i = rn δn,m , n, m ≥ 0 ; rn 6= 0 , n ≥ 0.
(15)
and
The form u is called regular if we can associate with it a polynomial sequence {Pn }n≥0 ,
deg Pn = n, such that
The polynomial sequence {Pn }n≥0 is then said orthogonal with respect to u. Necessarily, {Pn }n≥0 is an (OPS) where every polynomial can be supposed monic and it fulfils
the three-term recurrence relation

 P0 (x) = 1 , P1 (x) = x − β0 ,
(16)

Pn+2 (x) = (x − βn+1 ) Pn+1 (x) − γn+1 Pn (x) , n ≥ 0
hu,xP 2 i
hu,P 2
i
6= 0, n ≥ 0. Moreover, the regular form u,
with βn = hu,P 2ni and γn+1 = hu,Pn+1
2
n
ni
associated to the (MOPS) {Pn }n≥0 satisfying (16), is said to be positive definite if and
only if βn ∈ R and γn+1 > 0 for all n ≥ 0.
The form u is said to be normalized if (u)0 = 1. In this paper, we suppose that any
form will be normalized.
p(x)−p(c)
From the linear application p 7→ (θc p) (x) =
, p ∈ P , c ∈ C , we define
x−c
−1
(x − c) u by
D
E
(x − c)−1 u, p := hu, θcpi
(17)
and we have [30,33]
(x − c) (x − c)−1 u = u ;
(x − c)−1 (x − c)u = u − (u)0 δc , u ∈ P 0 , c ∈ C. (18)
214
D-Semiclassical Orthogonal Polynomials
Finally, we introduce the operator σ : P −→ P defined by (σf)(x) := f(x2 ) for all
f ∈ P. Consequently, we define σu by duality
hσu, fi := hu, σfi , f ∈ P , u ∈ P 0
(19)
and we have for f ∈ P , u ∈ P 0 [33]
f(x)σu = σ(f(x2 )u) ;
2
σu0 = 2(σ(xu))0 .
(20)
Overview about D-Semiclassical Forms
Now, we will expose the D-semiclassical character according to P. Maroni [26,27,28,30].
Let Φ monic and Ψ be two polynomials , deg Φ = t , deg Ψ = p ≥ 1. We suppose
that the pair (Φ, Ψ) is admissible , i.e. when p = t − 1 , writing Ψ (x) = ap xp + ..., then
ap 6= n + 1 , n ∈ N.
DEFINITION 2.1. A form u is called D-semiclassical when it is regular and satisfies
the functional equation
(Φu)0 + Ψu = 0
(21)
where the pair (Φ, Ψ) is admissible. The corresponding orthogonal sequence {Pn }n≥0
is called D-semiclassical.
REMARKS 2.1.
1. The D-semiclassical character is kept by shifting( see [30]). In fact, let
−n
a (ha ◦ τ−b Pn ) n≥0 , a 6= 0 , b ∈ C;
when u satisfies (21), then ha−1 ◦ τ−b u fulfils the equation
a−t Φ (ax + b) ha−1 ◦ τ−b u
01−t
Ψ (ax + b) ha−1 ◦ τ−b u = 0.
(22)
2. The D-semiclassical form u is said to be of class s = max(p − 1, t − 2) ≥ 0 if and
only if
Y n
o
(23)
Ψ(c) + Φ0 (c) + u, θcΨ + θc2 Φ > 0,
c∈ZΦ
where ZΦ is the set of zeros of Φ. The corresponding orthogonal sequence {Pn }n≥0 will
be known as of class s [30].
3. When s = 0, the form u is usually called D-classical Hermite, Laguerre, Bessel,
and Jacobi [34].
We can state characterizations of D-semiclassical orthogonal sequences. {Pn }n≥0
is D-semiclassical of class s, if and only if one of the following statements holds (see
[31] and q = 1 in section 2. of [12])
(1). The formal Stieltjes function of u satisfies a non homogeneous first order linear
differential equation
Φ(z)S 0 (u) (z) = C0 (z) S (u) (z) + D0 (z) ,
(24)
215
A. Ghressi and L. Khériji
where
C0 (z) = −Ψ(z) − Φ0 (z),
D0 (z) = −(uθ0 Φ)0 (z) − (uθ0 Ψ)(z).
(25)
Φ and Ψ are the same polynomials as in (21).
(2). {Pn }n≥0 satisfies the following structure relation
0
Φ (x) Pn+1
(x) =
1
(Cn+1 (x) − C0 (x)) Pn+1 (x) − γn+1 Dn+1 (x) Pn (x) , n ≥ 0, (26)
2
where
Cn+1 (x) = −Cn (x) + 2 (x − βn ) Dn (x) , n ≥ 0
(27)
γn+1 Dn+1 (x) = −Φ (x) + γn Dn−1 (x)
2
+ (x − βn ) Dn (x) − (x − βn ) Cn (x) , n ≥ 0,
(28)
Φ, Ψ, C0, D0 are the same parameters introduced in (1); βn , γn are the coefficients of
the three term recurrence relation (16). Notice that D−1 (x) := 0, deg Cn ≤ s + 1 and
deg Dn ≤ s, n ≥ 0.
(3). Each polynomial Pn+1 , n ≥ 0 satisfies a second order linear differential equation
0
00
(x) + K (x, n) Pn+1
(x) + L (x, n) Pn+1 (x) = 0 , n ≥ 0,
(29)
J (x, n) Pn+1
with
and

J (x, n) = Φ (x) Dn+1(x) ,




0

(x) Φ (x) ,
 K (x, n) = Dn+1 (x) Φ0 (x) + C0 (x) − Dn+1
0
(x) −
n) = 12 Cn+1 (x) − C0 (x) Dn+1

 L (x,



1
0
0
 − Cn+1 − C0 (x) Dn+1 (x) − Dn+1 (x) Σn (x) , n ≥ 0.
2
Σn (x) :=
n
X
k=0
Dk (x) , n ≥ 0.
(30)
(31)
Φ, Cn , Dn are the same in the previous characterization. Notice that deg J(., n) ≤
2s + 2 , deg K(., n) ≤ 2s + 1 and deg L(., n) ≤ 2s. In particular, when s = 0 that is to
say the D-classical case, the coefficients of the structure relation (26) become
 C (x)−C (x)
n+1
0

= 12 Φ00 (0)((n + 1)x − Sn )+

2

(32)
+(Ψ0 (0) − Φ00 (0)(n + 1))βn+1 + (Ψ(0) − Φ0 (0)(n + 1)),



Dn+1 (x) = 12 Φ00 (0)(2n + 1) − Ψ0 (0), n ≥ 0,
with Sn =
Pn
k=0 βk , n
≥ 0. Also we get for (30)


 J(x, n) = Φ(x) ,
K(x, n) = −Ψ(x) ,


L(x, n) = (n + 1)(Ψ0 (0) − 12 Φ00 (0)n) , n ≥ 0.
(33)
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D-Semiclassical Orthogonal Polynomials
In 1990, P. Maroni gives the following result about order of zero of a D-semiclassical
polynomial Pn+1 of class s [31].
REMARK 2.2. Taking into account the structure relation (26), the polynomial
sequence {Dn+1 }n≥0 gives us some information about zeros of the polynomial Pn+1 .
In fact, if c is a zero of order η of Pn+1 , n ≥ 1 with η ≥ 2, then η ≤ s + 1 and c is a
zero of order η − 1 of Dn+1 .
In 1996, J. Alaya and P. Maroni give the following result to a symmetric D-LaguerreHahn form [1] which remains valid in the symmetric D-semiclassical case
PROPOSITION 2.1. Let u be a symmetric D-semiclassical form of class s satisfying
(21). The following statements hold
i) When s is odd then the polynomial Φ is odd and Ψ is even.
ii) When s is even then the polynomial Φ is even and Ψ is odd.
Finally, about integral representation (P. Maroni [32] in 1995), let u be a Dsemiclassical form satisfying (21). We are looking for an integral representation of
u and consider
Z
+∞
hu, fi =
−∞
U (x)f(x)dx , f ∈ P,
(34)
where we suppose the function U to be absolutely continuous on R, and is decaying as
fast as its derivative U 0 . From (21) we get
Z
+∞
−∞
+∞
(ΦU )0 + ΨU f(x)dx − Φ(x)U (x)f(x) −∞ = 0 , f ∈ P.
Hence, from the assumptions on U , the following conditions hold
Z
+∞
Φ(x)U (x)f(x) −∞ = 0 , f ∈ P,
+∞
−∞
Condition (36) implies
(ΦU )0 + ΨU f(x)dx = 0 , f ∈ P.
(ΦU )0 + ΨU = ωg,
(35)
(36)
(37)
where ω 6= 0 arbitrary and g is a locally integrable function with rapid decay representing the null-form
Z +∞
xn g(x)dx = 0 , n ≥ 0.
(38)
−∞
Conversely, if U is a solution of (37) verifying the hypothesis above and the condition
Z
+∞
−∞
U (x)dx 6= 0,
then (35)-(36) are fulfilled and (34) defines a form u which is a solution of (21).
(39)
217
A. Ghressi and L. Khériji
In particular, from (Φu)0 + Ψu
Φ(x) =
t
X
n
= 0, n ≥ 0, writing
ck x k ;
Ψ(x) =
k=0
p
X
ak x k
k=0
and by taking into account (9), we have for the moments of u
(u)0 = 1 ;
p
X
k=0
ak (u)k = 0;
p+1
X
k=1
ak−1 (u)n+k − (n + 1)
t
X
k=0
ck (u)n+k = 0, n ≥ 0. (40)
In [37], P. Maroni and M. Ihsen Tounsi (2004) have described all (MOPS) which are
identical to their second associated sequence that is to say those satisfying (16) with
βn+2 = βn ;
γn+3 = γn+1 ,
n ≥ 0.
The resulting polynomials are D-semiclassical of class s ≤ 3. In fact there are five
canonical situations itemized (a)-...-(e) (see Proposition 3.5 in [37]). The characteristic
elements of the structure relation (26) and the second-order differential equation (29)
are given explicitly by using the quadratic decomposition [33]. Integral representations of the corresponding forms are also given by an other process which consists of
representing the corresponding Stieljes function (11). So, our interest in the following example is to describe the canonical situation (d) in that work but with processes
exposed in this section.
EXAMPLE 2.1. Let’s consider the D-semiclassical form u of class 1 and its (MOPS)
{Pn }n≥0 . We have [37]

1
n

 βn = (−1) , γn+1 = − 4 , n ≥ 0 ,
02
(41)

+ x + 2)u = 0.
 x(x2 − 1))u
For the moments of u, from (40)-(41) we get
(
(u)0 = 1,
(u)1 = 1 ,
−(n + 4)(u)n+2 + (u)n+1 + (n + 2)(u)n = 0 , n ≥ 0.
(42)
Consequently, it is easy to prove by induction that
(u)2n+1 = (u)2n , n ≥ 0.
(43)
Now, taking n ← 2n in the second equality in (42) and by virtue of (43) we get
(u)2n+2 =
2n + 3
(u)2n , n ≥ 0
2n + 4
from which we derive the following
(u)2n+1 = (u)2n =
1 (2n + 1)!
, n ≥ 0.
n + 1 22n (n!)2
(44)
218
D-Semiclassical Orthogonal Polynomials
For an integral representation of u of type (34), equation (37) with ω = 0 and hypothesis
becomes
1
1
1
0
2
2
U (x) + − +
−
U (x) = 0.
(45)
x x−1 x+1
A possible solution of (45) is the function

0
, x ≤ −1,


q
1+x
U (x) =
ρ x 1−x , −1 < x < 1,


0
, x ≥ 1.
(46)
First, condition (35) is fulfilled because
2
r
n
x(x − 1)x ρ x
Second, with the change of variable t =
−1
ρ
=
Z
1
−1
x
r
Z
1+x
dx =
1−x
where
Jk =
Z
1+x
1−x
q
+∞
0
+∞
0
1+x
1−x
+1
−1
= 0 , n ≥ 0.
the normalization constant ρ satisfies
4t2 (t2 − 1)
dt = 4J1 − 12J2 + 8J3
(t2 + 1)3
dt
, k ≥ 1.
(t2 + 1)k
But, upon integration by parts we get
Jk+1 =
In particular, J1 = π2 , J2 =
integral representation
π
4
2
hu, fi =
π
2k − 1
Jk , k ≥ 1.
2k
and J3 =
Z
1
−1
x
r
3π
16 .
Thus, ρ−1 =
π
2
and u has the following
1+x
f(x) dx , f ∈ P.
1−x
(47)
For the structure relation (26), we may write it as follows
1
0
x(x2 − 1)Pn+1
(x) = (n + 1)x2 + bn x + cn Pn+1 (x) +
dn x + en Pn (x) , n ≥ 0. (48)
4
The problem is then to determine the coefficients bn , cn , dn , en , n ≥ 0. Taking x = 0,
x = 1, x = −1 respectively in (48) we get

1

 cn Pn+1 (0) + 4 en Pn (0) = 0
(n + 1) + bn + cn Pn+1 (1) + 41 (dn + en )Pn (1) = 0
, n ≥ 0.
(49)


(n + 1) − bn + cn Pn+1 (−1) + 14 (−dn + en )Pn (−1) = 0
219
A. Ghressi and L. Khériji
Moreover, from (41) and (48), the formulas in (25)-(26) become
(
Cn (x) = (2n + 1)x2 + (2bn−1 − 1)x + 2cn−1 − 1
Dn (x) = dn−1x + en−1
,n ≥ 0
(50)
where b−1 := 0, c−1 := 0, d−1 := 2, e−1 := 2. Consequently, replacing in (27)-(28)
we get after identification

dn = 2(n + 2)




n

2b
n + 2bn−1 − 2en−1 = 2 1 − 2(−1) (n + 1)



 2c + 2c
n
n
n−1 + 2(−1) en−1 = 2
, n ≥ 0.
(51)

2bn−1 + en−1 = 1 + (−1)n (−2n + 1)





−2(−1)n bn−1 + 2cn−1 − 2(−1)n en−1 = 2n + 12 − (−1)n



−2(−1)n cn−1 + 41 en + 34 en−1 + = (−1)n+1
On the other hand, regarding (49) we are going to compute Pn (0), Pn (1), Pn (−1) for
any n ≥ 0.
For Pn (0), taking x = 0 in (16) and on account of (41) we obtain
1
Pn+2 (0) = −(−1)n+1 Pn+1 (0) + Pn (0), n ≥ 0 ; P0 (0) = 1 ; P1 (0) = −1.
4
Thus,
 
Pn+2 (0) +

(−1)n+1
Pn+1 (0)
2
=
(−1)n
2
Pn+1 (0) +
(−1)n
Pn (0)
2
P1 (0) + 12 P0 (0) = − 12 .
, n ≥ 0,
Therefore,
(n−1)n
(−1)n
(−1) 2
Pn+1 (0) = −
Pn (0) −
2
2n+1
from which we deduce that
Pn (0) =
, n≥0
n(n+1)
n+1
(−1) 2 , n ≥ 0.
2n
(52)
For Pn (1), taking x = 1 in (16) and on account of (41) we obtain
1
Pn+2 (1) = (1 − (−1)n+1 )Pn+1 (1) + Pn (1), n ≥ 0 ; P0 (1) = 1 ; P1 (1) = 0.
4
(53)
With n ← 2n and n ← 2n − 1 successively in (53) we deduce
(
P2n+2 (1) = 2P2n+1(1) + 14 P2n (1), n ≥ 0
P2n+1 (1) = 14 P2n−1 (1), n ≥ 1
which gives
P2n+1 (1) = 0, n ≥ 0 ;
P2n (1) =
1
, n ≥ 0.
22n
(54)
220
D-Semiclassical Orthogonal Polynomials
For Pn (−1), taking x = −1 in (16) and on account of (41) we obtain
(
Pn+2 (−1) = −(1 + (−1)n+1 )Pn+1 (−1) + 41 Pn (−1) , n ≥ 0,
P0 (−1) = 1 ; P1 (−1) = −2.
(55)
With n ← 2n and n ← 2n − 1 successively in (55) we deduce
(
P2n+2 (−1) = 14 P2n (−1), n ≥ 0
P2n+1 (−1) = 14 P2n−1(−1) −
1
22n−1 ,
n≥1
which gives
P2n(−1) =
1
, n≥0 ;
22n
P2n+1 (−1) = −
n+1
n ≥ 0.
22n−1
(56)
Now, replacing the numbers in (52), (54), (56) in the system (49) and by taking into
account the second system (51) we obtain

b2n = 1
;
b2n+1 = 0





 c2n = −(2n + 1) ; c2n+1 = −2(n + 1)
, n ≥ 0.
(57)

dn = 2(n + 2)




 e = −4(n + 1) ; e
2n
2n+1 = 2(2n + 3)
So, it is quite straightforward to get the expressions in (26) and (30) for the structure
relation and the second order linear differential equation.
3
3.1
The Standard Perturbation Revisited
On the standard perturbation
Let u be a regular form. Denoting by {Pn }n≥0 its (MOPS) satisfying (16). Let u
e ∈ P0
satisfying
(x − τ )e
u = λu , τ ∈ C, λ ∈ C − {0}.
(58)
According to (18) relation (58) is equivalent to
u
e = δτ + λ(x − τ )−1 u
with constraints (e
u)0 = 1, (u)0 = 1 and λ + τ = (e
u)1 .
e
Suppose u
e regular and let {Pn }n≥0 be its (MOPS)

 Pe0 (x) = 1 , Pe1 (x) = x − βe0 ,
 e
Pn+2 (x) = (x − βen+1 )Pen+1 (x) − e
γn+1 Pen (x) , n ≥ 0.
The connection between Pen and Pn is (see [31])
Pe0 (x) = 1 , Pen+1 (x) = Pn+1 (x) + an Pn (x) , n ≥ 0,
(59)
(60)
(61)
221
A. Ghressi and L. Khériji
with
(1)
an = −
where
Pn(1)(x)
:=
Pn+1 (τ ) + λPn (τ )
(1)
Pn (τ ) + λPn−1 (τ )
6= 0 , n ≥ 0,
Pn+1 (x) − Pn+1 (ξ)
u,
x−ξ
(62)
, n ≥ −1.
We have [30]
(1)
Pn+1 (x)Pn+1 (x) − Pn+2 (x)Pn(1)(x) =
n
Y
k=0
γk+1 , n ≥ 0.
(63)
Denoting
λn = −
Pn (τ )
(1)
Pn−1 (τ )
, n ≥ 1 , λ0 = 0.
(64)
Let us recall the fundamental result owing to P. Maroni (1990) [31]
PROPOSITION 3.1. Let u be a regular form. The following statements are equivalent
i) The form u
e = δτ + λ(x − τ )−1 u is regular.
ii) λ 6= λn , n ≥ 0.
We may write
γn+1
+ an+1 − βn+1 = −τ , n ≥ 0,
an
(65)
βe0 = β0 −a0 = τ +λ , βen+1 = βn+1 +an −an+1 , γen+1 = −an (an −βn +τ ) , n ≥ 0, (66)

 (x − τ )Pn (x) = Pen+1 (x) + (βn − an − τ )Pen (x) , n ≥ 0 ,
(67)

(x − τ )Pn+1 (x) = (x − an − τ )Pen+1 (x) + an (an − βn + τ )Pen (x) , n ≥ 0.
In particular, if the regular form u is symmetric then the form u
e = δ0 +λx−1 u is regular
for any λ 6= 0 and we have (γ0 := 1 ; γ−1 := 1) **

n
Q

γ2k

a2n = −λ

γ2k−1 , n ≥ 0,


k=0
(68)

n

Q
γ

2k+1
1
 a2n+1 =

λ
γ2k , n ≥ 0.
k=0
3.2
The D-semiclassical case
Suppose u be a D-semiclassical form of class s satisfying (21). Multiplying (21) by λ
and on account of (58) we get
0
e u + Ψe
e u = 0,
Φe
(69)
222
D-Semiclassical Orthogonal Polynomials
with
e (x) = (x − τ )Φ (x) , Ψ
e (x) = (x − τ )Ψ (x) .
Φ
(70)
Φ(τ ) 6= 0 , λ 6= λn , n ≥ 0 or Φ(τ ) = 0 , λ 6= λn , n ≥ −1,
(71)
Now, taking q = 1 in section 3.1 of [12], we recover the following result concerning the
class of u
e [31]
THEOREM 3.1. If the D-semiclassical form u is of class s then the form u
e is
D-semiclassical of class se = s + 1 for
where
λ−1 = −
Ψ(τ ) + Φ0 (τ )
.
hu, θτ Ψ + θτ2 Φi
(72)
REMARK 3.1. Also, taking q = 1 in section 3.2
n ofo [12], we recover again the
results in [31] concerning the structure relation of Pen
and the second order
n≥0
linear differential equation satisfied by Pen+1 , n ≥ 0.
Finally, if we suppose that the form u has the following integral representation:
Z +∞
Z +∞
hu, fi =
U (x) f (x) dx , f ∈ P ;
U (x) dx = 1
−∞
−∞
where U is a locally integrable function with rapid decay and continuous at the point
x = τ then in view of (59) we may write
Z +∞
Z +∞
U (x)
U (x)
he
u, fi = 1 − λP
dx f (0) + λP
f (x) dx , f ∈ P ,
(73)
−∞ x − τ
−∞ x − τ
where
P
Z
+∞
−∞
U (x)
dx := lim
x−τ
ε 0+
Z
τ−ε
−∞
U (x)
dx +
x−τ
Z
+∞
τ+ε
U (x)
dx .
x−τ
(74)
In particular, taking q = 1 in (5) of the document [12], the moments of u
e are given by
(e
u)0 = 1 ;
(e
u)n = τ n + λ
n
X
k=1
τ n−k (u)k−1 , n ≥ 1.
(75)
EXAMPLE 3.1. Let us consider u the D-semiclassical form of class 1 studied in
Example 2.1. and u
e = δ−1 + λ(x + 1)−1 u be its standard perturbed (τ = −1). Taking
x = −1 in (63), by virtue of (41) and the fact that Pn (−1) 6= 0, n ≥ 0 ( see (56)) we
get
Pn+2 (−1) (1)
(−1)n+1
(1)
Pn+1 (−1) =
Pn (−1) + 2n+2
, n≥0
Pn+1 (−1)
2
Pn+1 (−1)
which gives
(1)
Pn+1 (−1) =
n n
k Y
X
Y
Pk+2 (−1)
(−1)k+1
Pl+1 (−1)
1+
, n ≥ 0.
Pk+1 (−1)
22k+2 Pk+1 (−1)
Pl+2 (−1)
k=0
k=0
l=0
223
A. Ghressi and L. Khériji
Consequently, in accordance with (56) it follows
(1)
P2n+1 (−1) = 0
1
, n ≥ 0.
22n
(76)
1
, n≥0
4(2n + 2 − λ)
(77)
(1)
;
P2n (−1) =
Thus, (62) and (64) can be written as
a2n = 2n + 2 − λ
;
a2n+1 =
λ2n+1 = 2(n + 1) , n ≥ 0.
(78)
−1
By virtue of Proposition 3.1., The form u
e = δ−1 + λ(x + 1) u is regular if and only if
λ 6= 2(n + 1), n ≥ −1. In particular, from (41) and (77), the relations in (66) become


βe0 = λ − 1




1
 βe2n+1 = 2n + 1 − λ −
4(2n+2−λ)
, n ≥ 0.
(79)
1

− 2n − 3 + λ
βe2n+2 = 4(2n+2−λ)




 γe
1
= −(2n + 2 − λ)(2n − λ) ; e
γ
=−
2n+1
2n+2
16(2n+2−λ)2
In accordance with Theorem 3.1. u
e is D-semiclassical of class 2 for any λ 6= 2(n +
1), n ≥ −1 satisfying the functional equation
02
2
x(x − 1)(x + 1) )e
u
+ x + 2)e
u = 0.
(80)
For the moments of u
e, with (44) and (75) we get
(e
u)2n = 1 ;
(e
u)2n+1 = λ − 1 , n ≥ 0.
(81)
For an integral representation of u
e, in view of (47) and (73) and by the fact that
R1
√ x
dx
=
0
we
may
write
−1 1−x2
he
u, fi = f(−1) + λ
3.3
Z
1
−1
√
x
f(x) dx , f ∈ P.
1 − x2
(82)
The study of δ0 + λx−1 B(α)
Consider u = B(α), α 6= − n2 , n ≥ 0 the D-classical Bessel form. we have [32]

1−α
, n ≥ 0,
β0 = − α1 , βn+1 = (n+α)(n+α+1)




(n+1)(n+2α−1)
γn+1 = − (2n+2α−1)(n+α)2(2n+2α+1) , n ≥ 0,

0


 x2 B(α) − 2(αx + 1)B(α) = 0,
and for α ≥ 6
2 4
,f
π
∈P
hB(α), fi =
Tα−1
Z
0
+∞
1
x2
Z
x
+∞
x 2α
t
2 2
exp
−
s(t)dtf(x)dx,
t
x
(83)
(84)
224
D-Semiclassical Orthogonal Polynomials
with
Z +∞ 2α
1
x
2 2
Tα =
exp
−
s(t)dtdx,
x2 x
t
t
x
0

x ≤ 0,
 0,
s(x) =
1
1

exp(−x 4 ) sin x 4 , x > 0.
Z
+∞
(85)
(86)
where s represents the null-form and given in 1895 by T. J. Stieljes [42]. The integral
representation (84) of the D-classical Bessel form is given by P. Maroni (1995) in [32].
In fact, the history of the Bessel form is more tortured than that other D-classical
forms and the reason is certainly that the Bessel form is not positive definite for any
value of the parameter α. For others representations through a function, a distribution
or through an ultra-distribution see A. J. Duran (1993) [10], A. M. Krall (1981)[19]
and W. D. Evans et al. (1992)[11], S. S. Kim et al. (1991) [18] respectively.
Taking into account the functional equation in (83), it is easy to see that the
moments of B(α) are
(B(α))n = (−1)n 2n
Γ(2α)
, n ≥ 0,
Γ(n + 2α)
(87)
where Γ is the Gamma function.
Putting x = 0 in (26) and by virtue of (83) and (32) we get
Pn (0) = 2n
Γ(n + 2α − 1)
, n ≥ 0.
Γ(2n + 2α − 1)
(88)
Also, taking x = 0 in (63) and in accordance with(88) and (83) we obtain the recurrence
relation
(1)
Pn+1 (0) =
n + 2α
Γ(2α)
P (1)(0) + (−1)n+1 2n+1
, n ≥ 0.
(2n + 2α + 1)(n + α + 1) n
Γ(2n + 2α + 2)
Therefore, an easy computation yields the equality
n
Pn(1)(0) = 2n+1
Γ(2α)Γ(n + 2α) X (−1)k k!(k + α)
, n ≥ 0.
Γ(2n + 2α + 1)
Γ(k + 2α)
k=0
By induction, we can easily prove that
n
X
(−1)k k!(k + α)
k=0
Γ(k + 2α)
=
1 2α − 1 1
(n + 1)!
+ (−1)n
, n ≥ 0,
2 Γ(2α)
2
Γ(n + 2α)
which gives (compare with [31])
Pn(1)(0) = 2n
(2α − 1)Γ(n + 2α) + (−1)n (n + 1)!Γ(2α)
, n ≥ 0.
Γ(2n + 2α + 1)
Thus, for (62) and (64) we get for n ≥ 0
an = −2
Γ(2n + 2α − 1)
Γ(2n + 2α + 1)
(89)
225
A. Ghressi and L. Khériji
2Γ(n + 2α) + λ (2α − 1)Γ(n + 2α) + (−1)n (n + 1)!Γ(2α)
,
×
2Γ(n + 2α − 1) + λ (2α − 1)Γ(n + 2α − 1) + (−1)n−1 n!Γ(2α)
and
λn = −2
Γ(n + 2α − 1)
, n ≥ 1.
(2α − 1)Γ(n + 2α − 1) + (−1)n−1 n!Γ(2α)
(90)
(91)
Consequently, in accordance with theorem 3.1. and (69)-(72) the form u
e = δ0 +
2
λx−1 B(α) is D-semiclassical of class e
s = 1 for any λ 6= λn , n ≥ −1 with λ−1 = 1−2α
and fulfils the functional equation (69) with
e
Φ(x)
= x3 ,
e
Ψ(x)
= −2x(αx + 1).
(92)
Also, the recurrence coefficients of {Pen }n≥0 are given by (66) with the above results
(88) and (83).
From (75) and (87), the moments of u
e are
(e
u)n = λ(−1)n−1 2n−1
Γ(2α)
, n ≥ 1 , (e
u)0 = 1.
Γ(n + 2α − 1)
(93)
When α ≥ 6( π2 )4 and on account of (73)-(74), (84)-(86) let us consider the function
1
e
U(x)
= 3
x
Z
+∞
x 2α
ξ
x
We have for n ≥ 0,
n
x U
e (x)
≤
≤
=
exp
2
2
s(ξ)dξ , x > 0.
x
2
x
Z
+∞
ξ −2α exp
ne
x U (x) = Θn (x) + O x
n+2α−3
2
exp −
x
Z 1
2
2
Θn (x) = xn+2α−3 exp −
ξ −2α exp
s(ξ)dξ.
x
ξ
x
Hence,
Z
0
(94)
2
1
exp −ξ 4 dξ
ξ
x
Z +∞
2
1 1
2
1 1
xn+2α−3 exp(− ) exp(− x 4 )
ξ −2α exp
exp − ξ 4 dξ
x
2
ξ
2
x
1 1
o xn+2α−3 exp − x 4 .
2
xn+2α−3 exp −
For 0 < x ≤ 1 and n ≥ 0
with
ξ
−
1
|Θn (x)|dx ≤
Z
0
1
ξ
−2α
exp
2
ξ
|s(ξ)|
Z
0
ξ
x
n+2α−3
exp
2
−
dx dξ.
x
226
D-Semiclassical Orthogonal Polynomials
Moreover, upon integration by parts we have
Z
=
≤
and hence
ξ
2
dx
x
0
Z ξ
1 n+2α−1
2 1
2
ξ
exp − − (n + 2α − 1)
xn+2α−2 exp − dx
2
ξ
2
x
0
Z ξ
2
1
2
1 n+2α−1
ξ
exp − + (n + 2α − 1)ξ
xn+2α−3 exp − dx,
2
ξ
2
x
0
xn+2α−3 exp −
Z
ξ
x
n+2α−3
0
for all 0 ≤ ξ ≤
It results in
2
n+2α−1
n+2α−1
exp − 2ξ
2
1 ξ
exp − dx ≤
x
2 1 − 21 (n + 2α − 1)ξ
, n ≥ 0.
Z
1
|Θn (x)|dx < +∞.
0
Consequently, we get the natural integral representation of u
e: for α ≥ 6( π2 )4 and f ∈ P
+λTα−1
Z
with
4
4.1
Teα =
+∞
0
Z
0
he
u, fi = (1 − λTα−1 Teα )f(0)
1
x3
+∞
Z
1
x3
+∞
x 2α
t
x
Z
+∞
x
exp
x 2α
t
2
exp
t
−
2
t
2
s(t)dtf(x)dx,
x
−
2
s(t)dtdx.
x
(95)
(96)
The Symmetrization Process Revisited
On the symmetrization process
Let u
b be a symmetric regular form and {Pbn }n≥0 be its (MOPS). They satisfy a three
term recurrence relation

 Pb0 (x) = 1, Pb1 (x) = x ,
(97)
 b
b
b
Pn+2 (x) = xPn+1 (x) − γbn+1 Pn (x) , n ≥ 0.
It is very well known (see T.S. Chihara [6] and P. Maroni [33]) that
Pb2n (x) = Pen (x2 ),
Pb2n+1(x) = xPn (x2 ) , n ≥ 0,
(98)
where {Pen }n≥0 and {Pn }n≥0 are the two (MOPS) related to regular forms u
e and xe
u,
respectively, with
u
e = σb
u.
(99)
227
A. Ghressi and L. Khériji
The form u
b is said to be the symmetrized of the form u
e.
In fact [6,33]
u
b is regular ⇐⇒ u
e and xe
u are regular.
u
b is positive definite ⇐⇒ u
e and xe
u are positive definite.
Furthermore, taking into account the three term recurrence relations for {Pen }n≥0 and
{Pn }n≥0 which are (60) and (16) respectively, we get [6,33]

e
γ1 ,

 β0 = b
βen+1 = b
γ2n+2 + γb2n+3 , n ≥ 0,


γn+1 = b
e
γ2n+1 b
γ2n+2 , n ≥ 0.
and
(
Consequently,

 b
γ1 = βe0 ,
4.2
(100)
βn = b
γ2n+1 + γb2n+2 , n ≥ 0,
γn+1 = b
γ2n+2 γb2n+3 , n ≥ 0.
γ2 =
b
Qn
γ1
e
e0 ,
β
γ
 b
γ2n+1 = βe0 Qnk=1 eγkk ,
k=1
The D-semiclassical case
γ2n+2 =
b
Qn+1
γk
1 Qk=1 e
,
n
e
β0
k=1 γk
(101)
n ≥ 1.
(102)
Regarding section two in [2], a study of the D-semiclassical character and the class of u
b
is done by J. Arvesú, M. J. Atia and F. Marcellán (in 2002) after determination of the
non homogeneous first order linear differential equation satisfied by the formal Stieltjes
function of u
b which derived from that of u
e. In the following theorem, we are going to
resume these results in terms of the polynomials coefficients in the Pearson equation
of u
e.
THEOREM 4.1. Let u
e be a regular form and {Pen }n≥0 its (MOPS) such that
Pen+1 (0) 6= 0, n ≥ 0. Then its symmetrized form u
b is regular.
If u
e is D-semiclassical of class e
s satisfying the functional equation
e u)0 + Ψe
eu = 0
(Φe
(103)
b u)0 + Ψb
bu = 0
(Φb
(104)
then u
b is D-semiclassical of class sb satisfying the functional equation
with
i)
if
b
e 2)
Φ(x)
= (θ0 Φ)(x
,
e
Φ(0)
=0
b
e 2 ) + 2(θ0 Ψ)(x
e 2)
Ψ(x)
= x (θ02 Φ)(x
,
e 0 (0) + 2Ψ(0)
e
Φ
= 0,
(105)
(106)
228
D-Semiclassical Orthogonal Polynomials
and b
s = 2e
s for (105).
ii)
if
b
e 2)
Φ(x)
= x(θ0 Φ)(x
e
Φ(0)
=0
and b
s = 2e
s + 1 for (107).
iii)
b
e 2)
Φ(x)
= xΦ(x
,
,
if
,
b
e 2)
Ψ(x)
= 2Ψ(x
e 0 (0) + 2Ψ(0)
e
Φ
6= 0,
b
e 2 ) + x2 Ψ(x
e 2)
Ψ(x)
= 2 − Φ(x
e
Φ(0)
6= 0,
and b
s = 2e
s + 3 for (109).
(107)
(108)
(109)
(110)
REMARK 4.1. Some calculation allows to give the structure relation of {Pbn }n≥0
by taking into account the results of the components {Pen }n≥0, {Pn }n≥0 and Theorem
4.1.
Finally, let us suppose that the form u
e has the following integral representation:
Z +∞
Z +∞
e (x) f (x) dx , f ∈ P;
e (x) dx = 1
he
u, fi =
U
U
(111)
0
0
e is a locally integrable function with rapid decay. Consider now f ∈ P and let
where U
us split it up into its even and odd parts
f(x) = f e (x2 ) + xf o (x2 ).
From the symmetric character of the form u
b, (99) and (112) we get
hb
u, fi = hb
u, σf e i = hσb
u, f e i = he
u, f e i.
(112)
(113)
In view of (111) and (113), with the fact that
√
√
f( x) + f(− x)
f e (x) =
, x ≥ 0,
2
and after a change of variables, we obtain the following integral representation of u
b
Z +∞
e 2 )f(x)dx , f ∈ P
hb
u, fi =
|x|U(x
(114)
−∞
since
Z
0
+∞
1
e (x)dx < +∞ , n ≥ 0.
xn+ 2 U
In particular, the moments of u
b are
(b
u)0 = 1 ;
(b
u)2n+1 = 0 , (b
u)2n+2 = (e
u)n+1 , n ≥ 0.
(115)
(116)
229
A. Ghressi and L. Khériji
4.3
The symmetrized of δ0 + λx−1 B(α)
Let u
e = δ0 + λx−1 B(α) and its associated (MOPS) {Pen }n≥0 the D-semiclassical of
2
and λn , n ≥ 0are given by (91),
class se = 1 for any λ 6= λn , n ≥ −1 with λ−1 = 1−2α
taking x = 0 in the first equality of (67) we get
Pen+1 (0) = (an − βn )Pen (0), n ≥ 0.
Consequently, Pen+1 (0) 6= 0, n ≥ 0, which implies that the form u
b, symmetrized of u
e,
is regular.
On the other hand, from (92) and (106) we have
e
e 0 (0) + 2Ψ(0)
e
Φ(0)
=0 , Φ
= 0.
Then with (106) the form u
b is symmetric D-semiclassical of class 2 satisfying the
functional equation (104) with
b
b
Φ(x)
= x4 , Ψ(x)
= −x (4α − 1)x2 + 4 .
(117)
Taking into account (102) and (65)-(66), the recurrence coefficients of {Pbn }n≥0 are
γb1 = λ , b
γ2 = a0 , b
γ2n+1 =
a0
an−1
γn , b
γ2n+2 =
an
, n ≥ 1,
a0
(118)
where γn , n ≥ 1 and an , n ≥ 0 are given by (83) and (90) respectively.
The moments of u
b are
(b
u)2n+1 = 0 , (b
u)2n+2 = λ(−1)n 2n
Γ(2α)
, n ≥ 0 , (b
u)0 = 1.
Γ(n + 2α)
Regarding the natural integral representation (95) of u
e, for α ≥ 6
4
2
π
−
(119)
1
4
the con-
dition (115) is fulfilled and from (114) we obtain the following representation of u
b, for
f∈P
hb
u, fi = (1 − λTα−1 Teα )f(0)
Z +∞
Z
2 +∞ −2α
2
+λTα−1
|x|4α−5 exp − 2
t
exp
s(t)dtf(x)dx.
(120)
x
t
−∞
x2
REMARK 4.2. The representation (120) doesn’t exist in [8] which proves that the
list of integral representations given in [8] is not complete. Another integral representation analogous to (120) is given in [39].
4.4
Symmetrization after Perturbation when u is a Symmetric
D-Semiclassical Form
Let us suppose u to be a symmetric D-semiclassical form of class s satisfying the
Pearson equation (21) and u
e = δ0 + λx−1 u or equivalently xe
u = λu. It has been shown
in section 3 that u
e is D-semiclassical satisfying the equation (69) with (70) and u
e is of
230
D-Semiclassical Orthogonal Polynomials
class se = s + 1 when (71)-(72) are valid.
If u
b is the symmetrized of (e
u) then u
e = σb
u and from (58)-(59) we get
(
σb
u = δ0 + λx−1 u
xσb
u = λu.
(121)
Thus, σb
u and xσb
u are well known.
Now, we are able to give γbn+1 , n ≥ 0. From (102) and by virtue of (65)-(66) we
obtain
γ4n+1 = −a2n , γb4n+2 = a2n , γb4n+3 = −a2n+1 , b
b
γ4n+4 = a2n+1 , n ≥ 0.
From (11), (121) and the fact that u is symmetric, the moments of u
b are
(b
u)0 = 1 ; (b
u)4n+1 = (b
u)4n+3 = (b
u)4n+4 = 0 , (b
u)4n+2 = λ(u)2n , n ≥ 0.
(122)
(123)
On the other hand, in accordance with (70) we have
Therefore,
e
Φ(x)
= xΦ(x)
e
Φ(0)
=0
,
e
Ψ(x)
= xΨ(x).
,
e 0 (0) + 2Ψ(0)
e
Φ
= Φ(0).
Consequently, with Theorem 4.1., two cases arise
i)
b
b
Φ(x)
= Φ(x2 ),
Ψ(x)
= x (θ0 Φ)(x2 ) + 2Ψ(x2 )
(124)
if
Φ(0) = 0,
and b
s = 2e
s = 2s + 2
Or
ii)
if
(125)
for (124).
b
Φ(x)
= xΦ(x2 ),
b
Ψ(x)
= 2x2 Ψ(x2 )
Φ(0) 6= 0,
(126)
(127)
and b
s = 2e
s + 1 = 2s + 3 for (126).
EXAMPLE 4.1. Let u be the symmetric D-semiclassical form of class 1 of generalb of δ0 +λx−1 u
ized Hermite H(µ) (µ 6= 0 , µ 6= −n− 12 , n ≥ 0), then the symmetrized u
is of class sb = 4 satisfying
0
x2 u
b + x 4x4 − (4µ + 1) u
b = 0.
(128)
In this case (122) becomes

Γ(µ+ 1 )

γb4n+1 = λ Γ(n+µ+2 1 ) n!


2




Γ(µ+ 1 )

 γb4n+2 = −λ Γ(n+µ+2 1 ) n!
2
Γ(n+µ+ 3 ) 1


γb4n+3 = − λ1 Γ(µ+ 1 )2 n!


2



3

 γb4n+4 = 1 Γ(n+µ+1 2 ) 1
λ
Γ(µ+ ) n!
2
, n ≥ 0.
(129)
231
A. Ghressi and L. Khériji
The moments are

(b
u)0 = 1


(b
u)4n+1 = (b
u)4n+3 = (b
u)4n+4 = 0 , n ≥ 0.


λ Γ(µ+1)Γ(2n+2µ+1)
(b
u)4n+2 = 22n Γ(2µ+1)Γ(n+µ+1)
(130)
EXAMPLE 4.2. Let u be the symmetric D-semiclassical form of class 1 of generalized Gegenbauer G(α, β) (α 6= −n−1 , β 6= −n−1 , β 6= − 12 , α+β 6= −n−1 , n ≥ 0),
then the symmetrized u
b of δ0 + λx−1 u is of class b
s = 4 satisfying
0
x2 (x4 − 1)b
u + x −(4α + 4β + 7)x4 + 4β + 3 u
b = 0.
(131)
In this case (122) becomes

Γ(n+α+1)Γ(α+β+1)Γ(β+1)
λn!

γ4n+1 = 2n+α+β+1
b

Γ(α+1)Γ(n+α+β+1)Γ(n+β+1)




Γ(n+α+1)Γ(α+β+1)Γ(β+1)
λn!

 b
γ4n+2 = − 2n+α+β+1
Γ(α+1)Γ(n+α+β+1)Γ(n+β+1)
Γ(α+1)Γ(n+α+β+2)Γ(n+β+2)

1

b4n+3 = − λn!(2n+α+β+2)
γ

Γ(n+α+1)Γ(α+β+2)Γ(β+1)




Γ(α+1)Γ(n+α+β+2)Γ(n+β+2)
 b
1
γ4n+4 = λn!(2n+α+β+2)
Γ(n+α+1)Γ(α+β+2)Γ(β+1)
, n ≥ 0.
(132)
The moments are

(b
u)0 = 1


(b
u)4n+1 = (b
u)4n+3 = (b
u)4n+4 = 0 , n ≥ 0.


Γ(α+β+2)Γ(n+β+1)
(b
u)4n+2 = λ Γ(β+1)Γ(n+α+β+2)
(133)
EXAMPLE 4.3. Let u be the symmetric D-semiclassical form of class 1 of Bessel
kind B[ν] (ν 6= −n − 1 , n ≥ 0), then the symmetrized u
b of δ0 + λx−1 u is of class sb = 4
with
0
x6 u
b − x (4ν + 3)x4 + 1 u
b = 0.
(134)
In this case (122) becomes

n
n! Γ(ν+1)

γ4n+1 = λ (−1)
b

(2n+ν) Γ(n+ν)




(−1)n n! Γ(ν+1)

 b
γ4n+2 = −λ (2n+ν) Γ(n+ν)
The moments are
(−1)n
Γ(n+ν+1)

1

b4n+3 = 4λ
γ

(2n+ν+1)n!
Γ(ν+1)



n

(−1)
Γ(n+ν+1)
 b
1
γ4n+4 = − 4λ
(2n+ν+1)n!
Γ(ν+1)
, n ≥ 0.

(b
u)0 = 1


(b
u)4n+1 = (b
u)4n+3 = (b
u)4n+4 = 0 , n ≥ 0.


(−1)n Γ(ν+1)
(b
u)4n+2 = λ 22n Γ(n+ν+1)
(135)
(136)
232
D-Semiclassical Orthogonal Polynomials
Acknowledgments. Thanks are due to the Professor Sui Sun Cheng for his helpful
suggestions and comments on our work. The authors are very grateful to the referee for
their constructive recommendations and for L. Flah, English teacher, for her general
revision of English.
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