## 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 _{q}-derivative operator when

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 _{q}-derivative operator. For a comprehensive and easy reading, background results are provided in the Preliminaries of sections 2.1–2.5.

## 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 * n* then the set is called a simple set and is necessarily a basic set.

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.

** Theorem 2.2.1 (Cannon [**6

]).

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 * q* be a positive number different from 1. The

*–analogue of the positive integer*q

*is given by*n

Also, the * q*-analogue of

and the * q*-analogue of the binomial coefficient

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 * n* in

_{,}then we have from (27), (29) and (42), that

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:

** Theorem 3.1.1 ([Nassif and Adepoju [**15

], Theorem B)

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 * W* is the Whittaker constant. It is shown also that the result in this theorem is bes t possible.

Indeed, applying the result of Buckholtz ([10], formula 2), the following theorem which resolved the conjecture of Nassif ([8], p.138), is established.

** Theorem 3.1.2 ([**15

**)**], Theorem B

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 * u* on the points

_{n}

Comparing this relation with the identify

of Levinson [12], we infer that

Differentiating (50)* k* times,

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, * m* such that

such that

Moreover, from (20), the definition (15) ensures the existence of the points

Having fixed the integer * m* and the sequence

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 * q*-analogue of the binomial coefficient

*-analogue of the set*q

*-derivative operator. Our results show that effectiveness properties of the set.*Dq

We establish the following:

** Theorem 4.1.1 ([**17

], Theorem 1.1)

When the points * h* is as in (47).

** Theorem 4.1.2. ([**17

], Theorem 3.1)

Given

** Theorem 4.1.3 ([**17

], Theorem 1.2)

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 * Dq*-derivative with respect to

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 * m* for which

such that

Also, from the definition (46) of

With this choice of the integer * m* and the points

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 * D* is operating with respect to

_{q}

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 * t* be such that

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 * q*–analogue of

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 * Dq–*derivative operator have other properties of interest and worth recording. Hence, we present, in this section, more results regarding the Goncarov polynomials

*and whose points*Dq

When * Dq* leads, in the language of basic sets, to the following theorem:

** Theorem 6.1 ([**19

], Theorem 1).

The set of Gancarov polynomials * Dq* operator, with

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.

** Theorem 6.2 ([**19

], Theorem 2).

Suppose that * Dq*–derivative operator, will be effective in

To prove this theorem we put, as in the proof of Theorem (72),

so that * Dq* and

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 ^{th} derivative described in (33) where now, * a* and

*are any complex numbers. By considering possible variations of t and q, it was shown that except for the cases*t

### 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.

## References

- 1.
Nassif M. Zeros of simple sets of polynomials. Duke Math. J. 1952; 19 (1): 107-113. - 2.
Whittaker JM. Interpolatory Function Theory. Cambridge; 1945 - 3.
Buckholtz JD, Frank JL. Whittaker Constants. Proc. Lond. Math. Soc. 1971; 3(23): 348-371. - 4.
Cannon TW. On the convergence of series of polynomials. Proc. Lond. Math. Soc. 1937; 2(43): 248-364. - 5.
Whittaker JM. Sur les series des polynomes quelcuonques, Gauthier- Villars, Paris. 1949. - 6.
Cannon TW. On the convergence of integral functions by general basic series. Math. Zeit. 1949; 45:185-205 - 7.
Boas Jr. R P. Basic sets of polynomials I. Duke Math. J. 1948; 15: 717-724. - 8.
Nassif M. Products and Zeros of Basic sets. Univ. Assiut Monograph Series. 1958; 50 (2); p.138. - 9.
Nassif M. On the mode of increase of simple sets of polynomials of given zeros. Proc. Math. Phys. Soc. Egypt. 1952; 4: 29-36. - 10.
Buckholtz JD. The Whittaker constant and successive derivatives of entire functions. J. Approx. Theory, 1970; 3: 194-212. - 11.
Goncarov W. Recherches Sur. Les derives successive de fonctions analytiques’ Anall. Scient. de L’ EcoleNormale Superieures. 1930; 47:1-78. - 12.
Levinson N. The Gontcharoff polynomials. Duke Math. J. 1944; 11: 729-733. - 13.
Macintyre S S. On the Zeros of successive derivatives of integral functions. Trans. Amer. Math. Soc. 1949; 67: 241-251. - 14.
Nassif M. On a class of Generalized Goncarov polynomials. J. Approx. Theory. 18: 329- 345. - 15.
Nassif M, Adepoju JA. Zeros of polynomials belonging to simple sets. Natn. Acad. Sci. Lett. 1979; 1(16): 223-224. - 16.
Marden M. The Geometry of the Zeros of a polynomial in a complex variable. Amer. Math. Soc. Math. Surv. 3; 1958 :NewYork. - 17.
Nassif M, Adepoju J A. A polynomial set Related to Goncarov polynomials. Indian J. Pure Appl. Math 1980; 1 (12): 1665-1672. - 18.
AL- Salam, W. A., q- analogue of Cauchy formula. Proc. Amer. Math. Soc. 1966; 17: 616-621. - 19.
Adepoju JA. On the convergence of a class of Generalized Goncarov polynomials. Soochow J. Math. 1995; 21 (1): 71-80. - 20.
Kaz’min Yu A. Perturbed Appell polynomials and systems of functions Associated with them. Soviet. Math. Dokl. 1985; 31(3): 452-455. - 21.
Varga RS. Topics in polynomials and rational interpolation and Approximation. Sem. Math. Sup. 1982. Presses Univ. Montreal, Montreal. - 22.
Eweida, M; A note on Abel’s polynomials. Proc. Math. Phys. Soc. Egypt. 1959; 5 (3): 63-66 - 23.
Kaz’min Yu. Vestnic Moscow Univ. Ser. I. Math. Mekh. 1978; 2: 88-90, English Transl. in Moscow Univ. Math. Bull. 1978; 33: 1978. - 24.
Eweida MA. Proc. Math. Phys. Soc. Egypt. 1959; 22: 83-88. - 25.
Adeniran A, Yan C. Goncarov polynomials in partition lattice and and exponentials families. Advanced in Appl. Math. Elsevier. May 2021. - 26.
Lorentz R, et al. Generalized Goncarov polynomials. In Butler J. Cooper and G. Hierlbert (eds). Cambridge Univ. P. Press . 2018. - 27.
Wu R. Abel- Goncarov Type multiquadric Quasi- interpolation operators with higher approximation order. Hindawi J. Math. July 2021. - 28.
Anjorin A. Akinbode S. Chebychev polynomials of the first kind and the Whittaker Constant. Global J. Sci. Frontier Research: For Math. and Decision Sci. 2015 (2):Version 10 - 29.
Yacubovich S. On some properties of Abel- Goncerov polynomials and the Casas- Alvero problem. Integral Transforms and special functions. 2016; 27 (8).