Open access peer-reviewed chapter

Effectiveness of Basic Sets of Goncarov and Related Polynomials

Written By

Jerome A. Adepoju

Submitted: 03 July 2021 Reviewed: 12 July 2021 Published: 21 September 2021

DOI: 10.5772/intechopen.99411

From the Edited Volume

Recent Advances in Polynomials

Edited by Kamal Shah

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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 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

p0z=1;pnz=k=0nknannkzk;n1,E1

and the points an0 are given complex numbers with kn the q-analogue of the binomial coefficient kn. From the results reported, it is shown that the location of the points ak0 that leads to favorable effectiveness results depends on whether q<1 orq>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 Qnzz0z1zn1 belonging to the Dq-derivative operator whenq>1, leads to the result that, when the points zk0 lie in the unit disk U, the resulting polynomials fail to be effective.

Consequently, we provide some results on the polynomials Qnzz0z1zn1 when

zkqk;k0,E2

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 Qnzz0z1zn1 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.

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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 pnz of 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.

zn=k=0πn,kpkz;n0.E3

The polynomials pnz are linearly independent.

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

fz=n=0anzn=n=0ank=0πn,kpkz.

Formally rearranging the terms, we obtain the series

k=0pkzn=0anπn,k.

We write

kf=n=0anπn,k;k0.

Hence, we obtain the series

k=0kfpkz,

which is called the basic series associated with the function fz and the correspondence is written as

fzk=0kfpkz.E4

The coefficients Πkf is the basic coefficients of fz relative to the basic set pkz and is a linear functional in the space of functions fz.

If pnz is of degree n then the set is called a simple set and is necessarily a basic set.

The basic series (4) is said to represent fz in a disk zr where fz analytic, if the series is converges uniformly to fz in zr or that the basic set pnz represents fz in zr.

When the basic set pnz represents in zr every function analytic in zR,Rr, then the basic set is said to be effective in zr for the class H¯R of 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

wnr=k=0πn,kMkr,E5

where

Mkr=maxz=rpkz.E6

From (3), we have that wnrrn,

so that, if we write

λr=limnsupwnr1n,E7
λrrn.E8

The function λr is called the Cannon function of the set pnz in 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 pnz to be effective, in zr, is

λr=r.E9

2.2 Mode of increase of basic sets

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

w=limrlimsupnlogwnrnlogn.E10

where wnr is given by (5). The type γ is defined, when 0<w<, by

γ=limrewlimnsupwnr1nnw1w.E11

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 p type q is

limnsupepqn1pwnr1n1forallr>0.E12

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.zn1 associated with the sequence zn0 of points in the plane are defined through the relations, Buckholtz ([10], p. 194),

G0z=1,
znn!=k=0nznknk!Gkzz0zk1;n1.E13

These polynomials generate any function fz analytic at the origin through the Goncarov series

fzk=0fkzkGkzz0zk1,E14

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

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

A consideration of gz=sinπ41z, for which gn1n=0 and n=0gn1nGnz11..=0 cf. 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 zk0 and on the growth of the function fz.

Concerning the case where the points zk0 lie 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 fz is 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

Hn=maxGk0z0zn1,E15

where the maximum is taken over all sequences zk0n1 whose terms lie in U, Buckholtz ([10], Lemma 3) proved that limnHn1n exists and is equal to sup1n<Hn1n.

Moreover, if we put

limnHn1n=H=sup1n<Hn1n,E16

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

W=1H.E17

Employing an equivalent definition of the polynomials Gnzz0zn1 as originally given by Goncarov [11] in the form

Gnzz0zn1=z0zds1z1s1ds2,,zn1sn1dsn;n1,E18

and differentiating with respect to z, we can obtain

Gnkzz0zn1=Gnkzzkzn1;1kn1.E19

Writing

Gn0z0zn1=Fnz0zn1n1,E20

then (18) yields, among other results,

Gnzz0zn1=Fnzz0zn1Fnzz1zn1;n1,E21

and

Fn0z1zn1=0;n1.E22

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

Fkkz0z1zn1=Fnkz0zn1E23

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

Expanding Fnz0zn1 in powers of z0, in the form

Fnz0zn1=k=0nz0kk!Fnk0z1zn1,

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

Fnz0zn1=k=1nz0kk!Fnkzkzn1.E24

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

zkGnkzz0zn1=Gkzz0zk1Gnk1zkzk1zn1E25

for 0kn1.

2.5 The q-analogues and Dq derivatives

Let q be a positive number different from 1. The q–analogue of the positive integer n is given by

n=qn1q1.E26

Also, the q-analogue of n! is

n!=nn121;n1;0!=1,E27

and the q-analogue of the binomial coefficient kn is

kn=n!k!nk!;0kn.E28

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

Dqfz=fqzfzzq1,E29

so that when fz=zn, then according to (26), we have Dqzn=nzn1 and if fz=n=0anzn1 is any function analytic at the origin then

Dqfz=n=1nanzn1.E30

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

Dfz=n=1dnanzn1E31

associated with the sequence zk0, where en=d1d2dn1,e0=1 and dn1 is a non-decreasing sequence of numbers to obtain

p0z=1,enzn=k=0nenkzknkPkzz0.zk1;n1.E32

When dn=n, the relations (32) reduce to (6), hence the polynomials pnz reduce to the proper Goncarov polynomials Gnzz0zn1. Comparing (30) and (32), Nassif [14] investigated the class of generalized Goncarov polynomials Qnzz0zn1 belonging to the Dq- derivative operator when dn=n and en=1n! given by,

Q0z=1znn!=k=0nzknknk!Qkzz0.zk1;n1,E33

and the Goncarov series associated with the function fz=n=0anzn is

fzk=0DqkfkzkQkzz0zk1.E34

Writing

Rnz0zn1=Qn0z0zn1E35

so that

Rn0z1zn1=0,n1E36

then we have from, (32) that

Rnz0zn1=k=0n1znknk!Rkz0.zk1.E37

Also, Nassif ([14], Lemma 4.1), proved that

Qnzz0zn1=Rnz0zn1Rnzz1zn1.E38

We can verify, with Buckholtz ([10], Lemma 1), from the formulae (33), the following:

Qnλzλz0λzn1=λnQnzz0zn1;n1.E39
Qnz0z0zn1=0;n1.E40
DqQnzz0zn1=Qn1zz1zn1;n1.E41

And hence, by repeated application of Dq, we obtain

DqkQnzz0zn1=Qnkzzkzn1;1kn1.E42

Expressing Qnzz0zn1 as a polynomial of degree n in z, then we have from (27), (29) and (42), that

Qnzz0zn1=k=0nzkn!Rnkzk.zn1.E43

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

DqkRn0z1zn1=Rnkzk.zn1,E44

for 1kn1, where the differentiation is with respect to the first argument. Expanding Rnz0z1zn1 in powers of z0, then (36) and (44) imply that

Rnz0z1zn1=k=1nz0kk!Rnkzk.zn1.E45

Finally, if we put

hn=maxRnz0z1zn1,E46

where the maximum is taken over all sequences zk0n1 and the terms lie in the unit disk U, then Buckholtz and Frank ([3], Corollary 5.2), proved that

limnhn1n=h=sup1n<hn1n.E47

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

hh212=1+1212>3212>1.E48
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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 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 ε, a simple set pnz of polynomials, whose zeros all lie in U can 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 bn1 be a sequence of points lying in the unit disk and consider the set qnz of polynomials given by

q0z=1;qnz=z+bn1n;n1.E49

Suppose that zn admits the representation

zn=k=0nw˜n,kqnkz.E50

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

w˜n,0=k=1nnkbnkw˜nk,0;n1.

Write.

un=w˜n,0n!;n0,E51

then the above relation will give

un=k=1nbnkk!unk;n1.

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

unb1b2bn=k=1nbnkk!unkb1b2bnk.

Comparing this relation with the identify

Fnz0z1zn1=k=1nz0kk!Fnzkzn1,

of Levinson [12], we infer that

unb1b2bn=Fnbnbn1b1.E52

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

w˜n,k=nkw˜nk,0bk+1bk+2bn.E53

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

w˜n,kn!k!Hnk;0kn.E54

Observing that Mqkr1+rk for any value of r0, then the Cannon sum of the set qnz for z=r will, in view of (54), be

wnr=k=0nw˜n,kΜqkrn!Ηnexp1+rΗ.

It follows from (17) that the set qnz is 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.

πn,kn!k!Hnk.E55

Since pnz1+rn in zr, it follows that the set pnz is 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 m such that

m>logΗ/log1+2ΗE56

such that

Hm1m>H2.E57

Moreover, from (20), the definition (15) ensures the existence of the points ak1m lying in z1 such that

Hm=Fmamam1a1.E58

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

Lemma 3.2.1 ([15], Lemma 3.2).

For any integer j1, write

fjz1ƶ2ƶj=Fj+1m+jamaiƶjamaiƶj1amaiƶ1ama1E59

Then, the complex numbers ξk1 can be chosen so that

ξk=1;k1,E60

and

fjξ1ξ2ξj=Hmj+1;j1.E61

Proof.

The proof is by induction.

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

f1z1=F2m+1ama1z1ama1.

Then the value ξ1 will be chosen so that

ξ1=1;f1ξ1=supz11f1z1.E62

Applying the identify (25) of Macintyre to

F2m+1ama1z1ama1, we obtain

ddz1f1z1=Fmama1Gmz1ama1,

so that (20) and (58) imply that

f1''0=Hm2,

where the prime denotes differentiation with respect to z1.

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

f1ξ1Hm2,

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

The numbers ξk+1 will be fixed so that

ξk+1=1Fk+1ξ1ξ2ξk+1=supzk+11Fk+1ξ1ξ2ξkzk+1.E63

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

fk+11ξ1ξ2ξk0Hmk+2,E64

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

Fk+1ξ1ξ2ξkzk+1Hmk+2.

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

We now prove theorem 3.1.2.

The required simple set pnz of polynomials is constructed as follows:

P0z=1,pjm+1z=z+ξjjm+1;j1,pjm+1+iz=z+ajjm+1+i;1im;j0,E65

where the points ak1m are chosen to satisfy (63) and the numbers ξk1 are fixed as in the Lemma.

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

Also, if zn admits the unique linear representation.

zn=k=0nπn,kpkz,E66

and if we write

un=πn,0n!;n0,E67

then from the relation (52), we deduce from (59) and (65), that

uj+1m+j=fjξ1ξ2ξj;j1.E68

Now, in view of (66), the Cannon sum of the set pnz for z=r, is wnr>πn,0.

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

wj+1m+jrj+1m+j!H2j+1m;j1.

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

γH2mm+1H2m=1m.E69

In view of the inequality (56), we deduce from (69) that

γ>H

and Theorem 3.1.2 is established.

This settles the conjecture.

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4. Generalization

4.1 As a generalization of the above problem, we consider the simple set pnzn given by

p0z=1;pnz=pnza=k=0nnkannkzk;n1,E70

where nk is the q-analogue of the binomial coefficient nk and ak1 is a sequence of given complex numbers. The set pnz is in fact, the q-analogue of the set qnz in (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.

pnz depend on whether q<1 or q>1.

We establish the following:

Theorem 4.1.1 ([17], Theorem 1.1)

When the points ak1 all lie in the unit disk U, the corresponding set pnz for q<1,will be effective in zr for rh1q, where h is as in (47).

Theorem 4.1.2. ([17], Theorem 3.1)

Given >0, the points ak1 lying in z1 can the chosen so that the correspondence set pnz of (70) with q<1 will not be effective in z<r for r<h1q..

Theorem 4.1.3 ([17], Theorem 1.2)

When q>1 and

akqk;q1E71

the corresponding set pnz of (70) will be effective in zr for r>q1, where 1γ is the least root of the equation.

n=0qn2xn=2.E72

Theorem 4.1.2 shows that the result in Theorem 4.1.1 is best possible. Also, the restriction (71) on the sequence ak1 when q>1, is shown to be justified in the sense that if the restriction is not satisfied, the corresponding set pnz may be of infinite order and not effective.

Proof.

Proof of Theorem 4.1.1 is similar to the first part of Theorem 3.1.1.

Let zn admits the representation

zn=k=0nπn,ka1a2anpkz,E73

then multiplying the matrix of coefficients nkannk of the set pnz with the inverse matrix πn,k we obtain

k=0nnkannkπk,0a1a2ak=0;n1.

Putting

v0=1,vk=vka1a2..ak=1k!πk,oa1ank`,E74

the above relation yields

vna1..an=k=1nakk!vnka1ank`E75

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

vka1..ak=Rkak..a1.E76

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

Dpkzakq=Kpk1zak;k1.E77

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

πn,ka1.an=nkπn1,k1a1an,

which, on reduction, yields

πn,ka1.an=nkπnk,0aK+1an;0kn.E78

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

πn,ka1.an=n!k!RnkanaK+1;0kn.E79

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

Suppose q<1 and assume that

rh1q.E80

Since h>1 as in (47), and restricting the points ak1 to lie in the unit disk U as in the theorem, it follows from (28) and (80) that

Μpkrk+1rk;k0.E81

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

wnr=k=0nπn,kΜpkrn+12rn,E82

from which it follows that the set pnz is effective in zr for rh1q and the theorem is established.

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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 n1 and k0, the following identity holds.

Dq,zkQk+nzz0zk+n1=Qkzzozk1Qn1zkzk+1zk+n1,E83

where Dq,zk denote 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),

QK+1zz0zk=zk+1k+1!j=0k1zjk+1jk+1j!Qjzz0zj1zkQkzz0zj1.

Hence, operating Dq,zk on this equality, we have that

Dq,zkQk+1zz0zk1=Qkzz0zk1,

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+1 in the form,

Qk+m+1zz0zk+m=zk+m+1k+m+1!j=0k1zjk+m+1jk+m+1j!Qjzz0,,zj1zm+1km+1!Qkzz0zk1j=1mzk+jm+1jm+1j!Qk+jzz0zk+j1.

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

Dq,zkQk+m+1zz0zk+m=zmm!Qkzz0zk1+j=1mzk+jm+1jm+1j!QKzz0zK1×Qj1zkzk+1zK+j1.

Or equivalently,

Dq,zkQk+m+1zz0zk+m=Qkzz0zk1×zkmm!j=0m1zk+j+1m+jmj!Qjzkzk+1zK+j.

Hence, formulae (33) imply that

Dq,zkQk+m+1zz0zk+m=Qkzz0zk1Qmzkzk+1zK+m,

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>1 as in (39), the >0 of 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 m for which

m>logh/log1+2h,E84

such that

hm1m>h2.E85

Also, from the definition (46) of hm, the points αi1m lying in U can be chosen so that

hm=Rmαm.α1.E86

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

Lemma 5.1.2.

With the notation

ujz1z2,,zj=Rj+1m+jαmα1zjαmα1zj1αmα1αmα1,E87

we can choose a sequence ξj1m of points on z=1 such that

ujξ1ξ2ξjmj+1;j1.E88

Proof.

We first observe, from a repeated application of (30), that an analytic function fz regular at the origin, can be expanded in a certain disk z1 in a series of the form

fz=n=0znn!Dqnf0.

Hence, by Cauchy’s inequality, we have

ΜfrrDqf0.E89

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

u1z1=R2m+1αmα1ziαmα1

Hence the identity (83) yields

Dqu1z1=Dq,z1Q2m+10αmα1ziαmα1=Rmαmα1Qmziαmα1.

Therefore, we obtain

Dqu10=Rm2αmα1,E90

where the Dq is operating with respect to z1.

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

u1ξ1=supu1z1:z1=1;

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

u1ξ1hm2,

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 ak1 lying in U which define the required set pnz of polynomials (70), are chosen as follows:

ajm+1=ξjajm+1+i=αi;1im;j0,E91

where the points αi1m are fixed as in (86) and the sequence ξj0 of points is determined as in Lemma 5.1.2; and the integer m is chosen as in (84) and (85).

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

πj+1m+j,o=j+1m+i!ujξ1ξ2ξj,j1,E92

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

wj+1m+jr>j+1m+j!h2j+1m;r>0..E93

Since q<1, we have that

limnn!1n=11q.E94

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

λr=limsupnwnr1nlimsupjwj+1m+jr1j+1m+j11qh2mm+1;r>0.

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

λrh1q;r>0,

and pnz will not be effective in zr for r<h1q. This completes the proof.

5.2 Proof of Theorem 4.1.3

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

For this, we put

ak=tk;k1,E95

and let t be such that

t=β,1q<β<qE96

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

Now, in the identity (37), we let

zk=ank=tnk;0kn1,

to obtain

k=0ntnkk!Rnktnkt=0;n>0.E97

Put

Rjtjt=t12jj1uj;j1,E98

so that (97) yields

k=0nt12kk+1k!unk=0;n>0.E99

Hence, if we put

uz=n=0unzn,E100

then (97) implies that

uz=1φz,E101

where

ϕzt=n=0t12nn1n!zn.

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

Let

σ=infzφz=0<,E102

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

wnr>πn,0=n!β12nn1un.E103

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

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

c=q1,E104

then

c>1q1.E105

We then multiply the matrix nkannk with the inverseπn,k to get

πn,k=j=kn1nkannjπj,k:n>k;πk,k=1.E106

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

πk+1,kc.

Thus, the inequality

πmkcmk;mk,E107

is true for m=k,k+1.

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

njqjnjqq1nj;1jn.E108

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.

πn,kcnkj=1qcq1jqj2=cnk.

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

kj=qjkjkjq>1,

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

Μpkrrkj=0kkjqj2rjrkj=0kkjq12jj1qrj;q>1.

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

Μpkrrkj=1k1+1qjr;k1,r>0.E109

The Cannon sum of the set pnz for z=r can be evaluated from (107) and (109) in the form

wnrj=1n1+1qjrk=0ncnkrk.E110

Hence, when rc we should have

wnrn+1j=1n1+1qjrrn,

from which it follows that the set pnz is effective in zr and Theorem 4.1.3 is proved.

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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 Qnzz0zn1 as defined in (84) which belong to the derivative operator Dq and whose points zn0 lie in the unit disk U for which q<1 or q>1.

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

Theorem 6.1 ([19], Theorem 1).

The set of Gancarov polynomials Qnzz0zn1 belonging to the Dq operator, with q<1 and associated with the sequence of points zn0 in U, is effective in zr for rh1q.

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 q>1 the Goncarov polynomials fail to be effective and also, that if zqn, no favorable effectiveness results will occur, thus justifying the restriction zqn on the points zn0.

We also state and prove the following theorem.

Theorem 6.2 ([19], Theorem 2).

Suppose that q>1 and that the points zn0 satisfy the restriction (111). Then the Goncarov set Qnzz0zn1 belonging to the Dq–derivative operator, will be effective in zr for rhqq1 and this result is best possible.

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

q1=1q,E111

so that q1<1 and we differentiate between the Goncarov polynomials belonging to the operations Dq and Dq1 by adopting the notation.

Qnzz0zn1andpnzz0zn1,

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

Qnzz0zn1=znn!k=0n1zknknk!Qkzz0zk1,E112

and

Pnzz0zn1=znn!k=0n1zknknk!Pkzz0zk1,E113

where k! and k! are the respective q and q1 analogues of the factorial k. With this notation, the following Lemma is to be proved.

Lemma 6.1.

The following identity is true for n1 and q>1:

q12nn+1Qnqnzqnz0..qzn1=pnzz0.zn1.E114

Proof.

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

qn2n!=q12nn+1n!;n1,E115

and

qnk2+nkknk!=1nk!q12nn+112kk+1;0kn.E116

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

Qnqzqnz0qzn1=qn2n!znk=0n1qnk2+nkknk!zkn1Qkqkzqkz0qzk1.

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

q12nn+1Qnqnzqnz0qzn1=znn!k=0n1zknknk!q12kk+1Qkqkzqkz0qzk1.E117

Now, since

q1Q1qzqz0=zz0=p1zz0,

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

q12nn+1Qnqnzqnz0qzn1=znn!k=0n1zknknk!Pkzz0zk1=Pnzz0zn1,

and hence the Lemma is established.

Proof of Theorem 6.2.

Write

zk=qkak;k0,E118

so that the restriction (111) implies that

ak1;k0E119

Therefore, a combination of (37), (114), (118) yields

Qnzz0zk1=q12kk+1Pkzz0zk1.E120

Also, by actual calculation we have that

n!nk!qknk12kk+1=n!nk!;0knE121

Inserting (118), (120) and (121) into (33), we obtain

zn=k=0nn!nk!zknkQkzz0zk1
=k=0nn!nk!aknkPkza0ak1,

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

Μkr=supz=rQkz;0k1
mkr=supz=rPkza0ak1

and Ωnr and wnr are the respective Cannon sums of the sets Qnzz0zn1 and Pnza0an1, it follows that

Ωnr=k=0nn!nk!zknkΜkrE122
=k=0nn!nk!aknkmkr=wnr.

Since the points ak0 lie in U, from (119), then applying Theorem 6.1 we deduce from (122) that the set Qnzz0zn will be effective in zr for rh1q=qhq1 as 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 Pnza0an1 may not be effective in zr for r<qhq1.

In view of the relation (122), we may conclude that the set Qnzz0zn will not be effective in zr for r<qhq1 and Theorem 6.2 is fully established.

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

Nassif [14] studied the convergence properties of the class of Goncarov polynomials Qnzz0zn1 generated through the qth derivative described in (33) where now, zk=atk,k0 and a and t are any complex numbers. By considering possible variations of t and q, it was shown that except for the cases t1,q<1 and t>1q;q>1, all other cases lead to the effectiveness of the set Qnzaatatn1 in finite circles ([14]; Theorems 1.1, 1.2, 1.3, 3.2, 3.3).

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:

z+αnn,n=0,i,2;αn11.E123

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 a0 of complex numbers with an1, forms a QP- basis in the space 1σ, for 0<σ<W and 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].

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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.

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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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No conflict of interest

The author declares no conflict of interest.

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Written By

Jerome A. Adepoju

Submitted: 03 July 2021 Reviewed: 12 July 2021 Published: 21 September 2021