1. Introduction
Let
denote a nondecreasing real-valued, bounded function with a finite or an infinite number of points of increase in the interval,
. The latter interval may be infinite. We assume that moments of all orders exist, that is:
exists for all
, with the integration defined in the Lebesgue-Stieltjes sense.
Definition 1. The polynomial sequence,
,
is said to be orthogonal with respect to the measure,
,
if:where and is the Kronecker symbol.
To any sequence of moments,
is associated the linear functional,
, defined on
by:
The functional,
(also called moment functional), is said to be regular if there exists a family,
, of polynomials orthogonal with respect to
, that is:
It is well-known that the necessary and sufficient condition for
to be regular is that the principal submatrices of the infinite Hankel matrix,
, are nonsingular. Furthermore, to the measure
or to the corresponding regular functional,
, is associated the so-called Stieltjes function
, defined as:
If
is known, then the distribution function,
μ, can be recovered from
by means of the Stieltjes inversion formula [
1] (see, also, [
2,
3]):
Classical orthogonal polynomials of a continuous variable,
, can be characterized by the distributional differential Equation (usually called Pearson-type Equation) satisfied by their corresponding regular functional,
, [
4]:
where
and
ψ are polynomials with
and
, with
and while
and
are linear functionals defined by:
Equation (5) is equivalent to the following two Equations [
4,
5] (which also characterize classical orthogonal polynomials), namely, the second-order differential Equation satisfied by each
:
and the Riccati Equation satisfied by the Stieltjes function,
:
Properties, similar to Equations (5)–(7) exist for classical orthogonal polynomials of a discrete variable [
6,
7,
8], as well as for classical orthogonal polynomials of a
q-discrete variable [
7,
9,
10].
Semi-classical orthogonal polynomials of a continuous variable [
4] are defined as those for which the corresponding linear functional,
, satisfies a Pearson-type Equation (5), but with a more relaxed condition on the polynomials,
and
ψ, namely,
This is equivalent to the Riccati Equation (7), but with the constant term,
, replaced by a polynomial,
. Laguerre-Hahn orthogonal polynomials [
4] are also defined by general forms of Equations (5) and (7).
Classical orthogonal polynomials on a nonuniform lattice satisfy an Equation of the type: [
11,
12,
14]
where
and
ψ are polynomials of maximal degree, two and one, respectively,
is a constant depending on the integer,
n, and the leading coefficients,
and
, of
and
ψ:
and
is a nonuniform lattice defined by:
Here, Δ and ∇ are the forward and the backward operators:
and:
where
is the set of complex numbers. The Lattices (10) satisfy:
for
with the initial values:
where:
In addition, the sequences,
, satisfy the following relations:
and are given explicitly by [
11,
12]:
and:
By means of the companion operators,
and
[
13,
14], Equation (8) can be rewritten as:
where:
The operators,
and
, transform polynomials of degree
n in
into a polynomial of degree
and
n in the same variable, respectively. In addition, they fulfill important relations—called product and quotient rules—which read, taking into account the shift (compared to the definition in [
14]), in the definition of the above defined companion operators as:
Theorem 2. [14]- 1.
The operators and satisfy the product rules Iwhere ,
and is a constant term with respect to .
- 2.
The operators,
and ,
also satisfy the product rules II:where:
For illustration, the Askey-Wilson polynomials [
7]
are defined by:
By taking
, the lattice reads as
By using the orthogonality relation and the Pearson-type Equation satisfied by the weight of the Askey-Wilson polynomials, Foupouagnigni [
14] showed that the polynomials,
, satisfy a divided-difference equation of the type from (8) with:
Costas-Santos and Marcellán [
15], using the corresponding weight function, gave four equivalent characterizations for classical orthogonal polynomials on nonuniform lattices: More precisely, they proved the equivalence between the second-order divided-difference Equation (8), the orthogonality of the divided-difference derivatives, the Rodrigues-type formula, as well as the structure relation.
However, despite the important results by Costas-Santos and Marcellán, the characterization of classical orthogonal polynomials on nonuniform lattices is still incomplete. In fact, there are still missing many properties, in particular:
- ①
The characterization in terms of the Pearson-type Equation for the corresponding regular functional (similar to Equation (5));
- ②
The characterization in terms of the Riccati Equation for the Stieltjes function of the corresponding regular functional (similar to Equation (7));
- ③
The method to recover orthogonal polynomials,
, from the second-order divided-difference Equation (8) (similar to the one of Fuchs-Frobenius method used to solve Equation (6)); this problem has already been raised by Ismail (see [
16], page 518);
- ④
The building of a bridge between the theory of Magnus based mainly on the Riccati Equation satisfied by the formal Stieltjes function [
17,
18] and the theory of orthogonal polynomials on nonuniform lattices based on the functional approach. Such a bridge would enable the construction of concrete examples of semi-classical and Laguerre-Hahn orthogonal polynomials, since it is easier and more convenient to modify the functional rather than the formal Stieltjes function.
These characterizations are very important, since they are the key for the definition and the characterizations of new classes of orthogonal polynomials on nonuniform lattices, namely, the semi-classical and the Laguerre-Hahn orthogonal polynomials on nonuniform lattices.
To provide a solution to the four problems raised above and taking into consideration the expansions of
in the monomial basis
,
and the one of the formal Stieltjes function [
18]:
there is a strong need to control the actions of the operators,
and
, on the monomial,
, and also on the function,
. Unfortunately, the application of these operators to the monomial,
, produces a linear combination (with complicated coefficients) of all monomials of a degree less than or equal to
and
n, respectively. Also, the application of
and
to the function,
, produces a rational function of the variable,
, with coefficients being very difficult to handle [
14]. These constraints makes the monomial basis,
, not appropriate for the aforementioned operators and justifies our decision to search for an appropriate basis for the operators,
and
.
The aim of this work is:
- ①
To provide an appropriate basis for the companion operators, that is, a basis
, such that each
is a polynomial of degree,
n, in
fulfilling:
where
and
are given constants.
- ②
To provide an algorithmic method to solve Equation (15) as a series in terms of the new basis and to extend this result to solve arbitrary linear divided-difference equations with polynomial coefficients involving only products of operators, and .
- ③
To use another appropriate basis for the operators, and , to derive Representation (15) for the Askey-Wilson polynomials from the hypergeometric Representation (22) without making use of the weight function.
- ④
To solve explicitly an Equation of type (15) and to extend this result to solve arbitrary linear divided-difference equations with polynomial coefficients involving only products of operators and .
- ⑤
To provide a new representation of the formal Stieltjes function of a given linear functional on a nonuniform lattice and deduce from it various important properties connecting the functional approach and the one based on the Riccati Equation for the formal Stieltjes function.
The content of this paper is organized as follows. In
Section 1, we recall necessary preliminaries, while in the second section, we provide the basis,
, compatible with the companion operators,
and
. The third section deals with the algorithmic series solutions of divided-difference equations in terms of the basis,
. In
Section 4, we give the second basis,
, compatible not with the companion operators, but rather with their products,
and
, and use this basis in the fifth section to find the algorithmic series solutions of some divided difference equations in terms of the basis,
. In the last section, we apply the basis,
, to provide new representation of the formal Stieltjes series and deduce its corresponding properties. Basic exponential and basic trigonometric functions have also been expanded in terms of the basis,
.
2. A New Basis Compatible with the Companion Operators
Using the generalized power on the lattice,
, defined by Suslov [
12] (see, also, [
11]) as:
we define the function,
, by:
where
ϵ is the unique solution of the Equation (in the variable,
t):
provided that the coefficients,
, of (10) fulfill
or
for the quadratic lattice of the
q-discrete and discrete variable, respectively. More precisely,
ϵ fulfills:
for the
q-quadratic lattice,
, or the relation:
for the quadratic lattice,
.
Although the function, , from Relation (26) does not appear explicitly as a function of , as will be shown in the following theorem, it is a polynomial of degree n in the variable, . Therefore, we adopt the notation, , instead of , and obtain the following results:
In order to prove Theorem 3, we first prove the following lemmas:
Lemma 4. The following relation holds:where is the set of nonnegative integer numbers.
Lemma 5. The following relations are satisfied: where the function,
,
is defined as: Proof of Lemma 4
Proof: Relation (39) is obtained by iteration of the following relation:
given by Suslov ([
12], Equation 2.8, page 235). Therefore:
where the function,
, is defined as:
Proof of Lemma 5
Proof: Relation (40) is proven by combination of the definition of
, given by (16), and the following relation, given by Suslov ([
12], Equation (2.22), page 239):
but with
and
s replaced by
. Here,
means that the forward operator acts on the variable,
s.
The proof of Relation (41) will be done in the following steps:
In the first step, we apply operator
on both sides of Relation (42) and adjust the index,
j, to obtain:
In the second step, we use the relation:
obtained from (12) and the relation,
, to get:
In the third step, we use Relation (39) to transform the previous Equation into:
Therefore, using, again, Relation (39), we get:
In the fourth step, we use the following relation obtained from (12):
to Transform (44) as:
In the last step, we use the following relation obtained from (11):
to obtain the Equation:
and the proof is complete. ☐
Proof of Theorem 3
Proof: By replacing:
z by
in Equations (40) and (41), we obtain:
Therefore, for
to be a linear combination of
and
, it is necessary for the parameter,
z, to be a solution of the following Equation in the variable,
t:
This solution is unique, denoted by
ϵ (see Equation (27)), provided that the coefficients,
, of (10) fulfill
or
for the quadratic lattice of the
q-discrete and discrete variable, respectively. The resulting basis reads:
This basis fulfills Relations (30) and (31).
The proof of Relations (32) and (33) is similar to those of Relations (30) and (31).
For the proof of Relation (34), we use Relations (39), (42) and (43) for fixed nonnegative integer,
n, and the fact that
with
to obtain:
Relation (35) is satisfied, since, for integers,
and
k, such that
and
, we get, by direct computation using (
11), that:
Therefore,
.
Relation (36) is proven by induction on n. Relation (37) is obtained by Iterating (30). We split the proof of Relation (38) into three steps:
In the first step, for fixed
, we expand
in the basis
:
and use the following relation, due to (34):
to get
. Considering (45) for
and
, we get—using, again, (46):
Therefore,
, thanks to (35). Progressively, we obtain in a similar way for a fixed integer,
j, using (35), (45) and (46), that:
In the second step, we rewrite, accordingly, Relation (45):
and obtain, using (34):
where:
Use of Equation (47) for
gives:
taking into account Relation (46) and the fact that
. In the third step, we apply the operator,
(defined in (59)) to (47), and use the relation:
derived by direct computation, to obtain the relation:
from which we deduce using, again, (46) that:
The remaining coefficients,
, are obtained in the same way by successive application of
on (49) and use of the Equations,
. ☐
As corollary of Theorem 3 (Equation (34)), we would like to give, explicitly, a representation of the basis, .
Corollary 6.
- ①
The basis,
,
is explicitly defined on the lattices,
(with ), by:where ϵ is defined by (28) and .
- ②
In the particular case of the Askey-Wilson lattice, (,
the previous Equations read: - ③
The basis,
,
is explicitly defined on the lattices,
(with ), by:where the Pochhammer symbol, ,
is defined as - ④
In the particular case of the Racah lattice,
(,
the previous Equations read:
Remark 7. It should be mentioned that for the specific case of the Askey-Wilson lattice our basis,
,
coincides (up to a multiplicative factor) to the basis, ,
used by Ismail [19] (Equation (1.4), page 261):to provide the Taylor expansion of a polynomial in terms of the basis,
.
3. Algorithmic Series Solutions of Divided-Difference Equations
In this section, we provide an algorithmic method to solve a divided-difference equation in terms of the bases, and , with the latter, which is defined in Proposition 14, being more appropriate for the Askey-Wilson lattice.
3.1. Algorithmic Series Solutions of Divided-Difference Equations in Terms of
The basis, , is relevant for the companion operators and provides a method to obtain series solutions of divided-difference equations.
Theorem 8.
If:is a polynomial solution of the Equation:where λ is a constant, ϕ and ψ are polynomials a degree of at most two and one, respectively, and given by:then, the coefficients,
,
satisfy a second-order recurrence Equation:with:where: Proof: In the first step, we apply the companion operators to (56) and, taking into Account (30) and (31), we get:
In the next step, we use (60) and (61) in (57) and the following relations obtained by Iterating (34):
to get an Equation of type:
The proof is completed by transforming the previous Equation into:
and using the fact that
is a basis of
. ☐
As an application of Theorem 8, in the following proposition, we provide two linearly independent solutions of Equation (8) (or Equation (15)), where and ψ are those coefficients of the Askey-Wilson polynomials given by Equation (23) with and .
Proposition 9. For and ,
two linearly-independent solutions of the divided-difference Equation (8) for the Askey-Wilson polynomials are represented in terms of the basis,
,
as:where ,
is a polynomial and is not.
Proof: Following the method described in the previous theorem, we obtain for
and
ψ, given by (23), with
and
, the second-order difference Equation for the expansion coefficient,
(see (58)):
with the change of variable
.
To solve this recurrence Equation, we use the refined version of
q-Petkovšek’s algorithm published by Horn [
20,
21] and implemented in Maple by Sprenger [
23,
24]) (in his package
qFPS.mpl) by the command
qHypergeomsolveRE, which finds the
q-hypergeometric term solutions of
q-holonomic recurrence Equations. We obtain two solutions expressed in terms of
p-Pochhammer and
-Pochhammer terms. Details of this computation can be found in a Maple file made available on [
22]. Finally, we transform these solutions using the relation (see [
25], Equation 1.2.40, page 6):
and the relation obtained by direct computation:
to obtain the coefficients of
in the Expansions (62) and (63), given above, after replacing
p by
.
The q-hypergeometric representations of and are deduced by direct computation using the Expansions (62) and (63) and Proposition 6. These solutions are linearly-independent, since the first is a polynomial, while the second is an infinite series expansion in terms of . ☐
Remark 10.
- ①
For fixed positive integer, n, the polynomial solution given by (62) is proportional to the Askey-Wilson polynomial given by (22) with and .
- ②
The non-polynomial solution of Equation (64) given by (63), which is convergent, proves clearly that our method described in the previous theorem can be applied to recover convergent series solutions of divided difference Equations in terms of the basis, .
The previous theorem can be extended to solve divided-difference Equations of arbitrary order with polynomial coefficients, using Equations (37) and (38).
Theorem 11. The coefficients of a polynomial solution:of any divided-difference Equation of the form:where and and are polynomials of arbitrary (but fixed) degree in the variable,
,
are solution of a linear difference Equation. Proof: Equation (67) can be transformed into an Equation of the type:
where
and
are polynomials of arbitrary (but fixed) degree in the variable,
, using Relations (19) and (20). The proof of the theorem is completed in the same way as in Theorem 8, Substituting (66) in (68) and making use of Equations (37) and (38). ☐
Remark 12. The previous result generalizes the one given by Atakishiyev and Suslov [26] in which they provide a method to construct particular solutions to hypergeometric-type difference Equations on a nonuniform lattice. Remark 13. Theorems 8 and 11 provide a method of expanding solutions of Askey-Wilson operator Equations in terms of the basis, (see Remark 7), or ,
providing, therefore, the solution of the problem raised by Ismail ([16], page 518). The coefficients of the expansion of the polynomial solution given in given in (62) is not q-hypergeometric, but rather -hypergeometric. Therefore, there is the need of an additional transformation in order to prove that are effectively the known Askey-Wilson polynomials. In order to provide an explicit, direct and simple representation of polynomial solutions of divided-difference Equations, such as (15), we provide, in the next subsection, a second basis, which is not compatible with the operators, and , but rather, with and and are, therefore, very useful when searching for series solutions of divided-difference Equations with polynomials coefficients, involving linear combination of the products of and . This basis allows, for example, to recover from the divided-difference Equation (15), without the need of any further transformation, the defining representation of the Askey-Wilson polynomials given by given by (22).
3.2. Algorithmic Series Solutions of Divided-Difference Equations in Terms of
In the first step, we define the basis and study its properties.
Expressing the Askey-Wilson polynomials (22) in terms of
q-Pochhammer symbols:
and the fact that these polynomials fulfill (15) suggests the study of the action of the companion operators on the function:
which happens to be a polynomial of degree
n in
. By considering a more general situation, we get:
Proposition 14. The general q-quadratic lattice:and the corresponding polynomial basis:which we relabel as:fulfilling the relations:where: Proof: The proof is obtained by direct computation. ☐
Remark 15. It should, however, be noted that for ,
Relation (71) appears as exercise in [27], page 34. It also appears in [16], Equation (20.3.11), page 518. The following proposition provides the connection coefficients between the bases, and :
Proposition 16. The bases, and ,
are connected in the following ways:where:and: Proof: The proof is obtained first by applying the operator,
, to both members of (78) for fixed non-negative integers,
and
, and using (37) and (71), then by observing that:
☐
From Proposition 14, it appears clearly that because of the appearance of in Relations (71) and (72), the action of and on cannot be written as finite (number of terms not depending on n) linear combination of the elements of the basis . However, this problem is solved by using the operators, and , instead, to obtain Relations (73) and (74). Equation (15) can therefore be solved using the known coefficients, and ψ, of Askey-Wilson.
3.3. Algorithmic Series Solutions of Divided-Difference Equations in Terms of
Theorem 17.
If:is a polynomial solution of the Equation:where λ is a constant, ϕ and ψ are polynomials of a degree of at most two and one, respectively:then, the coefficients, ,
satisfy a second-order difference Equation:with: Proof: The proof is similar to the one of Theorem 8 using the properties of the basis given in Proposition 14. ☐
The previous theorem can be extended to divided-difference Equations of arbitrary order with polynomial coefficients, involving linear combinations of powers of the operators, and . Such operators can be rewritten, using Relations (19) and (20), as linear combination of and For this extension, we will need the following results, obtained by iteration of Relations (71)–(77).
Proposition 18. The basis, ,
satisfies the following relations:where: We now state the following theorem, which can be proven in the same way as Theorem 17, but using, instead, the Equations of the previous proposition.
Theorem 19. Ifis a polynomial solution of the divided-difference Equation:where and T are polynomials in the variable, ,
then the coefficients, ,
satisfy a linear difference Equation of maximal order,
.
As the corollary of Theorem 17, we have recovered the representation of the Askey-Wilson polynomials by solving Equation (
8).
In fact, we have replaced the coefficients,
and
ψ, of the Askey-Wilson polynomials (see (23)) in Equation (85) to obtain the following difference Equation of the expansion coefficient,
:
In [
28,
29]—see, also, [
30]—algorithms were presented to find all solutions of an arbitrary
q-holonomic difference Equation in terms of linear combinations of
q-hypergeometric terms. This algorithm was tuned and made much more efficient in [
20,
21], and a Maple implementation was made available in [
23]. For the purpose of solving the second order
q-difference Equation (87) in terms of hypergeometric terms, we have used the command,
qrecsolve, from the
q-sum package [
29] to obtain the expansion coefficients,
, of the expansion of the Askey-Wilson polynomial (69), with
Details of this computation can be found in a Maple file made available on [
22].