ETNA
Electronic Transactions on Numerical Analysis.
Volume 9, 1999, pp. 1-25.
Copyright  1999, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
NON–STANDARD ORTHOGONALITY FOR MEIXNER POLYNOMIALS∗
MARı́A ÁLVAREZ DE MORALES†, TERESA E. PÉREZ‡, MIGUEL A. PIÑAR‡ , AND ANDRÉ
RONVEAUX§
Abstract. In this work, we obtain a non–standard orthogonality property for Meixner polynomials
(γ,µ)
{Mn
}n≥0 , with 0 < µ < 1 and γ ∈ R , that is, we show that they are orthogonal with respect to some
discrete inner product involving difference operators. The non–standard orthogonality can be used to recover properties of these Meixner polynomials, e. g., linear relations for the classical Meixner polynomials.
Key words. Meixner polynomials, inner product involving difference operators, non–standard orthogonality.
AMS subject classifications. 33C45.
1. Introduction. Let γ and µ be real numbers such that γ > 0 and 0 < µ < 1. It is
(γ,µ)
well known that classical monic Meixner polynomials {Mn }n≥0 can be defined by their
explicit representation in terms of the hypergeometric function 2 F1 (see [8], section 2.7, p.
49 and [2], p. 42):
Mn(γ,µ) (x) = (γ)n
(1.1)
µ
µ−1
−n, −x
where x ∈ [0, +∞), 2 F1
γ
denotes the usual Pochhammer symbol
(b)0 = 1,
n
2 F1
1
−n, −x ,
1
−
γ
µ
k
+∞
X
(−n)k (−x)k
1
1 − 1 =
and (γ)n
1−
µ
(γ)k k!
µ
k=0
(b)n = b(b + 1) . . . (b + n − 1),
b ∈ R,
∀n ≥ 1.
Simplifying expression (1.1), we get
(1.2)
Mn(γ,µ) (x)
=
µ
µ−1
n X
k
n 1
n
, n ≥ 0.
(γ + k)n−k (x − k + 1)k 1 −
k
µ
k=0
We must notice that expression (1.2) is valid for every value of the real parameter γ and,
in this way, it can be used to define Meixner polynomials for all γ ∈ R .
From the explicit representation (1.2), we can deduce that Meixner polynomials
(γ,µ)
{Mn }n≥0 satisfy, for every real value of γ, the three-term recurrence relation
(γ,µ)
M−1 (x) = 0,
(1.3)
xMn(γ,µ) (x) =
(γ,µ)
M0
(γ,µ)
Mn+1 (x)
(x) = 1,
(γ,µ)
+ βn(γ,µ) Mn(γ,µ) (x) + γn(γ,µ) Mn−1 (x),
∗ Received
n ≥ 0,
November 1, 1998. Accepted for publicaton December 1, 1999. Recommended by F. Marcellán.
de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain (alvarezd@goliat.ugr.es). Supported by Junta de Andalucı́a, G. I. FQM 0229
‡ Departamento de Matemática Aplicada, Instituto Carlos I de Fı́sica Teórica y Computacional, Universidad
de Granada, 18071 Granada, Spain (tperez@goliat.ugr.es, mpinar@goliat.ugr.es). Supported by
Junta de Andalucı́a, G. I. FQM 0229, DGES PB 95-1205 and INTAS-93-0219-ext.
§ Facultés Universitaires N. D. de la Paix. Namur, Belgium
† Departamento
1
ETNA
Kent State University
etna@mcs.kent.edu
2
Non–standard orthogonality for Meixner polynomials
where
(1.4)
βn(γ,µ) =
n(1 + µ) + µγ
,
1−µ
γn(γ,µ) =
nµ(n − 1 + γ)
.
(µ − 1)2
(γ,µ)
6= 0, for all n ≥ 1, and Favard’s theorem
Whenever γ 6= 0, −1, −2, . . . , we have γn
(γ,µ)
(see Chihara [5], p. 21) ensures the orthogonality of the sequence {Mn }n≥0 with respect
to some quasi–definite linear functional. If γ > 0 the functional is positive definite and the
µx Γ(γ + x)
polynomials are orthogonal with respect to the weight function ρ(γ,µ) (x) =
Γ(γ)Γ(x + 1)
on the interval [0, +∞). For γ = 0, −1, −2, . . . , from expression (1.4), we deduce that
(γ,µ)
vanishes for some value of n. So, in this case, we can not deduce
the coefficient γn
orthogonality results from Favard’s theorem.
The main aim of this paper is to obtain orthogonality properties for the sequence of
(γ,µ)
Meixner polynomials {Mn }n≥0 , with γ ∈ R and 0 < µ < 1. In fact, we are going to
show that they are orthogonal with respect to a discrete inner product involving difference
operators.
Similar results for different families of classical polynomials, but in the continuous case,
have been obtained by several authors. For instance, K. H. Kwon and L. L. Littlejohn, in [6],
(−k)
established the orthogonality of the generalized Laguerre polynomials {Ln }n≥0 , k ≥ 1,
with respect to a Sobolev inner product of the form:


hf, gi = ( f (0), f 0 (0), . . . , f (k−1) (0) ) A 

g(0)
g 0 (0)
..
.


+

Z
g (k−1) (0)
+∞
0
f (k) (x)g (k) (x)e−x dx,
being A a symmetric k × k real matrix. In [7], the same authors showed that the Jacobi
(−1,−1)
}n≥0 , are orthogonal with respect to the inner product
polynomials {Pn
Z
(f, g)1 = d1 f (1)g(1) + d2 f (−1)g(−1) +
1
−1
f 0 (x)g 0 (x)dx,
where d1 and d2 are real numbers.
Later, in [9], T. E. Pérez and M. A. Piñar gave an unified approach to the orthogonality of
(α)
the generalized Laguerre polynomials {Ln }n≥0 , for any real value of the parameter α, by
proving their orthogonality with respect to a Sobolev non–diagonal inner product, whereas,
in [10], they have shown how to use this orthogonality to obtain different properties of the
generalized Laguerre polynomials.
M. Alfaro, M.L. Rezola, T.E. Pérez and M.A. Piñar, in [1], have studied sequences of
polynomials which are orthogonal with respect to a Sobolev bilinear form defined by


(N )
(1.5) BS (f, g) = ( f (c), f 0 (c), . . . , f (N −1) (c) ) A 

g(c)
g 0 (c)
..
.
g (N −1) (c)


 + hu, f (N ) g (N ) i,

ETNA
Kent State University
etna@mcs.kent.edu
M. Álvarez de Morales, T. E. Pérez, M. A. Piñar and A. Ronveaux
3
where u is a quasi–definite linear functional, c ∈ R , N is a positive integer, and A is a symmetric N × N real matrix such that each of its principal submatrices is regular. In particular,
(−N,β)
}n≥0 , for β +N not a negative integer, are orthey deduced that Jacobi polynomials {Pn
thogonal with respect to (1.5), for u the Jacobi functional Jacobi corresponding to the weight
function ρ(0,β+N ) (x) = (1 + x)β+N and c = 1.
In a recent paper [3], M. Álvarez de Morales, T.E. Pérez and M.A. Piñar have studied
(−N + 12 )
}n≥0 , for N ≥ 1 a positive
the sequence of the monic Gegenbauer polynomials {Cn
integer. They have shown that this sequence is orthogonal with respect to a Sobolev inner
product of the form
(2N )
(f, g)S
(1.6)
=
T
Z
= ( F (1)|F (−1) ) A ( G(1)|G(−1) ) +
1
−1
f (2N ) (x)g (2N ) (x)(1 − x2 )N dx,
where
( F (1)|F (−1) ) = ( f (1), f 0 (1), . . . , f (N −1) (1), f (−1), f 0 (−1), . . . , f (N −1) (−1) ) ,
A = Q−1 D(Q−1 )T , Q is a regular matrix whose elements are the consecutives derivatives of
the Gegenbauer polynomials evaluated at the points 1 and −1, and D is an arbitrary diagonal
positive definite matrix.
The structure of the paper is as follows. In Section 2, from the explicit representation
(γ,µ)
of monic Meixner polynomials, {Mn }n≥0 , for γ ∈ R , we deduce some of their usual
properties, namely the three-term recurrence relation, the difference property, the second
order difference equation, etc. In Section 3, we define an inner product involving difference
operators of the form
(1.7)
(K,γ+K)
(f, g)S
=
+∞
X
F (x)Λ(K) G(x)T ρ(γ+K,µ) (x),
x ∈ [0, +∞),
x=0
where K ≥ 0 is a non negative integer,
F (x) = ( f (x), ∆f (x),
. . . , ∆K f (x) ) ,
∆ and ∇ are, respectively, the forward and backward difference operators defined by
∆f (x) = f (x + 1) − f (x),
∇f (x) = f (x) − f (x − 1),
ρ(γ+K,µ) denotes the weight function associated with the classical Meixner polynomials
(γ+K,µ)
}n≥0 , and Λ(K) is a real symmetric and positive definite (K + 1) × (K + 1) ma{Mn
(γ,µ)
trix. We show that the sequences of polynomials, {Mn }n≥0 , for γ ∈ R and 0 < µ < 1,
is a sequence of monic orthogonal polynomials (MOPS) with respect to the inner product
(K,γ+K)
, where K ≥ max{0, [−γ + 1]}.
(., .)S
Section 4 of the paper is devoted to the study of a difference operator, F (K) , which is
defined on the space of the real polynomials, P, and is symmetric with respect to the inner
ETNA
Kent State University
etna@mcs.kent.edu
4
Non–standard orthogonality for Meixner polynomials
product (1.7). From the expression of this operator, we can establish several relations between
(γ,µ)
(γ+K,µ)
}n≥0 .
Meixner polynomials {Mn }n≥0 , and classical Meixner polynomials {Mn
(−N,µ)
}n≥0 , N =
Finally, in Section 5, we study the sequence of Meixner polynomials {Mn
0, 1, 2, . . ..
2. The Meixner polynomials. Let γ and µ be real numbers such that γ > 0 and 0 <
µ < 1. The explicit representation of the n-th classical monic Meixner polynomials is given
by
(2.1)
Mn(γ,µ) (x)
=
µ
µ−1
n X
k
n 1
n
, n ≥ 0.
(γ + k)n−k (x − k + 1)k 1 −
k
µ
k=0
Notice that for every real value of the parameter γ, expression (2.1) defines a monic
polynomial of exact degree n. In this way, for γ ∈ R , we can define a family of monic
(γ,µ)
polynomials {Mn }n≥0 , which is a basis of the linear space of real polynomials, P. These
polynomials will be called generalized Meixner polynomials.
Very simple and straightforward manipulations of the explicit representation show that
the main algebraic properties of the classical Meixner polynomials remains for the generalized Meixner polynomials.
P ROPOSITION 2.1. Let γ be an arbitrary real number and 0 < µ < 1. Then, the
(γ,µ)
generalized Meixner polynomials {Mn }n≥0 satisfy the following properties:
i) Three-term recurrence relation
(γ,µ)
M−1 (x) = 0,
(2.2)
xMn(γ,µ) (x)
=
(γ,µ)
M0
(γ,µ)
Mn+1 (x)
(x) = 1,
(γ,µ)
+ βn(γ,µ) Mn(γ,µ) (x) + γn(γ,µ) Mn−1 (x),
n ≥ 0,
where
βn(γ,µ) =
(2.3)
n(1 + µ) + µγ
,
1−µ
γn(γ,µ) =
nµ(n − 1 + γ)
.
(µ − 1)2
ii) For any integer k, 0 ≤ k ≤ n, we have
(γ+k,µ)
(x),
(γ+k,µ)
(x − k).
(2.4)
ii.1) ∆k Mn(γ,µ) (x) = (n − k + 1)k Mn−k
(2.5)
ii.2) ∇k Mn(γ,µ) (x) = (n − k + 1)k Mn−k
iii) Structure relations
γ+n−1
(γ,µ)
∆Mn(γ,µ) (x) = Mn(γ,µ) (x) +
Mn−1 (x),
1−µ
µ
x
(γ,µ)
∇Mn(γ,µ) (x) = Mn(γ,µ) (x) +
(1 − γ − n)Mn−1 (x).
n
µ−1
(2.6)
iii.1)
(2.7)
iii.2)
x+γ
n
iv) Second order difference equation
ETNA
Kent State University
etna@mcs.kent.edu
M. Álvarez de Morales, T. E. Pérez, M. A. Piñar and A. Ronveaux
5
x∆∇y + [(µ − 1)x + µγ]∆y + (1 − µ)ny = 0,
(2.8)
(γ,µ)
where y = Mn (x).
v) ∆-representation
Mn(γ,µ) (x) =
(2.9)
1
µ
(γ,µ)
∆Mn+1 (x) +
∆Mn(γ,µ) (x).
n+1
1−µ
vi) ∇-representation
Mn(γ,µ) (x) =
(2.10)
1
1
(γ,µ)
∇Mn+1 (x) +
∇Mn(γ,µ) (x).
n+1
1−µ
P ROPOSITION 2.2. For the same conditions of Proposition 2.1, we have
i)
Mn(γ,µ) (x) =
i
k X
µ
k
∆i Mn(γ−k,µ) (x) =
i
µ
−
1
i=0
= I+
(2.11)
ii)
Mn(γ,µ) (x)
=
k X
k
i=0
i
1
µ−1
i
k
µ
∆ Mn(γ−k,µ) (x).
µ−1
∇i Mn (γ−k,µ) (x + k) =
=
(2.12)
I+
k
1
∇ Mn(γ−k,µ) (x + k).
µ−1
Proof. i) The case k = 0 is trivial. Using expression (2.9) and property (2.4), we obtain
I+
µ
µ
∆ Mn(γ−1,µ) (x) = Mn(γ−1,µ) (x) +
∆Mn(γ−1,µ) (x) =
µ−1
µ−1
1
(γ−1,µ)
∆Mn+1 (x) = Mn(γ,µ) (x).
=
n+1
Finally, the result follows from the identity
(2.13)
I+
n
j
n X
µ
µ
n
∆ f (x) =
∆j f (x).
j
µ−1
µ
−
1
j=0
ii) It is analogous to i) but now using the property
(2.14)
I+
n
j
n X
1
1
n
∇ f (x) =
∇j f (x).
j
µ−1
µ
−
1
j=0
ETNA
Kent State University
etna@mcs.kent.edu
6
Non–standard orthogonality for Meixner polynomials
3. Non-standard orthogonality. Let K ≥ 0 be an integer. Let us define a lower triangular (K + 1) × (K + 1) matrix L(K) =

1
µ
K


1

µ−1
2
 
µ
K


2
µ−1


..

 .
K
 K
µ
K
µ−1
K −1
1
K −1
K −1
0
0
1
0
..
.
µ
µ−1
µ
µ−1
... 0
1
K−1
K −2
K −2
..
.
µ
µ−1
K−2


... 0 



... 0 


.. 
..
. . 


... 1
From this, we can define a real symmetric matrix Λ(K) by means of
Λ(K) := L(K)L(K)T .
(3.1)
K
If we denote Λ(K) = ( λm,k )m,k=0 , then
min{m,k}
λm,k =
X
(−1)m+k
p=0
K −p
m−p
K −p
k−p
µ
1−µ
m+k−2p
,
0 ≤ m, k ≤ K.
Obviously, Λ(K) is positive definite since expression
(3.1) constitutes the Cholesky factorization for Λ(K) , (see [12], p. 174), and det Λ(K) = 1.
Letting K ≥ 0 be an integer and γ > −K a real number, we define an inner product
involving the difference operators by means of the expression
(3.2)
(K,γ)
(f, g)∆
=
+∞
X
F (x)Λ(K) G(x)T ρ(γ+K,µ) (x),
x ∈ [0, +∞),
x=0
where F (x) are G(x) are two vectors, defined by
F (x) = ( f (x),
∆f (x),
. . . , ∆K f (x) ) ,
G(x) = ( g(x),
∆g(x),
. . . , ∆K g(x) ) ,
and
ρ(γ+K,µ) (x) =
µx Γ(γ + K + x)
,
Γ(γ + K)Γ(x + 1)
is the Meixner weight function.
Since γ + K > 0, the series (3.2) converges and, as consequence of the positive definite
(K,γ)
is an inner product. By
character of the symmetric matrix Λ(K) , we conclude that (., .)∆
analogy with the Sobolev inner products, the inner product (3.2) will be called ∆–Sobolev
inner product.
ETNA
Kent State University
etna@mcs.kent.edu
M. Álvarez de Morales, T. E. Pérez, M. A. Piñar and A. Ronveaux
7
R EMARK 1. In the case K = 0 and, therefore, γ > 0, the inner product (3.2) is the
standard inner product associated to the classical weight function ρ(γ,µ) , i. e.,
(0,γ)
(f, g) = (f, g)∆
=
+∞
X
f (x)g(x)ρ(γ,µ) (x),
x ∈ [0, +∞).
x=0
R EMARK 2. Substituting the explicit expression for the elements of Λ(K) in (3.2), we
obtain
(K,γ)
(f, g)∆
(3.3)
=
K
+∞ X
X
λm,k ∆m (f (x)) ∆k (g(x)) ρ(γ+K,µ) (x).
x=0 m,k=0
From now on, we denote by ρ the Meixner classical weight function ρ(γ+K,µ) .
From the explicit expression of the matrix L(K), it is possible to obtain a representation
of the inner product (3.2) in terms of the forward difference operator ∆.
P ROPOSITION 3.1. Let γ and µ be real numbers such that 0 < µ < 1, and let K ≥ 0 be
an integer with γ + K > 0. Then, for two arbitrary polynomials f and g, the inner product
(3.2) can be written in the form:
(K,γ)
(f, g)∆
=
K +∞ X
X
I+
=
(3.4)
x=0 j=0
K−j
K−j
µ
µ
j
∆
∆
∆ f (x) I +
∆j g(x)ρ(x).
µ−1
µ−1
Proof. Using expression (2.13), the product F (x)L(K) transforms into
( f (x),
=
∆f (x),
K
µ
∆
f (x),
I+
µ−1
I+
. . . , ∆K f (x) ) L(K) =
K−1
µ
K
∆
∆f (x), . . . , ∆ f (x) .
µ−1
In the following Proposition, we establish a recurrent expression for the inner product
defined in (3.2).
P ROPOSITION 3.2. In the above conditions, the inner product (3.2) can be written in the
following recurrent form
(K,γ)
(f, g)∆
(3.5)
=
I+
(K−1,γ) X
+∞
µ
µ
∆ f, I +
∆ g
+
∆K f (x)∆K g(x)ρ(x).
µ−1
µ−1
∆
x=0
Proof. From (3.4), we have
ETNA
Kent State University
etna@mcs.kent.edu
8
Non–standard orthogonality for Meixner polynomials
(K,γ)
(f, g)∆
=
K +∞ X
X
=
+∞ K−1
X
X
x=0 j=0
I+
x=0 j=0
+
+∞
X
I+
K−j
K−j
µ
µ
∆
∆
∆j f (x) I +
∆j g(x)ρ(x) =
µ−1
µ−1
K−j
K−j
µ
µ
∆
∆
∆j f (x) I +
∆j g(x)ρ(x) +
µ−1
µ−1
∆K f (x)∆K g(x)ρ(x) =
x=0
+∞ K−1
X
X
=
I+
x=0 j=0
K−j−1
K−j−1
µ
µ
µ
j
∆
∆ f (x)
I+
∆
∆
.
I+
µ−1
µ−1
µ−1
+∞
X
µ
∆ g(x) ρ(x) +
∆K f (x)∆K g(x)ρ(x) =
µ−1
x=0
(K−1,γ) X
+∞
µ
µ
∆ f, I +
∆ g
+
∆K f (x)∆K g(x)ρ(x).
=
I+
µ−1
µ−1
∆
x=0
I+
∆
j
In the following theorem we establish the orthogonality of generalized Meixner polyno(γ,µ)
mials {Mn }n≥0 , for γ ∈ R and 0 < µ < 1, with respect to the inner product (3.4).
T HEOREM 3.3. Let γ and µ be real numbers such that 0 < µ < 1. The sequence of
(γ,µ)
generalized Meixner polynomials {Mn }n≥0 is a MOPS with respect to the ∆–Sobolev
(K,γ)
inner product (., .)∆ , where K ≥ max{0, [−γ + 1]}.
(γ,µ)
Proof. We compute the inner product of two generalized Meixner polynomials Mn
(γ,µ)
and Mm . From relations (2.4) and (2.11), we get
(γ,µ)
Mn(γ,µ) , Mm
(K,γ)
∆
=
K +∞ X
X
I+
x=0 j=0
K−j
µ
∆
∆j Mn(γ,µ) (x)
µ−1
K−j
µ
(γ,µ)
∆
∆j Mm
(x)ρ(x) =
µ−1
K−j
K
+∞ X
X
µ
(γ+j,µ)
∆
(n − j + 1)j (m − j + 1)j I +
Mn−j (x)
=
µ
−
1
x=0 j=0
K−j
µ
(γ+j,µ)
∆
Mm−j (x)ρ(x) =
I+
µ−1
I+
=
K
+∞ X
X
(γ+K,µ)
(γ+K,µ)
(n − j + 1)j (m − j + 1)j Mn−j
(x)Mm−j
(x)ρ(x),
x=0 j=0
(γ+K,µ)
= 0, for i < 0. The result follows from the orthogonality of
where we assume Mi
(γ+K,µ)
}i≥0 with respect to the weight function ρ.
classical Meixner polynomials {Mi
R EMARK 1. Using the same matrix Λ(K) , we can obtain an alternative version of the
above result for the inner product given by
ETNA
Kent State University
etna@mcs.kent.edu
9
M. Álvarez de Morales, T. E. Pérez, M. A. Piñar and A. Ronveaux
(K,γ)
(f, g)∇
(3.6)
=
+∞
X
F̃ (x)Λ(K) G̃(x)T ρ(γ+K,µ) (x),
x=0
where F̃ (x) and G̃(x) are two vectors defined by
F̃ (x) = ( f (x),
∇f (x + 1), . . . , ∇K f (x + K) ) ,
G̃(x) = ( g(x),
∇g(x + 1), . . . , ∇K g(x + K) ) .
R EMARK 2. In the case of the operator ∇ and using a different matrix, Λ̂(K) , we can
consider the inner product
(K,γ)
(f, g)∇
(3.7)
=
+∞
X
F̂ (x)Λ̂(K) Ĝ(x)T ρ(γ+K,µ) (x − K),
x=0
where F̂ (x) and Ĝ(x) are the vectors
F̂ (x) = ( f (x), ∇f (x), . . . , ∇K f (x) ) ,
Ĝ(x) = ( g(x), ∇g(x), . . . , ∇K g(x) ) ,
and Λ̂(K) is a real symmetric and positive definite matrix whose elements are defined by
min{i,j}
λ̂i,j =
X
(−1)i+j
p=0
K −p
i−p
K −p
j−p
1
1−µ
i+j−2p
,
0 ≤ i, j ≤ K.
4. The difference operator F (K) . In this section, we will define a difference operator
F , defined on the linear space of real polynomials P, symmetric with respect to the ∆–
Sobolev inner product (3.4). Using this operator we will deduce the existence of several
(γ,µ)
relations involving the sequence of generalized Meixner polynomials {Mn }n≥0 , and the
(γ+K,µ)
}n≥0 .
sequence of classical Meixner polynomials {Mn
(K)
by means of
We define the difference operator F
(K)
(4.1)
F
(K)
K
Φ(x; K) X
:=
(−1)m λm,k ∇m ρ(x)∆k ,
ρ(x)
m,k=0
µx Γ(γ + K + x)
, with x ∈ [0, +∞), γ,
Γ(γ + K)Γ(x + 1)
µ ∈ R , such that 0 < µ < 1, and K ≥ 0 an integer.
where Φ(x; K) = µK (x + γ)K and ρ(x) =
Expanding the expression of F (K) , we can write (4.1) in the form:
(4.2)
F (K)
 ρ(x)I 
(K)  ρ(x)∆ 
Φ(x; K)
K

.
=
Λ
I, −∇, . . . , (−∇)
..


ρ(x)
.
ρ(x)∆K
ETNA
Kent State University
etna@mcs.kent.edu
10
Non–standard orthogonality for Meixner polynomials
If we now substitute the elements of the matrix Λ(K) and use relations (2.13) and (2.14) in
(4.2), we obtain a simple expression for the operator F (K) :
(4.3) F (K) =
K
Φ(x; K) X
(−∇)j
ρ(x) j=0
I+
K−j
K−j
µ
µ
∇
∆
ρ(x) I +
∆j ,
1−µ
µ−1
Expression (4.3) can be written in recurrent form, as we show in the following Proposition.
P ROPOSITION 4.1. Let K ≥ 0 be a given integer and let γ and µ be real numbers such
that 0 < µ < 1. Then,
F (0) = I,
F (K) = [(1 − µ)x − µγ] F (K−1) ∆ − x∇ F (K−1) ∆ +
K
K
µ
µ
Φ(x; K)
∇
∆
ρ(x) I +
,
I+
+
ρ(x)
1−µ
µ−1
(4.4)
K ≥ 1.
Proof. For K = 0, we get F (0) = I from (4.3). Now, we deduce the recurrence
expression. From (4.3), we have
(K)
F (K) = F1
(K)
+ F2
,
where
(K)
F1
(K)
F2
K−j
K−j
K
Φ(x; K) X
µ
µ
∇
∆
(−∇)j I +
ρ(x) I +
∆j ,
ρ(x) j=1
1−µ
µ−1
K
K
Φ(x; K)
µ
µ
∇
∆
=
ρ(x) I +
.
I+
ρ(x)
1−µ
µ−1
=
Then,
(K)
F1
=


K−j−1
K−j−1
K−1
µ
µ
Φ(x; K)  X
∇
∇
∆
(−∇)j I +
ρ(x) I +
∆j+1 
=−
ρ(x)
1
−
µ
µ
−
1
j=0

K−j−1
K−1
µ
Φ(x + 1; K) X
j

∇
(−∇) I +
ρ(x).
= −∇
ρ(x + 1) j=0
1−µ
!
K−j−1
Φ(x + 1; K)
µ
j+1
∆
∆
+∇
I+
µ−1
ρ(x + 1)
K−j−1
K−1
K−j−1
X
µ
µ
j
∇
∆
(−∇) I +
ρ(x) I +
∆j+1 =
1
−
µ
µ
−
1
j=0
ETNA
Kent State University
etna@mcs.kent.edu
M. Álvarez de Morales, T. E. Pérez, M. A. Piñar and A. Ronveaux
11
Φ(x; K)
= −∇ (x + 1)F (K−1) ∆ + ∆
ρ(x)
K−j−1
K−1
K−j−1
X
µ
µ
∇
∆
(−∇)j I +
ρ(x) I +
∆j+1 =
1
−
µ
µ
−
1
j=0
Φ(x + 1; K) Φ(x; K) −
= −∇ (x + 1)F (K−1) ∆ +
ρ(x + 1)
ρ(x)
K−j−1
K−1
K−j−1
X
µ
µ
∇
∆
(−∇)j I +
ρ(x) I +
∆j+1 =
1
−
µ
µ
−
1
j=0
= −∇ (x + 1)F (K−1) ∆ + [x + 1 − µ(x + γ)] F (K−1) ∆ =
= [(1 − µ)x − µγ] F (K−1) ∆ − x∇ F (K−1) ∆ .
P ROPOSITION 4.2. We have
F (K) xn = F (n, K)xn + . . . ,
(4.5)
n ≥ K,
where
K
X
n!
(1 − µ)
F (n, K) =
(n
− i)!
i=0
i
µ
1−µ
K−i
(γ + n)K−i > 0,
denotes the leading coefficient of the polynomial F (K) xn , for γ, µ ∈ R , 0 < µ < 1, and
K ≥ 0 an integer.
Proof. We will prove the result by induction. From Proposition 4.1, we get
(K)
F (K) = F1
(K)
+ F2
,
where
(K)
F1
(4.6)
(K)
F2
= [(1 − µ)x − µγ] F (K−1) ∆ − x∇ F (K−1) ∆ ,
K
K
Φ(x; K)
µ
µ
∇
∆
=
ρ(x) I +
.
I+
ρ(x)
1−µ
µ−1
If K = 0, then since F (0) = I, the result is trivial. For K = 1, we have
F (1) =
µ
(1 − γ)µ
I−
([(µ − 1)x + µ(γ − 1)] ∇ + µ(x − 1 + γ)∆∇) ;
µ−1
(µ − 1)2
thus, F (1) preserves the degree.
(K)
We assume that the result is true for K − 1. Then, by induction, the operator F1
preserves the degree of the polynomials. Therefore, we have only to show that the operator
(K)
F2 preserves it.
ETNA
Kent State University
etna@mcs.kent.edu
12
Non–standard orthogonality for Meixner polynomials
(0)
In the case K = 0, the result is trivial since F2
deduce that
is the identity operator. For K = 1, we
µ
Φ(x; 1)
µ
∇ ρ(x) I +
∆ =
=
I+
ρ(x)
1−µ
µ−1
#
"
2
µ
µ
µ
(x + γ)
ρ(x)∆ +
∇(ρ(x)I) −
∇(ρ(x)∆) =
ρ(x)I +
=µ
ρ(x)
µ−1
1−µ
1−µ
2
2
µ
µ
µ
µγ
x∇ −
γ∆ +
x∆∇,
I−
=
1−µ
µ−1
1−µ
1−µ
(1)
F2
(1)
so the operator F2 preserves the degree. We assume that the result is true for K − 1 and we
are going to prove it for K. We have
K
K
Φ(x; K)
µ
µ
∇
∆
ρ(x) I +
=
I+
ρ(x)
1−µ
µ−1
K−1
K−1 µ
µ
µ
Φ(x; K)
∇
∆
∆
ρ(x) I +
I+
I+
=
ρ(x)
1−µ
µ−1
µ−1
K−1
K−1 !
µ
µ
µ Φ(x; K)
µ
∇
I+
∇
∆
∆
ρ(x) I +
+
I+
1 − µ ρ(x)
1 − mu
µ−1
µ−1
K−1
µ
Φ(x + 1; K)
µ
µ
(K−1)
∆ +
∇
∇
I+
= µ(x + γ)F2
I+
µ−1
1−µ
ρ(x + 1)
1−µ
K−1 !
Φ(x + 1; K)
µ
µ
µ
∆
∆
+
∇
ρ(x) I +
I+
µ−1
µ−1
µ−1
ρ(x + 1)
K−1
K−1 µ
µ
µ
∇
∆
∆ =
ρ(x) I +
I+
I+
1−µ
µ−1
µ−1
µ
µ
µ
(K−1)
(K−1)
∆ +
∇ (x + 1)F2
∆
= µ(x + γ)F2
I+
I+
µ−1
1−µ
µ−1
K−1
K−1 Φ(x; K)
µ
µ
µ
µ
∆
∇
∆
∆
ρ(x) I +
+
I+
I+
µ−1
ρ(x)
1−µ
µ−1
µ−1
µ
µ
µ
(K−1)
(K−1)
∆ +
∇ (x + 1)F2
∆
= µ(x + γ)F2
I+
I+
µ−1
1−µ
µ−1
µ
µ
(K−1)
[x + 1 − µ(x + γ)] F2
∆ =
+
I+
µ−1
µ−1
µ
1
µ
(K−1)
x+
µ(x + γ) F2
∆
=
I+
µ−1
1−µ
µ−1
µ
µ
(K−1)
x∇ F2
∆
=
+
I+
1−µ
µ−1
µ
µγ
µ
µ
(K−1)
(K−1)
F2
∆ +
x∇ F2
∆
.
=
I+
I+
(1 − µ)
µ−1
1−µ
µ−1
(K)
F2
=
Finally, identifying the leading coefficients and using a recurrence reasoning, we deduce that
ETNA
Kent State University
etna@mcs.kent.edu
M. Álvarez de Morales, T. E. Pérez, M. A. Piñar and A. Ronveaux
F (n, K) =
K
X
(1 − µ)i
i=0
n!
(n − i)!
µ
1−µ
13
K−i
(γ + n)K−i > 0.
Next, we are going to show that the difference operator F (K) is symmetric with respect
to the inner product (3.2). First, we need the following lemma.
L EMMA 4.3. Let f be an arbitrary polynomial of the space P and let n ≥ 0 be an
integer. Then,
+∞
X
(4.7)
∆n f (x)ρ(x) = (−1)n
x=0
+∞
X
f (x)∇n ρ(x),
x=0
µx Γ(γ + K + x)
, with x ∈ [0, +∞), γ, µ ∈ R , with 0 < µ < 1, and
Γ(γ + K)Γ(x + 1)
K ≥ 0 an integer.
Proof. For n = 0 the result is trivial. If n = 1, then using the relations ∆ (f (x)g(x)) =
∆f (x)g(x) + f (x + 1)∆g(x), ∆f (x) = ∇f (x + 1) and ρ(−1) ≡ 0, we get
where ρ(x) =
+∞
X
∆f (x)ρ(x) =
x=0
=−
+∞
X
[∆ (f (x)ρ(x − 1)) − f (x)∆ρ(x − 1)] =
x=0
+∞
X
f (x)∆ρ(x − 1) = −
x=0
+∞
X
f (x)∇ρ(x).
x=0
Now, we assume that the result is true for n − 1 and we prove it for n.
+∞
X
∆n f (x)ρ(x) =
x=0
=−
+∞
X
∆ ∆n−1 f (x) ρ(x) =
x=0
+∞
X
∆n−1 f (x)∇ρ(x) = −(−1)n−1
x=0
+∞
X
f (x)∇n−1 (∇ρ(x)) .
x=0
From expression (4.1), we can obtain a representation of the ∆–Sobolev inner product in
terms of the inner product associated to the weight function ρ.
P ROPOSITION 4.4. Let f and g be two real polynomials of P. Then,
(K,γ)
(Φ(x; K)f, g)∆
=
+∞
X
f (x)F (K) g(x)ρ(x),
x=0
µx Γ(γ + K + x)
, with x ∈ [0, +∞), γ,
Γ(γ + K)Γ(x + 1)
µ ∈ R , such that 0 < µ < 1, and K ≥ 0 an integer.
where Φ(x; K) = µK (x + γ)K and ρ(x) =
ETNA
Kent State University
etna@mcs.kent.edu
14
Non–standard orthogonality for Meixner polynomials
Proof. Using expression (4.1) and relation (4.7) in the definition of the inner product
(3.3), we get
(K,γ)
(Φ(x; K)f, g)∆
=
K
+∞ X
X
=
K
+∞ X
X
λm,k ∆m (Φ(x; K)f (x)) ∆k (g(x)) ρ(x) =
x=0 m,k=0
(−1)m λm,k Φ(x; K)f (x)∇m ρ(x)∆k (g(x)) =
x=0 m,k=0
=
+∞
X
f (x)
x=0
=
+∞
X
K
X
(−1)m λm,k
m,k=0
Φ(x; K) m
∇ ρ(x)∆k (g(x)) ρ(x) =
ρ(x)
f (x)F (K) g(x)ρ(x).
x=0
T HEOREM 4.5. The difference operator F (K) is symmetric with respect to the inner
product (3.3), that is,
(K,γ) (K,γ)
= f, F (K) g
.
F (K) f, g
∆
∆
Proof. From expression (4.1), Proposition 4.4, relation (4.7) and the definition of the
inner product (3.3), we conclude that
(K,γ)
K
(K,γ)
X
φ(x; K) m
∇ ρ(x)∆k f (x) , g(x)
=
(−1)m λm,k
=
F (K) f, g
ρ(x)
∆
∆
m,k=0
=
K
+∞ X
X
(−1)m λm,k ∇m ρ(x)∆k f (x) F (K) g(x) =
x=0 m,k=0
=
K
+∞ X
X
λm,k ∆k f (x)∆m F (K) g(x) ρ(x) =
x=0 m,k=0
= f, F (K) g
(K,γ)
∆
.
P ROPOSITION 4.6. Let γ and µ be real numbers such that 0 < µ < 1, and let K ≥ 0 be
an integer. For every value of the integer n such that n ≥ K, we have
(4.8)
F (K) Mn(γ,µ) (x) = F (n, K)Mn(γ,µ) (x).
(γ,µ)
Proof. Writing the polynomial F (K) Mn
(γ,µ)
}i≥0 , we get
als {Mi
F (K) Mn(γ,µ) (x) =
in terms of generalized Meixner polynomi-
n
X
i=0
(γ,µ)
γn,i Mi
(x),
ETNA
Kent State University
etna@mcs.kent.edu
15
M. Álvarez de Morales, T. E. Pérez, M. A. Piñar and A. Ronveaux
where the coefficients γn,i are given by
γn,i
(K,γ)
(K,γ)
(γ,µ)
(γ,µ)
(γ,µ)
(γ,µ)
F (K) Mn , Mi
Mn , F (K) Mi
∆
∆
=
=
.
(K,γ)
˜i
(γ,µ)
(γ,µ)
k
, Mi
Mi
∆
Here, the symmetry of the operator F (K) has been used. From Proposition 4.2 (see (4.5)), and
(γ,µ)
with respect to the ∆–Sobolev
the orthogonality of generalized Meixner polynomial Mn
inner product, we deduce that γn,i = 0, for 0 ≤ i ≤ n − 1.
The following Proposition establishes several relations between generalized Meixner
(γ,µ)
(γ+K,µ)
}n≥0 .
polynomials {Mn }n≥0 and classical Meixner polynomials {Mn
P ROPOSITION 4.7. Let γ and µ be real numbers such that 0 < µ < 1, and let K ≥ 0 be
an integer. The following relations hold:
(4.9)
µK (x + γ)K Mn(γ+K,µ) (x) =
i)
n+K
X
i=n
where αn,n+K = µK ,
(4.10)
ii)
αn,n = F (n, K)
kn
k̃n
(x),
n ≥ 0,
;
n
X
F (K) Mn(γ,µ) (x) =
(γ,µ)
αn,i Mi
(γ+K,µ)
βn,i Mi
(x),
n ≥ K,
i=n−K
where βn,n = F (n, K),
βn,n−K = µK
k̃n
kn−K
.
Proof.
(γ+K,µ)
(x) in terms of the generalized Meixner
i) Expanding the polynomial µK (x + γ)K Mn
(γ,µ)
}i≥0 , we obtain
polynomials {Mi
µK (x + γ)K Mn(γ+K,µ) (x) =
n+K
X
i=0
(γ,µ)
αn,i Mi
(x),
where the coefficients αn,i are
αn,i =
K
(γ+K,µ)
(γ,µ)
γ)K Mn
, Mi
µ (x +
(K,γ)
(γ,µ)
(γ,µ)
, Mi
Mi
+∞
X
(K,γ)
∆
=
(γ,µ)
Mn(γ+K,µ) (x)F (K) Mi
x=0
k˜i
(x)ρ(x)
.
∆
(γ+K,µ)
, we deduce that αn,i = 0,
Now, using the orthogonality of classical polynomial Mn
for 0 ≤ i ≤ n − 1.
(γ,µ)
as a linear combination of the classical polynomials
ii) Writing the polynomial F (K) Mn
(γ+K,µ)
}i≥0 , we get
{Mi
ETNA
Kent State University
etna@mcs.kent.edu
16
Non–standard orthogonality for Meixner polynomials
F (K) Mn(γ,µ) (x) =
n
X
i=0
(γ+K,µ)
βn,i Mi
(x).
The coefficients can been computed again, from Proposition 4.4, and so, we get
+∞
X
βn,i =
(γ+K,µ)
Mi
(x)F (K) Mn(γ,µ) (x)ρ(x)
x=0
+∞
X
=
(K,γ)
(γ+K,µ)
(γ,µ)
µK (x + γ)K Mi
, Mn
∆
ki
(γ+K,µ)
(γ+K,µ)
Mi
(x)Mi
(x)ρ(x)
x=0
(γ,µ)
Finally, from the orthogonality of generalized polynomial Mn
for 0 ≤ i ≤ n − K − 1.
.
, we conclude that βn,i = 0,
(γ,µ)
The following Proposition, concerning the zeros of generalized polynomial Mn , is a simple consequence of the orthogonality.
P ROPOSITION 4.8. Let γ and µ be real numbers such that 0 < µ < 1. For every
(γ,µ)
has at least (n − K) real
n > K = max{0, [−γ + 1]}, the generalized polynomial Mn
zeros of odd multiplicity contained in the interval [0, +∞).
Proof. Using Proposition 4.4, relation (4.8), and the orthogonality of generalized
(γ,µ)
(K,K+α)
with respect to (., .)∆
, we have
Meixner polynomial Mn
µK (x + γ)K , Mn(γ,µ)
(K,K+α)
∆
=
+∞
X
F (K) Mn(γ,µ) ρ(x) =
x=0
= F (n, K)
+∞
X
Mn(γ,µ) ρ(x) = 0,
x=0
(γ,µ)
changes its sign in the interval [0, +∞).
and then the polynomial Mn
Let x1 , x2 , . . . , xr be the real and positive zeros of odd multiplicity of the polynomial
(γ,µ)
Mn , and denote by q(x) the polynomial
q(x) =
r
Y
(x − xi ).
i=1
Then,
µK (x + γ)K q(x), Mn(γ,µ)
(K,K+α)
∆
=
+∞
X
q(x)F (K) Mn(γ,µ) ρ(x) =
x=0
= F (n, K)
+∞
X
q(x)Mn(γ,µ) ρ(x) 6= 0,
x=0
(γ,µ)
≥ 0, ∀x ∈ [0, +∞). So, from the orthogonality of generalized Meixner
since q(x)Mn
(γ,µ)
(K,K+α)
with respect to (., .)∆
, we deduce that r ≥ n − K.
polynomial Mn
ETNA
Kent State University
etna@mcs.kent.edu
17
M. Álvarez de Morales, T. E. Pérez, M. A. Piñar and A. Ronveaux
(−N,µ)
}n≥0 . This section is devoted to the study of
5. The Meixner polynomials {Mn
the generalized Meixner polynomials in the special case when the parameter γ = −N , for
(−N,µ)
}n≥0 ,
N = 0, 1, 2, . . .. The special characteristics of monic Meixner polynomials {Mn
with 0 < µ < 1, and the ∆–Sobolev inner product defined in (3.2), allow us to deduce
properties for these polynomials.
Meixner polynomials satisfy the properties given in Propositions 2.1 and 2.2, respectively. Moreover, from the explicit representation of these polynomials, we can deduce some
new properties, as it is shown in the following Proposition.
P ROPOSITION 5.1. Let N ≥ 0 be an integer. Then, the monic Meixner polynomials
(−N,µ)
}n≥0 , with 0 < µ < 1, satisfy the following properties:
{Mn
n
µ
, n ≥ 0;
i) Mn(−N,µ) (0) = (−N )n
µ−1
ii) Mn(−N,µ) (0) = 0,
n ≥ N + 1;
iii) ∆k Mn(−N,µ)(0) = (n − k + 1)k (−N + k)n−k
iv) ∆k Mn(−N,µ) (0) = 0,
0 ≤ k ≤ N,
µ
µ−1
n−k
,
n ≥ k;
n ≥ N + 1;
(−N,µ)
MN +1 (x)
= (x − N )N +1 .
Proof.
i) We need only to replace γ = −N and x = 0 in the explicit representation (2.1).
v)
ii) If n ≥ N + 1, then (−N )n = 0, and replacing it in i), the result follows.
iii) From relations i) and (2.4), for n ≥ 0, we deduce that
k
∆
Mn(−N,µ) (0)
=
(−N +k,µ)
(n − k + 1)k Mn−k
(0)
= (n − k + 1)k (−N + k)n−k
µ
µ−1
n−k
.
iv) If we consider 0 ≤ k ≤ N and n ≥ N + 1, then the Pochhammer symbol (−N + k)n−k
(−N,µ)
vanishes, and using iii), we conclude ∆k Mn
(0) = 0.
v) Taking n = N + 1 in (2.1), with γ = −N , we obtain
(−N,µ)
MN +1 (x)
k
1
=
(−N + k)N +1−k (x − k + 1)k 1 −
µ
k=0
N +1
N +1
µ
1
=
(x − (N + 1) + 1)N +1 1 −
= (x − N )N +1 .
µ−1
µ
µ
µ−1
N +1 NX
+1 N +1
k
(−N,µ)
P ROPOSITION 5.2. For every n ≥ N + 1, the Meixner polynomial Mn
relation
(5.1)
(N +2,µ)
Mn(−N,µ)(x) = (x − N )N +1 Mn−N −1 (x − N − 1).
satisfies the
ETNA
Kent State University
etna@mcs.kent.edu
18
Non–standard orthogonality for Meixner polynomials
Proof. The proof uses induction again. For n = N + 1 in (5.1), we get
(−N,µ)
MN +1
(N +2,µ)
(x) = (x − N )N +1 M0
(x − N − 1).
(−N,µ)
If n = N + 2, from the recurrence relation (2.2) for the polynomials MN +1 , using
N +1+µ
(−N,µ)
(−N,µ)
(N +2,µ)
= N +1+β0
, and the property v) of Proposition
γN +1 = 0, βN +1 =
1−µ
5.1, we get
(N +2,µ)
(−N,µ)
(x) = x − N − 1 − β0
MN +1 (x) =
(N +2,µ)
(N +2,µ)
(x − N − 1).
= (x − N )N +1 M1
= (x − N )N +1 x − N − 1 − β0
(−N,µ)
MN +2
We will consider now the case n ≥ N + 2. Writing the three-term recurrence relation
(N +2,µ)
for the classical polynomials Mn−N −1 (x − N − 1) and using
(N +2,µ)
βn(−N,µ) = N + 1 + βn−N −1 ,
(N +2,µ)
γn(−N,µ) = γn−N −1 ,
expression (2.2) gives
(N +2,µ)
Mn−N
(N +2,µ)
(N +2,µ)
(x − N − 1) = x − N − 1 − βn−N −1 Mn−N −1 (x − N − 1)−
(N +2,µ)
(N +2,µ)
− γn−N −1 Mn−N −2 (x − N − 1) =
(N +2,µ)
(N +2,µ)
= x − βn(−N,µ) Mn−N −1 (x − N − 1) − γn(−N,µ) Mn−N −2 (x − N − 1).
(5.2)
Now, using an inductive reasoning, from (5.2) we deduce that the three-term recurrence rela(N +2,µ)
tion for the polynomials (x − N )N +1 Mn−N −1 (x − N − 1) coincides with the one for the
(−N,µ)
polynomials Mn
, and then the result follows.
R EMARK . From relation (5.1), we deduce that, for every value of n ≥ N + 1, the
(−N,µ)
has N + 1 real zeros at the form x = 0, 1, 2, . . . , N , as well
Meixner polynomial Mn
as n − N − 1 real zeros of odd multiplicity contained in the interval [0, +∞).
(−N,µ)
Finally, we will prove a difference relation between the Meixner polynomial Mn
(N +2,µ)
n ≥ N + 1, and the classical Meixner polynomial Mn−N −1 .
P ROPOSITION 5.3. For every n ≥ N + 1, we have
(5.3)
I+
, with
N +1
µ
(N +2,µ)
∆
∇N +1 Mn(−N,µ) (x) = (n − N )N +1 Mn−N −1 (x − N − 1).
µ−1
Proof. Using difference property (2.5), for k = N + 1 and γ = −N , and using relation
(2.11), we get
I+
N +1
N +1
µ
µ
∆
∆
∇N +1 Mn(−N,µ)(x) = I +
.
µ−1
µ−1
(1,µ)
(N +2,µ)
(n − N )N +1 Mn−N −1 (x − N − 1) = (n − N )N +1 Mn−N −1 (x − N − 1).
ETNA
Kent State University
etna@mcs.kent.edu
M. Álvarez de Morales, T. E. Pérez, M. A. Piñar and A. Ronveaux
19
5.1. ∆–Sobolev inner product. Let N ≥ 0 be a given integer, and consider the se(−N,µ)
}n≥0 , with 0 < µ < 1. We know from Section
quence of Meixner polynomials {Mn
3, that this sequence is orthogonal with respect to the inner product defined in (3.4), for
K ≥ max{0, [−γ + 1]} = N + 1. When K = N + 1, the ∆–Sobolev inner product (3.4)
reads
(N +1,−N )
=
+1
+∞ N
X
X
(5.4) =
I+
(f, g)∆
x=0 j=0
N +1−j
N +1−j
µ
µ
j
∆
∆
∆ f (x) I +
∆j g(x)µx .
µ−1
µ−1
Our purpose, in this particular case, is to obtain a simpler expression for the inner product
(5.4).
L EMMA 5.4. Let f be an arbitrary polynomial and 0 < µ < 1 a real number. We have
µ I+
x
n
1
µ
∆ f (x) =
∆n (µx f (x)) ,
µ−1
(µ − 1)n
n ≥ 0.
Proof. Let use induction again. If n = 0, then the result is trivial. For n = 1, using the
relations ∆ (f g) = ∆(f )g + f ∆(g) + ∆(f )∆(g) and ∆(µx ) = (µ − 1) µx , we get
1
1
∆ (µx f (x)) =
[µx ∆(f (x)) + (µ − 1) µx f (x) + (µ − 1) µx ∆(f (x))] =
µ−1
µ−1
µx
[(µ − 1) f (x) + µ∆(f (x))] =
=
µ−1
µ
∆ f (x).
= µx I +
µ−1
Next, we assume that the result is true for n−1 and, we will prove it for n. From the induction
hypothesis and the proof in the case n = 1, we deduce
n
n−1 µ
µ
µ
x
∆ f (x) = µ I +
∆
∆ f (x) =
I+
µ−1
µ−1
µ−1
1
1
µ
∆ f (x) =
∆n−1 µx I +
∆n (µx f (x)) .
=
(µ − 1)n−1
µ−1
(µ − 1)n
µ I+
x
L EMMA 5.5. With the above conditions,
µ I+
x
n
n
µ
1
∇ f (x) =
∇n (µx f (x)) ,
µ−1
µ−1
n ≥ 0.
Proof. The proof is analogous to the proof of Lemma 5.4.
L EMMA 5.6. Let n ≥ 0 be an integer, 0 < µ < 1 and let f and g be two real polynomials. We have
µ I+
x
µ
1
µ
µ
x
x
∆ f I+
∆ g=
µ ∆f ∆g .
∆ (µ f g) +
µ−1
µ−1
µ−1
µ−1
ETNA
Kent State University
etna@mcs.kent.edu
20
Non–standard orthogonality for Meixner polynomials
Proof. We have
µ
µ
µ
x
∆ f I+
∆ g = µ fg +
[∆(f g) − ∆f ∆g] +
µ−1
µ−1
µ−1
#
2
µ
µ
µ
x
x
∆ fg +
∆f ∆g = µ I +
+
2 µ ∆f ∆g =
µ−1
µ−1
(µ − 1)
µ
1
µ
1
x
x
x
∆ (µx f g) +
µ
µ
∆f
∆g
=
f
g)
+
∆f
∆g
.
∆
(µ
=
µ−1
µ−1
µ−1
(µ − 1)2
µ I+
x
L EMMA 5.7. Let N ≥ 0 be a given integer, 0 < µ < 1, and let f and g be two
polynomials. Then the inner product (5.4) can be written in the form:
(N +1,−N )
(f, g)∆
1
µ
(N )
(N,−N +1)
(∆f, ∆g)∆
=
(f, g)D +
1−µ
µ−1
+∞
X
∆N +1 f (x)∆N +1 g(x)µx ,
+
x=0
where

g(0)
 ∆g(0) 
= ( f (0), ∆f (0), . . . , ∆N f (0) ) Λ(N ) 
,
...
N
∆ g(0)

(N )
(f, g)D
and Λ(N ) is defined in (3.1).
Proof. Applying Lemma 5.6 to the inner product (5.4), we get
(N −1,−N )
(f, g)∆
=
x=0 j=0
=
N +1−j
N +1−j
µ
µ
j
∆
∆
∆ f (x) I +
∆j g(x)µx
I+
µ−1
µ−1
+1 +∞ N
X
X
N +∞ X
X
=
I+
x=0 j=0
+
+∞
X
∆N +1 f (x)∆N +1 g(x)µx
x=0
=
N +1−j
N +1−j
µ
µ
j
∆
∆
∆ f (x) I +
∆j g(x)µx
µ−1
µ−1
N +∞ X
X
I+
x=0 j=0
N −j
µ
µ
µ
∆
I+
∆
∆
∆j f I +
µ−1
µ−1
µ−1
N −j
+∞
X
µ
∆
∆j gµx +
∆N +1 f (x)∆N +1 g(x)µx
µ−1
x=0


N −j
N −j
+∞
N
X
X
1
µ
µ
∆
∆
=
∆
∆j f I +
∆j gµx 
I+
1 − µ x=0
µ
−
1
µ
−
1
j=0
I+
ETNA
Kent State University
etna@mcs.kent.edu
21
M. Álvarez de Morales, T. E. Pérez, M. A. Piñar and A. Ronveaux
+
+
N −j
N −j
+∞ X
N X
µ
µ
µ
j+1
∆
∆
∆
f
I
+
∆j+1 gµx
I
+
(µ − 1)2 x=0 j=0
µ−1
µ−1
+∞
X
∆N +1 f (x)∆N +1 g(x)µx
x=0
x=+∞
N −j
N −j
N 1 X µ
µ
j
j
x
∆
∆
=
∆ f I+
∆ gµ I+
1 − µ j=0 µ−1
µ−1
x=0
+∞
X
µ
(N,−N +1)
(∆f, ∆g)∆
+
∆N +1 f (x)∆N +1 g(x)µx
(µ − 1)2
x=0
X
+∞
1
µ
(N )
(N,−N +1)
(∆f, ∆g)∆
=
∆N +1 f (x)∆N +1 g(x)µx .
(f, g)D +
+
1−µ
µ−1
x=0
+
P ROPOSITION 5.8. With the above conditions, the inner product (5.4) can be expressed
as:
(N +1,−N )
(f, g)∆
N
1 X
=
1 − µ i=0

N
+1
X
+
(5.5)
i=0
µ
2
(µ − 1)
!i
µ
2
(µ − 1)
(N −i)
(∆i f, ∆i g)D
!i  +∞
X

∆N +1 f (x)∆N +1 g(x)µx .
x=0
Proof. Iterating in Lemma 5.7, the inner product (5.4) reads
(N +1,−N )
(f, g)∆
X
+∞
1
µ
(N )
(N,−N +1)
(∆f, ∆g)∆
=
∆N +1 f (x)∆N +1 g(x)µx
(f, g)D +
+
1−µ
µ−1
x=0
1
(N )
(f, g)D
1−µ
µ
µ
1
(N −1)
2
2 (N −1,−N +2)
(∆f,
∆g)
(∆
+
+
f,
∆
g)
D
∆
(µ − 1)2 1 − µ
µ−1
#
+∞
+∞
X
X
∆N +1 f (x)∆N +1 g(x)µx +
∆N +1 f (x)∆N +1 g(x)µx
+
=
x=0
x=0
2
µ
1
µ
(N )
(N −1)
(∆f,
∆g)
(f, g)D +
+
D
1−µ
(µ − 1)2
(µ − 1)2
X
+∞
µ
(N −1,−N +2)
(∆2 f, ∆2 g)∆
+ 1+
∆N +1 f (x)∆N +1 g(x)µx
(µ − 1)2 x=0
!i
N
1 X
µ
(N −i)
= ... =
(∆i f, ∆i g)D
1 − µ i=0 (µ − 1)2
=
ETNA
Kent State University
etna@mcs.kent.edu
22
Non–standard orthogonality for Meixner polynomials

+
N
+1
X
µ
i=0
(µ − 1)2
!i  +∞
X

∆N +1 f (x)∆N +1 g(x)µx .
x=0
(−N,µ)
R EMARK . It is well known that the Meixner polynomials {Mn
}, with 0 < µ < 1,
for n = 0, 1, . . . , N , are the Kravchuk polynomials {Knp (x, N + 1)}, where the parameter p
is given by the relation
1
1
=1− .
p
µ
In this form, the previous result allows us to give properties of ∆–Sobolev orthogonality for
the Kravchuk polynomials, since the term corresponding to the infinite sum in the ∆–Sobolev
product (5.5), vanishes for all polynomial of degree less or equal to N .
5.2. The operator F (N +1) . In the particular case γ = −N , it is possible to find a
compact expression for the operator F (N +1) defined in (4.3).
Let us remember that when K = N + 1 and γ = −N , we have
ρ(x) = µx ,
Φ(x; N + 1) = µN +1 (x − N )N +1 ,
0 < µ < 1.
So, if we use Lemmas 5.4 and 5.5, expression (4.3) can be written in the form
F (N +1)
N +1−j
N +1−j
N +1
µN +1 (x − N )N +1 X
µ
µ
j
x
∇
∆
=
(−∇) I +
µ I+
∆j
µx
1
−
µ
µ
−
1
j=0
N +1−j
N +1
µN +1 (x − N )N +1 X
µ
j
∇
(−1)
I
+
µx
1−µ
j=0
!
N +1−j
µ
∆
∇j µx I +
∆j
µ−1
N +1−j
N +1
µ
µN +1 (x − N )N +1 X
j
∇
(−1) I +
=
µx
1−µ
j=0
!
N +1−j
µ
j
x−j
j
∆
∆ µ
∇
I+
µ−1
N +1−j
N +1
j N +1 µ
µ
µN +1 (x − N )N +1 X 1 − µ
x
∇
∆
µ
∇j
I
+
I
+
=
µx
µ
1
−
µ
µ
−
1
j=0
=
=
j NX
i
+1−j N +1 µ
µN +1 (x − N )N +1 X 1 − µ
N +1−j
i
µx
µ
1−µ
j=0
i=0
!
N +1
µ
∆
∇i µx I +
∇j
µ−1
ETNA
Kent State University
etna@mcs.kent.edu
M. Álvarez de Morales, T. E. Pérez, M. A. Piñar and A. Ronveaux
23
i
j NX
+1−j N +1 1
µN +1 (x − N )N +1 X 1 − µ
N +1−j
i x
∇
µ
(−1)
I
+
i
µx
µ
µ−1
j=0
i=0
N +1
µ
∆
∇j
I+
µ−1
N +1 N
j N +1−j
+1 X
1−µ
1
µ
N +1
∆
=µ
(x − N )N +1 I +
∇N +1−j ∇j
µ−1
µ
1
−
µ
j=0


N +1
N
+1
N +1−j
X
µ
µ
j

∆
=
(1 − µ) (x − N )N +1 I +
∇N +1 .
1−µ
µ−1
j=0
=
In this way, the difference operator F (N +1) could be expressed as
(5.6)
F
(N +1)
= m(µ, N )(x − N )N +1 I +
N +1
µ
∆
∇N +1 ,
µ−1
where
N +1
m(µ, N ) = (1 − µ)
N
+1
X
i=0
!i
µ
2
(µ − 1)
.
R EMARK . The operator F (N +1) vanishes for every polynomial of degree less or equal
to N .
From relation (4.5), we know that the operator F (N +1) preserves the degree of the polynomials. In fact, we have
(5.7)
F (N +1) xn = F (n, N + 1)xn + . . . ,
n ≥ N + 1,
where F (n, N + 1) denotes the leading coefficient of the polynomial F (N +1) xn , and
F (n, N + 1) = m(µ, N )
n!
.
(n − N − 1)!
From Theorem 4.5, we deduce that the operator F (N +1) is symmetric with respect to the
inner product defined in (5.5). And from Proposition 4.4, we can obtain a representation of
the ∆–Sobolev inner product in terms of the inner product associated to the weight function
µx .
As a consequence of the relation (5.7) and the symmetric character of the operator
(−N,µ)
}n≥N +1 , with
F (N +1) , we deduce that the generalized Meixner polynomials {Mn
(N +1)
, as we state in the following
0 < µ < 1, are the eigenfunctions of the operator F
Proposition.
P ROPOSITION 5.9. Let N ≥ 0 be a given integer and 0 < µ < 1. Then, for every
n ≥ N + 1, we have
ETNA
Kent State University
etna@mcs.kent.edu
24
Non–standard orthogonality for Meixner polynomials
F (N +1) Mn(−N,µ) (x) = m(µ, N )
(5.8)
n!
M (−N,µ)(x).
(n − N − 1)! n
Proof. Using the relation (4.8) for K = N + 1 and Theorem 4.5, and identifying the
leading coefficients, the result follows.
From Proposition 4.4, we can deduce several relations between the generalized Meixner
polynomials and the classical Meixner polynomials.
P ROPOSITION 5.10. Let N ≥ 0 be a given integer and 0 < µ < 1. Then,
(5.9) i)
(−N,µ)
(x − N )N +1 Mn(1,µ) (x) = Mn+N +1 (x) +
n+N
X
i=n
(−N,µ)
αn,i Mi
(x),
n ≥ 0;
ii) For n ≥ N + 1, we get
F (N +1) Mn(−N,µ)(x) =
= m(µ, N )
(5.10)
n!
M (1,µ) (x) +
(n − N − 1)! n
n−1
X
(1,µ)
βn,i Mi
(x).
i=n−N −1
From the property of ∆–Sobolev orthogonality for the generalized Meixner polynomials
we can obtain certain properties of these polynomials. In particular, using expression (5.6) of
the difference operator F (N +1) , we can recover the properties (5.1) and (5.3).
P ROPOSITION 5.11. Let n ≥ N + 1 be an integer. Then, we have
i)
ii)
(N +2,µ)
Mn(−N,µ) (x) = (x − N )N +1 Mn−N −1 (x − N − 1);
I+
N +1
n!
µ
(N +2,µ)
∆
M
∇N +1 Mn(−N,µ)(x) =
(x − N − 1).
µ−1
(n − N − 1)! n−N −1
(N +2,µ)
Proof. i) Writing the polynomial (x − N )N +1 Mn−N −1 (x − N − 1) as a linear combi(−N,µ)
nation of the generalized Meixner polynomials {Mi
(N +2,µ)
(x − N )N +1 Mn−N −1 (x − N − 1) =
n
X
i=0
}i≥0 :
(−N,µ)
an,i Mi
(x − N − 1).
Using Proposition 4.4 and the expression of the operator F (N +1) , the coefficients an,i are
an,i
(N +1,−N )
(N +2,µ)
(−N,µ)
(x − N − 1)
(x − N )N +1 Mn−N −1 (x − N − 1), Mi
∆
=
(N +1,−N )
(−N,µ)
(−N,µ)
(x − N − 1), Mi
(x − N − 1)
Mi
∆
ETNA
Kent State University
etna@mcs.kent.edu
M. Álvarez de Morales, T. E. Pérez, M. A. Piñar and A. Ronveaux
1
=
=
µN +1
(N +2,µ)
(−N,µ)
Mn−N −1 (x − N − 1)F (N +1) Mi
(x − N − 1)µx
x=N +1
k̃i
1
m(µ, N )
µN +1
I+
+
+∞
X
25
+∞
X
(N +2,µ)
Mn−N −1 (x − N − 1)(x − N )N +1
x=N +1
k̃i
N +1
µ
(−N,µ)
∆
∇N +1 Mi
(x − N − 1)µx−N −1
µ−1
,
k̃i
(N +2,µ)
and, applying the orthogonality for the classical Meixner polynomial Mn−N −1 (x − N − 1)
(x − N )N +1 µx−N −1
, we obtain that an,i = 0, for 0 ≤ i < n.
with respect to
(N + 1)!
ii) From i) and using relation (5.8) as well as expression (5.6), we deduce that
n!
(N +2,µ)
(x − N )N +1 Mn−N −1 (x − N − 1) =
(n − N − 1)!
n!
M (−N,µ) (x) = F (N +1) Mn(−N,µ) (x) =
= m(µ, N )
(n − N − 1)! n
N +1
µ
∆
∇N +1 Mn(−N,µ) (x).
= m(µ, N )(x − N )N +1 I +
µ−1
m(µ, N )
Finally, we arrive at the result simplifying the factor m(µ, N )(x − N )N +1 .
REFERENCES
[1] M. A LFARO , M. L. R EZOLA , T. E. P ÉREZ AND M. A. P I ÑAR, Sobolev orthogonal polynomials: the
discrete–continuous case, (submitted).
[2] R. Á LVAREZ -N ODARSE, Polinomios hipergeométricos y q-polinomios, Monografı́as de la Academia de
Ciencias de Zaragoza (1999), (to appear).
[3] M. Á LVAREZ DE M ORALES , T. E. P ÉREZ AND M. A. P I ÑAR, Sobolev orthogonality for the Gegenbauer
(−N+ 1
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
2
polynomials {Cn
}n≥0 , J. Comput. Appl. Math., 100 (1998), pp. 111–120.
I. A REA , E. G ODOY, AND F. M ARCELL ÁN, Inner product involving differences: The Meixner–Sobolev
polynomials, J. Differ. Equations Appl., (to appear).
T. S. C HIHARA, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.
(−k)
K. H. K WON AND L. L. L ITTLEJOHN, The orthogonality of the Laguerre polynomials {Ln (x)} for
positive integer k, Ann. Numer. Math., 2 (1995), pp. 289–304.
, Sobolev orthogonal polynomials and second–order differential equations. Rocky Mountain J. Math.,
28 (2) (1998), pp. 547–594.
A. F. N IKIFOROV, S. K. S USLOV AND V. B. U VAROV, Classical Orthogonal Polynomials of a Discrete
Variable, Springer Ser. Comput. Phys., Springer-Verlag, Berlin, 1991.
T. E. P ÉREZ AND M. A. P I ÑAR, On Sobolev orthogonality for the generalized Laguerre polynomials, J.
Approx. Theory, 86 (1996), pp. 278–285.
, Sobolev orthogonality and properties of the generalized Laguerre polynomials, in Orthogonal Functions, Moment Theory and Continued Fractions: Theory and Applications, William B. Jones and A. Sri
Ranga, eds., Marcel Dekker, New York, 18 (1997), pp. 375–385.
A. RONVEAUX AND L. S ALTO, Discrete orthogonal polynomials and polynomial modification of a classical
functional, (submitted).
J. S TOER AND R. B URLIRSCH, Introduction to Numerical Analysis, Springer-Verlag, New York, 1980.
G. S ZEG Ő, Orthogonal Polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. Soc., Providence, RI, 1975.