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.
- Basic sets
- Simple sets
- Whittaker constant
- Goncarov polynomials
- Dq operator
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 , whose zeros lie in the closed unit disk . It is a complement of a theorem of Nassif  which resolved his conjecture on the value of the Whittaker constant . We provide also the relation between this problem and the theory of the proper Goncarov polynomials.
Next are results on a generalization of the problem where the polynomials are of the form
and the points are given complex numbers with the q-analogue of the binomial coefficient . From the results reported, it is shown that the location of the points that leads to favorable effectiveness results depends on whether or. The relation of this problem to the generalized Goncarov polynomials belonging to the Dq-derivative operator is also recorded.
It is shown that applying the results of Buckholtz and Frank  on the generalized Goncarov polynomials belonging to the Dq-derivative operator when, leads to the result that, when the points lie in the unit disk , the resulting polynomials fail to be effective.
Consequently, we provide some results on the polynomials when
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 belonging to the Dq-derivative operator. For a comprehensive and easy reading, background results are provided in the Preliminaries of sections 2.1–2.5.
We record here some background information for easy reading of the contents of the presentation.
2.1 Basic sets and effectiveness
A sequence of polynomials is said to be basic if any polynomial and, in particular, the polynomials , can be represented uniquely by a finite linear combination of the form.
The polynomials are linearly independent.
Formally rearranging the terms, we obtain the series
Hence, we obtain the series
which is called the basic series associated with the function and the correspondence is written as
The coefficients is the basic coefficients of relative to the basic set and is a linear functional in the space of functions.
If is of degree
The basic series (4) is said to represent in a disk where analytic, if the series is converges uniformly to in or that the basic set represents in.
When the basic set represents in every function analytic in, then the basic set is said to be effective in for the class of functions analytic in .
When , the basic set represents, in , every function which is analytic there and we say that the basic set is effective in .
To obtain conditions for effectiveness, we form the Cannon sum
From (3), we have that
so that, if we write
The function is called the Cannon function of the set in .
A necessary and sufficient condition for a Cannon set to be effective, in , is
2.2 Mode of increase of basic sets
The mode of increase of a basic set is determined by the order and type of the set. If is a Cannon set, its order is defined, Whittaker , by
where is given by (5). The type is defined, when , by
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 type is
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 , Nassif , Eweida ).
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 associated with the sequence of points in the plane are defined through the relations, Buckholtz (, p. 194),
These polynomials generate any function analytic at the origin through the Goncarov series
which represents in a disk , if it uniformly converges to in .
In this case, if , the Goncarov series (14) vanishes and .
A consideration of , for which andcf. Nassif , shows that the Goncarov series does not always represent the associated function and hence certain restrictions have to be imposed on the points and on the growth of the function .
If is an entire function of exponential type less than c and if each of has a zero in U then .
Buckholtz  obtained an exact determination of the constant W. In fact, if we write
where the maximum is taken over all sequences whose terms lie in , Buckholtz (, Lemma 3) proved that exists and is equal to
Moreover, if we put
Employing an equivalent definition of the polynomials as originally given by Goncarov  in the form
and differentiating with respect to z, we can obtain
then (18) yields, among other results,
for , where the differentiation is with respect to the first argument.
Expanding in powers of , in the form
q-analogues and derivatives
Moreover, the – derivative operator, corresponding to the number q is defined as follows: is any function of z, then
so that when , then according to (26), we have and if is any function analytic at the origin then
associated with the sequence , where and is a non-decreasing sequence of numbers to obtain
When , the relations (32) reduce to (6), hence the polynomials reduce to the proper Goncarov polynomials . Comparing (30) and (32), Nassif  investigated the class of generalized Goncarov polynomialsbelonging to the Dq- derivative operator when and given by,
and the Goncarov series associated with the function is
then we have from, (32) that
Also, Nassif (, Lemma 4.1), proved that
And hence, by repeated application of , we obtain
Finally, if we put
where the maximum is taken over all sequences and the terms lie in the unit disk , then Buckholtz and Frank (, Corollary 5.2), proved that
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.(, 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 , where
Given a positive number , a simple set of polynomials, whose zeros all lie in can be constructed such that the increase of the set is not less than order 1 type H–.
For completeness, we give the proof of Theorem 3.1.1 as a revised version of Theorem A.
Let be a sequence of points lying in the unit disk and consider the set of polynomials given by
Suppose that admits the representation
Then multiplying the matrix of coefficients with its inverse , we obtain
then the above relation will give
And to show the dependence of
Comparing this relation with the identify
of Levinson , we infer that
Observing that for any value of , then the Cannon sum of the set for will, in view of (54), be
It follows from (17) that the set is of increase not exceeding order 1 type . The proof is now completed by applying the results of Walsh and Lucas, cf. Marden (, pp. 15,46), with (54) and following exactly the same lines of argument as in (, pp.109–110), to arrive at the inequality.
Since in , it follows that the set is of increase not exceeding order 1 type H = .
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, .(In fact, according to Macintyre (; p. 241), we have H > ). Hence it follows from (16) that corresponding to , there exists an integer
Having fixed the integer
Lemma 3.2.1 (, Lemma 3.2).
For any integer , write
Then, the complex numbers can be chosen so that
The proof is by induction.
When , we have from (59) that
Then the value will be chosen so that
Applying the identify (25) of Macintyre to
where the prime denotes differentiation with respect to .
Hence, in view of (62), Cauchy’s inequality yields
The numbers will be fixed so that
where the prime denotes differentiation with respect to .
Hence, by induction, the inequality (61) of the Lemma is established.
We now prove theorem 3.1.2.
The required simple set of polynomials is constructed as follows:
where the points are chosen to satisfy (63) and the numbers are fixed as in the Lemma.
It follows that the zeros of the polynomials all lie in the unit disk
Also, if admits the unique linear representation.
and if we write
Now, in view of (66), the Cannon sum of the set for , is .
vIt follows from this inequality and Theorem 3.1 that the order of the set is exactly 1 and since , the type of the set will be
and Theorem 3.1.2 is established.
This settles the conjecture.
4.1 As a generalization of the above problem, we consider the simple set given by
where is the
depend on whether or .
We establish the following:
When the points all lie in the unit disk , the corresponding set for ,will be effective in for , where
Given , the points lying in can the chosen so that the correspondence set of (70) with will not be effective in for .
the corresponding set of (70) will be effective in for , where is the least root of the equation.
Theorem 4.1.2 shows that the result in Theorem 4.1.1 is best possible. Also, the restriction (71) on the sequence when , is shown to be justified in the sense that if the restriction is not satisfied, the corresponding set may be of infinite order and not effective.
Proof of Theorem 4.1.1 is similar to the first part of Theorem 3.1.1.
Let admits the representation
then multiplying the matrix of coefficients of the set with the inverse matrix we obtain
the above relation yields
which, on reduction, yields
Identify (79) is the bridge relation between the set and the Goncarov polynomials mentioned earlier.
Suppose and assume that
from which it follows that the set is effective in for and the theorem is established.
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:
For and , the following identity holds.
where denote the
Proof of Lemma
The proof is by induction.
For , we have from the construction formulae (33),
Hence, operating on this equality, we have that
Hence, the derivative operating on this equation gives, in view of (83),
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.
Also, from the definition (46) of , the points lying in can be chosen so that
With this choice of the integer
With the notation
we can choose a sequence of points on such that
We first observe, from a repeated application of (30), that an analytic function regular at the origin, can be expanded in a certain disk in a series of the form
Hence, by Cauchy’s inequality, we have
Applying the usual induction process, we obtain, from (87) for the case , that
Hence the identity (83) yields
Therefore, we obtain
Pick the number , with , such that
and the inequality (88) is satisfied for . The similarity with the proof of Lemma 3.2.1 shows that the proof of this Lemma can be completed in the same manner as that for ealier Lemma.
We can now prove Theorem 5.1.4.
We note that the points lying in which define the required set of polynomials (70), are chosen as follows:
Since , we have that
and will not be effective in for . This completes the proof.
5.2 Proof of Theorem 4.1.3
For this, we put
We claim that, in this case, the corresponding set will be of infinite order and hence the effectiveness properties of the set will be violated.
Now, in the identity (37), we let
so that (97) yields
Hence, if we put
then (97) implies that
Since , the function is entire of zero order and hence it will have zeros in the finite part of the plane.
To prove Theorem 4.1.3 we first note, from (72), that if we put
We then multiply the matrix with the inverseto get
Thus, the inequality
is true for .
To prove (107), in general, we observe that, since ,
Hence, it follows by induction, that the inequality (107) is true for Noting that
Appealing to a result of Al-Salam (; formula 2.5), we deduce that
Hence, when we should have
from which it follows that the set is effective in and Theorem 4.1.3 is proved.
6. Other related results
6.1 The Goncarov polynomials belonging to the
Dq–derivative operator have other properties of interest and worth recording. Hence, we present, in this section, more results regarding the Goncarov polynomials as defined in (84) which belong to the derivative operator Dqand whose points lie in the unit disk for which or
When , the result of Buckoltz and Frank (; Theorem 1.2) applied to the derivative operator
The set of Gancarov polynomials belonging to the
Theorem 1.5 of Buckholtz and Frank  shows that the result of Theorem 6.1.1 above is best possible. They also showed that when the Goncarov polynomials fail to be effective and also, that if , no favorable effectiveness results will occur, thus justifying the restriction on the points .
We also state and prove the following theorem.
Suppose that and that the points satisfy the restriction (111). Then the Goncarov set belonging to the
To prove this theorem we put, as in the proof of Theorem (72),
so that and we differentiate between the Goncarov polynomials belonging to the operations
for these respective polynomials. Thus, the constructive formulae (33) for these polynomials will be
where and are the respective and analogues of the factorial . With this notation, the following Lemma is to be proved.
The following identity is true for and
We finish note, from the definition of the analogue and , that
the identity (114) is satisfied for .
and hence the Lemma is established.
Proof of Theorem 6.1.2.
so that the restriction (111) implies that
Also, by actual calculation we have that
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.
and and are the respective Cannon sums of the sets and , it follows that
To show that the result of the Theorem is best possible we appeal to Theorem 1.5 of Buckholtz and Frank  to deduce that the set may not be effective in for .
In view of the relation (122), we may conclude that the set will not be effective in for and Theorem 6.1.2 is fully established.
6.2 The case of Goncarov polynomials with
Nassif  studied the convergence properties of the class of Goncarov polynomials generated through the
6.3 Quasipower basis (QP-basis)
Kazmin  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. (; Corollaries 3, 4).
Of interest is his results that the system in (123), for arbitrary sequence of complex numbers with , forms a QP- basis in the space , for and in the space , for where W = 0.7377 is the Whittaker constant. This value of W = 0.7377 is attributed to Varga . He also added that Corollaries 3 and 4 contain known results in [5, 9, 15, 22, 23].
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.
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.
No conflict of interest
The author declares no conflict of interest.