Open access peer-reviewed chapter - ONLINE FIRST

Effectiveness of Basic Sets of Goncarov and Related Polynomials

By Jerome A. Adepoju

Submitted: April 28th 2021Reviewed: July 12th 2021Published: September 21st 2021

DOI: 10.5772/intechopen.99411

Downloaded: 19


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
  • 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 U. It is a complement of a theorem of Nassif [1] which resolved his conjecture on the value of the Whittaker constant [2]. 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 an0are given complex numbers with knthe q-analogue of the binomial coefficient kn. From the results reported, it is shown that the location of the points ak0that leads to favorable effectiveness results depends on whether q<1orq>1. 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 [3] on the generalized Goncarov polynomials Qnzz0z1zn1belonging to the Dq-derivative operator whenq>1, leads to the result that, when the points zk0lie in the unit disk U, the resulting polynomials fail to be effective.

Consequently, we provide some results on the polynomials Qnzz0z1zn1when


with the obtained results justifying the restriction (2) on the points zk0.

Finally, we provide other relevant and related results on the properties of the generalized Goncarov polynomials Qnzz0z1zn1belonging to the Dq-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 pnzof polynomials is said to be basic if any polynomial and, in particular, the polynomials 1,z,z2,,zn,, can be represented uniquely by a finite linear combination of the form.


The polynomials pnzare linearly independent.

In the representation (3), let fz=n=0anznbe an analytic function about the origin. Substituting (3) into fz, we have


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 fzand the correspondence is written as


The coefficients Πkfis the basic coefficients of fzrelative to the basic set pkzand is a linear functional in the space of functionsfz.

If pnzis of degree nthen the set is called a simple set and is necessarily a basic set.

The basic series (4) is said to represent fzin a disk zrwhere fzanalytic, if the series is converges uniformly to fzin zror that the basic set pnzrepresents fzinzr.

When the basic set pnzrepresents in zrevery function analytic inzR,Rr, then the basic set is said to be effective in zrfor the class H¯Rof functions analytic in zR.

When R=r, the basic set represents, in zr, every function which is analytic there and we say that the basic set is effective in zr.

To obtain conditions for effectiveness, we form the Cannon sum




From (3), we have that wnrrn,

so that, if we write


The function λris called the Cannon function of the set pnzin zr.

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 pnzto be effective, in zr, is


2.2 Mode of increase of basic sets

The mode of increase of a basic set pnzis determined by the order and type of the set. If pnzis a Cannon set, its order is defined, Whittaker [2], by


where wnris given by (5). The type γis defined, when 0<w<, by


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 ptype qis


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 Gnzz0.zn1associated with the sequence zn0of points in the plane are defined through the relations, Buckholtz ([10], p. 194),


These polynomials generate any function fzanalytic at the origin through the Goncarov series


which represents fzin a disk zr, if it uniformly converges to fzin zr.

In this case, if fkzk=0,k0, the Goncarov series (14) vanishes and f0.

A consideration of gz=sinπ41z, for which gn1n=0andn=0gn1nGnz11..=0cf. Nassif [8], shows that the Goncarov series does not always represent the associated function and hence certain restrictions have to be imposed on the points zk0and on the growth of the function fz.

Concerning the case where the points zk0lie in the unit disk U, the Whittaker constant W (cf. Whittaker, Buckholtz, [2, 10]), is defined as the supremum of the number c with the following property:

If fzis an entire function of exponential type less than c and if each of f,f',f",..has a zero in U then fz0.

Buckholtz [10] obtained an exact determination of the constant W. In fact, if we write


where the maximum is taken over all sequences zk0n1whose terms lie in U, Buckholtz ([10], Lemma 3) proved that limnHn1nexists and is equal to sup1n<Hn1n.

Moreover, if we put


Buckholtz ([10], formula 2) further showed that


Employing an equivalent definition of the polynomials Gnzz0zn1as originally given by Goncarov [11] in the form


and differentiating with respect to z, we can obtain




then (18) yields, among other results,




Applying (21) and (22) to (19) we obtain


for 1kn1, where the differentiation is with respect to the first argument.

Expanding Fnz0zn1in powers of z0, in the form


we arrive through (22) and (23) to the formulae of Levinson [12],


Also, differentiating (18) with respect to zk, we obtain with Macintyre ([13], p. 243),


for 0kn1.

2.5 The q-analogues and Dqderivatives

Let qbe a positive number different from 1. The q–analogue of the positive integer nis given by


Also, the q-analogue of n!is


and the q-analogue of the binomial coefficient knis


Moreover, the Dq– derivative operator, corresponding to the number q is defined as follows: Iffzis any function of z, then


so that when fz=zn, then according to (26), we have Dqzn=nzn1and if fz=n=0anzn1is any function analytic at the origin then


In [3] we have a generalization of the Goncarov polynomials as in (13) belonging to the operator D such that for fz=n=0anzn,


associated with the sequence zk0, where en=d1d2dn1,e0=1and dn1is a non-decreasing sequence of numbers to obtain


When dn=n, the relations (32) reduce to (6), hence the polynomials pnzreduce to the proper Goncarov polynomials Gnzz0zn1. Comparing (30) and (32), Nassif [14] investigated the class of generalized Goncarov polynomialsQnzz0zn1belonging to the Dq- derivative operator when dn=nand en=1n!given by,


and the Goncarov series associated with the function fz=n=0anznis




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 Dq, we obtain


Expressing Qnzz0zn1as a polynomial of degree nin z, 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 1kn1, where the differentiation is with respect to the first argument. Expanding Rnz0z1zn1in powers of z0, then (36) and (44) imply that


Finally, if we put


where the maximum is taken over all sequences zk0n1and the terms lie in the unit disk U, then Buckholtz and Frank ([3], Corollary 5.2), proved that


Also, in view of the formulae (33), we can verify that, when q<1,


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 1W, where Wis 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 ε, a simple set pnzof polynomials, whose zeros all lie in Ucan be constructed such that the increase of the set is not less than order 1 type H–ε.

For completeness, we give the proof of Theorem 3.1.1 as a revised version of Theorem A.

Proof of Theorem 3.1.1 (Proof of alternative form of Theorem A)

Let bn1be a sequence of points lying in the unit disk and consider the set qnzof polynomials given by


Suppose that znadmits the representation


Then multiplying the matrix of coefficients nkbnnkwith its inverse w˜n,k, we obtain




then the above relation will give


And to show the dependence of unon the points bn, this relation can be rewritten as


Comparing this relation with the identify


of Levinson [12], we infer that


Differentiating (50)ktimes, k=1,2,,n1, we obtain that


Hence, a combination of (15), (16), (20), (51)-(53) leads to the inequality.


Observing that Mqkr1+rkfor any value of r0, then the Cannon sum of the set qnzfor z=rwill, in view of (54), be


It follows from (17) that the set qnzis of increase not exceeding order 1 type 1W. The proof is now completed by applying the results of Walsh and Lucas, cf. Marden ([16], pp. 15,46), with (54) and following exactly the same lines of argument as in ([1], pp.109–110), to arrive at the inequality.


Since pnz1+rnin zr, it follows that the set pnzis of increase not exceeding order 1 type H = 1W.

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, ε<H1.(In fact, according to Macintyre ([13]; p. 241), we have H > 10.7378). Hence it follows from (16) that corresponding to ε, there exists an integer msuch that


such that


Moreover, from (20), the definition (15) ensures the existence of the points ak1mlying in z1such that


Having fixed the integer mand the sequence ak1m, the following Lemma is to be first established.

Lemma 3.2.1 ([15], Lemma 3.2).

For any integer j1, write


Then, the complex numbers ξk1can be chosen so that





The proof is by induction.

When j=1, we have from (59) that


Then the value ξ1will be chosen so that


Applying the identify (25) of Macintyre to

F2m+1ama1z1ama1,we obtain


so that (20) and (58) imply that


where the prime denotes differentiation with respect to z1.

Hence, in view of (62), Cauchy’s inequality yields


and the inequality (61) is satisfied for j=1. Suppose then that, for some value j=k,the complex numbers ξ1,ξ2,,ξkhave been chosen satisfying (60) and (61).

The numbers ξk+1will be fixed so that


Proceeding in a similar manner as for the Case j=1and applying the identity (25) of Macintyre with (58), (59) and (61),we can obtain the inequality.


where the prime denotes differentiation with respect to zk+1..

Applying Cauchy’s inequality to the polynomial Fk+1ξ1ξ2ξkzk+1,we can deduce, using (63) and (64), that


Hence, by induction, the inequality (61) of the Lemma is established.

We now prove theorem 3.1.2.

The required simple set pnzof polynomials is constructed as follows:


where the points ak1mare chosen to satisfy (63) and the numbers ξk1are fixed as in the Lemma.

It follows that the zeros of the polynomials pnzall lie in the unit disk U.

Also, if znadmits the unique linear representation.


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 pnzfor z=r, is wnr>πn,0.

Hence, combining (57), (61), (67) and (68) yields


vIt follows from this inequality and Theorem 3.1 that the order of the set Pnzis exactly 1 and since H2>1, the type of the set will be


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 pnzngiven by


where nkis the q-analogue of the binomial coefficient nkand ak1is a sequence of given complex numbers. The set pnzis in fact, the q-analogue of the set qnzin (49). This study is motivated by the fact that this set is related to the generalized Goncarov polynomials belonging to the Dq-derivative operator. Our results show that effectiveness properties of the set.

pnzdepend on whether q<1or q>1.

We establish the following:

Theorem 4.1.1 ([17], Theorem 1.1)

When the points ak1all lie in the unit disk U, the corresponding set pnzfor q<1,will be effective in zrfor rh1q, where his as in (47).

Theorem 4.1.2. ([17], Theorem 3.1)

Given >0, the points ak1lying in z1can the chosen so that the correspondence set pnzof (70) with q<1will not be effective in z<rfor r<h1q..

Theorem 4.1.3 ([17], Theorem 1.2)

When q>1and


the corresponding set pnzof (70) will be effective in zrfor r>q1, where 1γis 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 ak1when q>1, is shown to be justified in the sense that if the restriction is not satisfied, the corresponding set pnzmay 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 znadmits the representation


then multiplying the matrix of coefficients nkannkof the set pnzwith the inverse matrix πn,kwe obtain




the above relation yields!vnka1ank`E75

Comparing the formulae (45) and (75) we infer that


Moreover, operating Dqon the polynomials pnz, we can deduce, from (28) and (29), that


Hence, when the operator Dqacts on the representation (73), then (77) leads to the equality


which, on reduction, yields


Applying (74), (76) and (78), we obtain


Identify (79) is the bridge relation between the set pnzand the Goncarov polynomials mentioned earlier.

Suppose q<1and assume that


Since h>1as in (47), and restricting the points ak1to lie in the unit disk Uas in the theorem, it follows from (28) and (80) that


The Cannon sum of the set pnzfor z=r,is evaluated from (46), (47), (79), (80) and (81) to obtain


from which it follows that the set pnzis effective in zrfor rh1qand the theorem is established.


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 n1and k0, the following identity holds.


where Dq,zkdenote the Dq-derivative with respect to zk.

Proof of Lemma

The proof is by induction.

For n=1,k0, we have from the construction formulae (33),


Hence, operating Dq,zkon this equality, we have that


so that the identity (83) is satisfied for n=1,k0. Suppose that (83) is satisfied forn=1,2,,m;k0. The formulae (33) can be written for k+m+1in the form,


Hence, the derivative Dq,zkoperating on this equation gives, in view of (83),


Or equivalently,


Hence, formulae (33) imply that


and the relation (83) is also valid for n=m+1

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 h>1as in (39), the >0of Theorem 4.1.2 can always be picked less than h1. Also, from (39) it follows that, corresponding to the number ∈, there exists an integer mfor which


such that


Also, from the definition (46) of hm, the points αi1mlying in Ucan be chosen so that


With this choice of the integer mand the points αi1m,the Lemma to be established is the following:

Lemma 5.1.2.

With the notation


we can choose a sequence ξj1mof points on z=1such that



We first observe, from a repeated application of (30), that an analytic function fzregular at the origin, can be expanded in a certain disk z1in a series of the form


Hence, by Cauchy’s inequality, we have


Applying the usual induction process, we obtain, from (87) for the case j=1, that


Hence the identity (83) yields


Therefore, we obtain


where the Dqis operating with respect to z1.

Pick the number ξ1, with ξ1=1, such that


hence, a combination of (86), (89) and (90) yields


and the inequality (88) is satisfied for j=1. 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 ak1lying in Uwhich define the required set pnzof polynomials (70), are chosen as follows:


where the points αi1mare fixed as in (86) and the sequence ξj0of points is determined as in Lemma 5.1.2; and the integer mis chosen as in (84) and (85).

If znadmits the representation (86), then applying (79), (87) and (91) we have that


so that, for the Cannon sum of the set pnzfor z=r,we obtain, from (85), (88) and (92),


Since q<1, we have that


Hence, (93) and (94) yield, for the Cannon function,


Noting that h2>1, we conclude, from (84), as in the proof of Theorem (50), that


and pnzwill not be effective in zrfor r<h1q. This completes the proof.

5.2 Proof of Theorem 4.1.3

Let pnzbe the basic set in (70) with q>1. We first justify the statement that if the restriction (71) is not satisfied the corresponding set pnzmay be of infinite order.

For this, we put


and let tbe such that


We claim that, in this case, the corresponding set pnzwill be of infinite order and hence the effectiveness properties of the set will be violated.

Now, in the identity (37), we let


to obtain




so that (97) yields


Hence, if we put


then (97) implies that




Since t=β<q, the function ϕztis entire of zero order and hence it will have zeros in the finite part of the plane.



then from (100) and (101), we have limsupnun1n=1σ>0.

Thus, for the Cannon sum of the set pnz, we have, from (79), (96) and (98), that


Since q>1and β>1qthen, in view of (102), we deduce from (103) that the set pnzis of infinite order; as claimed.

To prove Theorem 4.1.3 we first note, from (72), that if we put




We then multiply the matrix nkannkwith the inverseπn,kto get


Now, imposing the restriction (71) on the points ak1, we have from (105) and (106) that


Thus, the inequality


is true for m=k,k+1.

To prove (107), in general, we observe that, since q>1,


Assume that (107) is satisfied for m=k,k+1,,n1;then a combination of (71), (72), (104), (106), (107) and (108) leads to the inequality.


Hence, it follows by induction, that the inequality (107) is true for mk.Noting that


where kjis the q–analogue of kj, q1=1q<1, we then deduce from (70) and (71), that


Appealing to a result of Al-Salam ([18]; formula 2.5), we deduce that


The Cannon sum of the set pnzfor z=rcan be evaluated from (107) and (109) in the form


Hence, when rcwe should have


from which it follows that the set pnzis effective in zrand 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 Qnzz0zn1as defined in (84) which belong to the derivative operator Dqand whose points zn0lie in the unit disk Ufor which q<1or q>1

When q<1, the result of Buckoltz and Frank ([3]; Theorem 1.2) applied to the derivative operator Dqleads, in the language of basic sets, to the following theorem:

Theorem 6.1.1 ([19], Theorem 1).

The set of Gancarov polynomials Qnzz0zn1belonging to the Dqoperator, with q<1and associated with the sequence of points zn0in U, is effective in zrfor rh1q.

Theorem 1.5 of Buckholtz and Frank [3] shows that the result of Theorem 6.1.1 above is best possible. They also showed that when q>1the Goncarov polynomials fail to be effective and also, that if zqn, no favorable effectiveness results will occur, thus justifying the restriction zqnon the points zn0.

We also state and prove the following theorem.

Theorem 6.1.2 ([19], Theorem 2).

Suppose that q>1and that the points zn0satisfy the restriction (111). Then the Goncarov set Qnzz0zn1belonging to the Dq–derivative operator, will be effective in zrfor rhqq1and this result is best possible.

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


so that q1<1and we differentiate between the Goncarov polynomials belonging to the operations Dqand Dq1by adopting the notation.


for these respective polynomials. Thus, the constructive formulae (33) for these polynomials will be




where k!and k!are the respective qand q1analogues of the factorial k. With this notation, the following Lemma is to be proved.

Lemma 6.1.1.

The following identity is true for n1and q>1:



We finish note, from the definition of the analogue k!and k!, that




Hence, applying the relations (37) and (112) toQNqnzqnz0qzn, we get


Hence, the relations (115) and (116) can be introduced to yield


Now, since


the identity (114) is satisfied for n=1.

Moreover, if (114) is valid for k=1,2,,n1, the relations (113) and (117) will give


and hence the Lemma is established.

Proof of Theorem 6.1.2.



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 Ωnrand wnrare the respective Cannon sums of the sets Qnzz0zn1and Pnza0an1, it follows that


Since the points ak0lie in U, from (119), then applying Theorem 6.1.1 we deduce from (122) that the set Qnzz0znwill be effective in zrfor rh1q=qhq1as to be proved.

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 Pnza0an1may not be effective in zrfor r<qhq1.

In view of the relation (122), we may conclude that the set Qnzz0znwill not be effective in zrfor r<qhq1and Theorem 6.1.2 is fully established.

6.2 The case of Goncarov polynomials with Zk=atk,k0

Nassif [14] studied the convergence properties of the class of Goncarov polynomials Qnzz0zn1generated through the qth derivative described in (33) where now, zk=atk,k0and aand tare any complex numbers. By considering possible variations of t and q, it was shown that except for the cases t1,q<1and t>1q;q>1, all other cases lead to the effectiveness of the set Qnzaatatn1in finite circles ([14]; Theorems 1.1, 1.2, 1.3, 3.2, 3.3).

6.3 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 a0of complex numbers with an1, forms a QP- basis in the space 1σ, for 0<σ<Wand in the space 1σ, for 0<σW,where W = 0.7377 is the Whittaker constant. This value of W = 0.7377 is attributed to Varga [21]. He also added that Corollaries 3 and 4 contain known results in [5, 9, 15, 22, 23].


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.



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.


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Jerome A. Adepoju (September 21st 2021). Effectiveness of Basic Sets of Goncarov and Related Polynomials [Online First], IntechOpen, DOI: 10.5772/intechopen.99411. Available from:

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