Abstract
The Chapter presents diverse but related results to the theory of the proper and generalized Goncarov polynomials. Couched in the language of basic sets theory, we present effectiveness properties of these polynomials. The results include those relating to simple sets of polynomials whose zeros lie in the closed unit disk U=z:z≤1. They settle the conjecture of Nassif on the exact value of the Whittaker constant. Results on the proper and generalized Goncarov polynomials which employ the q-analogue of the binomial coefficients and the generalized Goncarov polynomials belonging to the Dq- derivative operator are also given. Effectiveness results of the generalizations of these sets depend on whether q<1 or q>1. The application of these and related sets to the search for the exact value of the Whittaker constant is mentioned.
Keywords
- Basic sets
- Simple sets
- Effectiveness
- Whittaker constant
- Goncarov polynomials
- Dq operator
1. Introduction
The Chapter is on the effectiveness properties of the Goncarov and related polynomials of a single complex variable. It is essentially a compendium of certain results which seem diverse but related to the theory of the proper and generalized Goncarov polynomials.
Our first set of results deals with simple sets of polynomial [1], whose zeros lie in the closed unit disk
Next are results on a generalization of the problem where the polynomials are of the form
and the points
It is shown that applying the results of Buckholtz and Frank [3] on the generalized Goncarov polynomials
Consequently, we provide some results on the polynomials
with the obtained results justifying the restriction (2) on the points
Finally, we provide other relevant and related results on the properties of the generalized Goncarov polynomials
2. Preliminaries
We record here some background information for easy reading of the contents of the presentation.
2.1 Basic sets and effectiveness
A sequence
The polynomials
In the representation (3), let
Formally rearranging the terms, we obtain the series
We write
Hence, we obtain the series
which is called the basic series associated with the function
The coefficients
If
The basic series (4) is said to represent
When the basic set
When
To obtain conditions for effectiveness, we form the Cannon sum
where
From (3), we have that
so that, if we write
The function
Theorems about the effectiveness of basic sets are due to Cannon and Whittaker (cf. [2, 4, 5]).
A necessary and sufficient condition for a Cannon set
2.2 Mode of increase of basic sets
The mode of increase of a basic set
where
The order and type of a set define the class of entire functions represented by the set.
The necessary and sufficient conditions for the Cannon set of polynomials to be effective for all entire functions of increase less than order
2.3 Zeros of simple sets of polynomials
The relation between the order of magnitude of the zeros of polynomials belonging to simple sets and the mode of increase of the sets has led to many convergence results, just as that between the order of magnitude of the zeros and the growth of the coefficients has. In the case of the zeros and mode of increase, the approach to achieve effectiveness is to determine the location of the zeros while that between the zeros and the coefficients is to determine appropriate bounds (cf. Boas [7], Nassif [8], Eweida [9]).
2.4 Properties of the Goncarov polynomials
We record in what follows certain properties of the proper and generalized Goncarov polynomials together with the definitions of the q-analogues and the Dq-derivative operator.
The proper Goncarov polynomials
These polynomials generate any function
which represents
In this case, if
A consideration of
Concerning the case where the points
If
Buckholtz [10] obtained an exact determination of the constant W. In fact, if we write
where the maximum is taken over all sequences
Moreover, if we put
Buckholtz ([10], formula 2) further showed that
Employing an equivalent definition of the polynomials
and differentiating with respect to z, we can obtain
Writing
then (18) yields, among other results,
and
Applying (21) and (22) to (19) we obtain
for
Expanding
we arrive through (22) and (23) to the formulae of Levinson [12],
Also, differentiating (18) with respect to
for
2.5 The q -analogues and D q derivatives
Let
Also, the
and the
Moreover, the
so that when
In [3] we have a generalization of the Goncarov polynomials as in (13) belonging to the operator D such that for
associated with the sequence
When
and the Goncarov series associated with the function
Writing
so that
then we have from, (32) that
Also, Nassif ([14], Lemma 4.1), proved that
We can verify, with Buckholtz ([10], Lemma 1), from the formulae (33), the following:
And hence, by repeated application of
Expressing
The identities (39) and (43) have been obtained, in their general form, in ([3]; formulae (2.5), (2.9)). Also, a combination of (38) and (42) yields
for
Finally, if we put
where the maximum is taken over all sequences
Also, in view of the formulae (33), we can verify that, when
3. Results on the zeros of simple sets
3.1 Zeros of simple sets of polynomials and the conjecture of Nassif on the Whittaker constant are discussed here
The following result is known for simple sets of polynomials whose zeros all lie in the unit disk.
Theorem A.([1], Theorem 1).
When the zeros of polynomials belonging to a simple set all lying within or on the unit circle the set will be of increase not exceeding order 1 type 1.378.
Using known contributions in the theory of Goncarov polynomials, we show that the alternative form of the above theorem is as follows:
When the zeros of the polynomials belonging to a simple set all lying in the unit disk, the set will be of increase not exceeding order 1 type
Indeed, applying the result of Buckholtz ([10], formula 2), the following theorem which resolved the conjecture of Nassif ([8], p.138), is established.
Given a positive number
For completeness, we give the proof of Theorem 3.1.1 as a revised version of Theorem A.
Let
Suppose that
Then multiplying the matrix of coefficients
Write.
then the above relation will give
And to show the dependence of
Comparing this relation with the identify
of Levinson [12], we infer that
Differentiating (50)
Hence, a combination of (15), (16), (20), (51)-(53) leads to the inequality.
Observing that
It follows from (17) that the set
Since
This completes the proof of the theorem.
3.2 Background and the proof of the conjecture
Before the proof of Theorem 3.2.1, we note that we can take,
such that
Moreover, from (20), the definition (15) ensures the existence of the points
Having fixed the integer
Lemma 3.2.1 ([15], Lemma 3.2).
For any integer
Then, the complex numbers
and
Proof.
The proof is by induction.
When
Then the value
Applying the identify (25) of Macintyre to
so that (20) and (58) imply that
where the prime denotes differentiation with respect to
Hence, in view of (62), Cauchy’s inequality yields
and the inequality (61) is satisfied for
The numbers
Proceeding in a similar manner as for the Case
where the prime denotes differentiation with respect to
Applying Cauchy’s inequality to the polynomial
Hence, by induction, the inequality (61) of the Lemma is established.
We now prove theorem 3.1.2.
The required simple set
where the points
It follows that the zeros of the polynomials
Also, if
and if we write
then from the relation (52), we deduce from (59) and (65), that
Now, in view of (66), the Cannon sum of the set
Hence, combining (57), (61), (67) and (68) yields
vIt follows from this inequality and Theorem 3.1 that the order of the set
In view of the inequality (56), we deduce from (69) that
and Theorem 3.1.2 is established.
This settles the conjecture.
4. Generalization
4.1 As a generalization of the above problem, we consider the simple set p n z n given by
where
We establish the following:
When the points
Given
When
the corresponding set
Theorem 4.1.2 shows that the result in Theorem 4.1.1 is best possible. Also, the restriction (71) on the sequence
Proof of Theorem 4.1.1 is similar to the first part of Theorem 3.1.1.
Let
then multiplying the matrix of coefficients
Putting
the above relation yields
Comparing the formulae (45) and (75) we infer that
Moreover, operating
Hence, when the operator
which, on reduction, yields
Applying (74), (76) and (78), we obtain
Identify (79) is the bridge relation between the set
Suppose
Since
The Cannon sum of the set
from which it follows that the set
5. Proof
5.1 Proof of Theorem 4.1.2
We argue as in the Proof of Theorem 3.1.2. We first obtain an identity similar to (25) of Macintyre using the following Lemma:
Lemma 5.1.1.
For
where
Proof of Lemma
The proof is by induction.
For
Hence, operating
so that the identity (83) is satisfied for
Hence, the derivative
Or equivalently,
Hence, formulae (33) imply that
and the relation (83) is also valid for
The Lemma is thus proved by induction. Now, following similar lines paralleling those of the proof of Theorem 3.1.2, we need to establish a Lemma similar to that used for Theorem 3.1.2.
Indeed, observing that
such that
Also, from the definition (46) of
With this choice of the integer
Lemma 5.1.2.
With the notation
we can choose a sequence
Proof.
We first observe, from a repeated application of (30), that an analytic function
Hence, by Cauchy’s inequality, we have
Applying the usual induction process, we obtain, from (87) for the case
Hence the identity (83) yields
Therefore, we obtain
where the
Pick the number
hence, a combination of (86), (89) and (90) yields
and the inequality (88) is satisfied for
We can now prove Theorem 5.1.4.
We note that the points
where the points
If
so that, for the Cannon sum of the set
Since
Hence, (93) and (94) yield, for the Cannon function,
Noting that
and
5.2 Proof of Theorem 4.1.3
Let
For this, we put
and let
We claim that, in this case, the corresponding set
Now, in the identity (37), we let
to obtain
Put
so that (97) yields
Hence, if we put
then (97) implies that
where
Since
Let
then from (100) and (101), we have
Thus, for the Cannon sum of the set
Since
To prove Theorem 4.1.3 we first note, from (72), that if we put
then
We then multiply the matrix
Now, imposing the restriction (71) on the points
Thus, the inequality
is true for
To prove (107), in general, we observe that, since
Assume that (107) is satisfied for
Hence, it follows by induction, that the inequality (107) is true for
where
Appealing to a result of Al-Salam ([18]; formula 2.5), we deduce that
The Cannon sum of the set
Hence, when
from which it follows that the set
6. Other related results
The Goncarov polynomials belonging to the
When
The set of Gancarov polynomials
Theorem 1.5 of Buckholtz and Frank [3] shows that the result of Theorem 6.1 above is best possible. They also showed that when
We also state and prove the following theorem.
Suppose that
To prove this theorem we put, as in the proof of Theorem (72),
so that
for these respective polynomials. Thus, the constructive formulae (33) for these polynomials will be
and
where
Lemma 6.1.
The following identity is true for
Proof.
We finish note, from the definition of the analogue
and
Hence, applying the relations (37) and (112) to
Hence, the relations (115) and (116) can be introduced to yield
Now, since
the identity (114) is satisfied for
Moreover, if (114) is valid for
and hence the Lemma is established.
Proof of Theorem 6.2.
Write
so that the restriction (111) implies that
Therefore, a combination of (37), (114), (118) yields
Also, by actual calculation we have that
Inserting (118), (120) and (121) into (33), we obtain
in the sense that each term in the sum on the left hand side of this relation is equal to the corresponding term in the sum on the right hand side.
Hence, if
and
Since the points
To show that the result of the Theorem is best possible we appeal to Theorem 1.5 of Buckholtz and Frank [3] to deduce that the set
In view of the relation (122), we may conclude that the set
6.1 The case of Goncarov polynomials with Z k = at k , k ≥ 0
Nassif [14] studied the convergence properties of the class of Goncarov polynomials
6.2 Quasipower basis (QP-basis)
Kazmin [20] announced results on some systems of polynomials that form a quasipower basis, (QP-basis), in specified spaces. These include the systems of Goncarov polynomials and of polynomials of the form:
For full details of QP-basis and some of the results announced, cf. ([20]; Corollaries 3, 4).
Of interest is his results that the system in (123), for arbitrary sequence
7. Conclusions
The chapter presents a compendium of diverse but related results on the convergence properties of the Goncarov and Related polynomials of a single complex variable. Most of the results of the author (or joint), have appeared in print but are here presented in considerable details in the proofs and in their development, for easy reading and assimilation. The results of other authors are summarized with related and relevant ones mentioned to complement the thesis of the chapter. Some recent works related to the Goncarov and related polynomials, cf. [24, 25, 26, 27, 28, 29], which provides further applications are included in the references.
The comprehensiveness of the presentation is for the needs of those who may be interested in the subject of the Goncarov polynomials in general and also in their application to the problem of the determination of the exact value of the Whittaker constant, a problem that is still topical and challenging.
Acknowledgments
I acknowledge the mentorship of Professor M. Nassif, (1916-1986), who taught me all I know about Basic Sets. I thank Dr. A. A. Mogbademu and his team for typesetting the manuscript at short notice and also the Reviewer for helpful comments which greatly improved the presentation.
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