On Algebraic Basic Sets of polynomials. II

MATHEMATICS

ON ALGEBRAIC BASIC SETS OF POLYNOMIALS. II BY

RAGY H. MAKAR (Communicated by Prof. J. F.

KOKSMA

at the meeting of October 31, 1953)

In the paper, algebraic basic sets of polynomials I, I have given four main results on the convergence properties of algebraic basic sets. In this paper I apply these results to obtain further results on these basic sets. The aggregate of power sets We prove here the following result. "Let {p11 (z)} be an algebraic simple monic set of polynomials, then if one or other of the properties l.

(i) effectiveness in a circle !z! ~ R, (ii) effectiveness at the origin, (iii) effectiveness for every

i~tegral

function,

is possessed by any one of the sets where a0 +~ + ... +a.* 0, and s is any integer ?: l, then this property is possessed by all these sets, provided the initial set is of the same degree as {p11(z)}." Before we prove this result we prove the following lemma: Lemma : If P is an algebraic lower semi-matrix in which all elements in the leading diagonal are unity and U =aoP" +~P"-1 + ... +a~I, a0 +~ + ... +a.#O, r?: l, is any polynomial matrix in P, then P can be expressed as a polynomial matrix in U of degree atmost m- l, where m is the degree of P, and U is also of degree m. Since P is algebraic of degree m we can write (1)

where

0 if r < m. From ( l) and the algebraic

rxJ +
70 We thus have the transformation I

u

I p

um-1

pm-1

lXo;= 0 for j ?:: l and lX00 = l. But the determinant llXiil=.;i:O, for if llXiil = 0 there exists a relation

where

koUm-1+k1Um-2+ ... .+km-11 =0

implying that U is algebraic of degree < m, which is a contradiction to hypothesis. We thus have the reciprocal transformation from which (2)

P = fJ10I + Pn u + ... + fJ1.m-1 um-1.

We now deduce our result. We consider for example the effectiveness in a circle lzl .:::;; R. It is sufficient to prove that the effectiveness in a circle lzl .:::;; R of any of the two sets {p,.(z)} and {u,.(z)}=a0 {p,.(z)}' +~{p,.(z)}'- 1 + + ... +a,{z"}, a0 +a1 + ... +a,::;t=O, r ?:: l, implies that of the other. If {p,.(z)} is effective in lzl .:::;; R, then the power sets {p,.(z)}2 , {p,.(z)}3 , ••• , {p,.(z)}' are all effective 1 ) in lzl .:::;; R. Hence, by the corollary of § 2 of ref. 4, the sum set {u,.(z)}, which is algebraic, is effective in lzl .:::;; R. Also if {u,.(z)} is effective in lzl .:::;; R, then applying the lemma we can write {p,.(z)}=b0{u,.(z)}m-1+b1{u,.(z)}m-2+ ... +bm_1{z"} and then {p,.(z)} is effective in lzl .:::;; R. This proves our result. Similarly for other convergence properties. We note here that if the simple monic set {p,.(z)} is not algebraic the result does not hold for property (i). This has been shown in ref. 5 § 3, when studying the effect of the addition of the unit set to a simple monic set of polynomials. We have considered there the basic set p0 (z) = l, p,.(z)=ez"-1 +z", n;;;::: l which is effective in lzl .:::;; R for all R;;;::: e, whilst the set {u,.(z)} = 3{p,.(z)}- 2{z"}, is effective in lzl .:::;; R only for R ;;;::: 3£?. But for properties (ii) and (iii) it is not decided yet whether the above result does not hold for non-algebraic simple monic sets. I am inclined to believe that it does not hold though I have no examples at hand to justify my belief. It may even be discovered in future that it does hold. I hope this will be settled in due time. Algebraic basic sets with given zeros The problem of basic sets of polynomials whose zeros are of given order of magnitude has received the interest of some writers on the subject

2.

1) This is an immediate corollary of the theorem of ref. 10 on the effectiveness of the product set of basic sets of polynomials. The theorem states that the product set of two simple monic sets which are effective in lzl.:::;; R, is effective in lzl.:::;; R.

71 [2, 6 § 6, 9 and 11]. We study here the same problem but when the basic sets are algebraic ones. One result is already known [6, Th. 3, p. 535]. This is "Let {p..(z)} be a simple monic set of polynomials in which the zeros of p ..(z) all lie in the circle lzl :::=;; Kn" where f-t>O, then {p..(z)} is effective for every integral function of order 2 ) < 1/f-l· It follows immediately that in case f-t=O, the algebraic simple monic. set {p..(z)} is effective for every integral function of finite order. We give here the more general result. (I) Let {p..(z)} be an algebraic basic set of polynomials satisfying the condition (3) ...

and let p ..(z) where the zeros a 1 , a 2 ,

=

k .. (z__:_a 1 ) (z- a 2 ) ••• (z-a,.)

••• ,

a,. all lie in a fixed circle and the numbers

k .. satisfy the condition lim

n-->oo

lk.. l11" =

M < oo, then the set {p..(z)} is

effective for every integral function. Given a basic set of polynomials {p..(z)}, each polynomial p,(z) may be divided by an arbitrary number a., and the convergence properties of the set {p,(z)} are not affected. For dividing each polynomial p,(z) by an arbitrary number a.. and calling the new polynomial q.,(z) we get a new basic set {q,(z)} whose matrix of coefficients is Q = AP where A is the diagonal matrix

0

..J

The matrix of operators of the set {q.. (z)} is then .Q=P-1A-1 • Thus[) is obtained from II by multiplying the elements of each column c~ by a,. Hence any expression w..iqi(z) +wn.i+Iqi+I(z) + ... +w..iqi(z)

is identical with the corresponding expression n,...pi(z) + nn.i+l Pi+I(z) + ··· + nniPi(z).

Hence the expression F,(R) (or w..(R)) associated with the basic set {p,..(z)} is identical with the corresponding expression associated with {q..(z)}. Thus the two sets {p,.(z)} and {q,.(z)} though differ in form are equivalent in their convergence properties; But unfortunately the division of the polynomials p,.(z) by arbitrary numbers a.. may destroy the algebraic property of the basic set. This is The corresponding result for non-algebraic sets is that {p,.(z)} is effective for every integral function of order < 1/(1 +.u) where .u> -1, [2, Th. I and 9]. 1)

72 simply illustrated by considering the algebraic set of degree 2, Pn(z) = zn when n is even, Pn(z) = zn + nzn- 1 when n is odd, and the non-algebraic set, qn(z) =zn when n is even, qn(z) =(1/n)zn +zn- 1 when n is odd. This phenomenon does not allow us to divide the polynomials Pn(z) by

arbitrary numbers when we are dealing with algebraic sets. More expressively, we cannot perform such division and deduce properties on basis that the resulting set is algebraic. This explains why the numbers kn should appear in the statement of our result I and the other results to follow. Returning now to the statement of I, we have

!ail

i ~ u(n) . (R+K}u(n) ·ikni

~ K , 0 ~

••• .An(R} ~

.•. .A(R)

~

(R+KY · M.

Hence .A(R) is finite for all finite values of R and the required result follows by theorem (4) of ref. 4. We emphathise here the fact that no further progress can be done with these basic sets whose zeros lie in a fixed circle. We give an example in which the set is neither effective at the origin nor effective in a circle lzl ~ R or a circle lzl < R. Example ( 1):

l

P2n(z)=z2n

P2n+l(z)= (z 2+k 2)2n+l+z2n+l,

k~

1,

n=O, 1, 2, ...•

It can be easily verified that the set is algebraic of degree 2. Now consider the two functions cp(z) = z2n+l f(z)

~

(z2+k2)2n+l.

The function f(z) has no zeros on the circle

I I

lzl =

2k and on this circle

.p(z) (2k)2n+1 f(Z) ~ (3k 2)2 n+ 1 < 1 for all n.

Hence f(z) +cfo(z) has the same number of zeros inside lzl = 2k as /(z) (3, ch. 7, th. 2). Thus the zeros of p 2n+ 1(z) all lie inside a fixed circle of radius 2k, for all n. Also, the set satisfies condition (3) with tX = 2, but we have W2n+1 (R} > .A2n+l (R} > (R2 + k2)2n-H

so that

J.(R)

~

(R 2 + k 2 ).

Thus J.(R)>R for all Rand J.(O+)=k 2 ::;i:O. statement.

This emphathises our

73

We must also note that if condition (3) is not satisfied, even the property of effectiveness for every integral function may not be possessed by the algebraic basic set {p,.(z)}. Example (2): p 0 (z)= l, PJ.(z)=z, p 3 (z)=z 3 ) P:m(z) = z2", n ;?: l P:m+I(z)=(z2 +l)"'+z:m+I, n;?: 2.

The set is algebraic of degree 2. Taking have on the circle Jz/ = 2

I I~ --an' < cf>(z) /(z)

22n+l

~(z)=z:m+I

and f(z)= (z 2 + l)"' we

for all n ;?: 2,

l

so that the zeros of p,.(z) all lie inside Jz/ = 2 for all n> 0. But W2n+l (R) > A:m+l (R) > (R 2

+ l )"'

so that A(R)=oo for all R>O, and hence A(O+)=oo. When we consider the case f-l < 0 we get the following result. (II) Let {p,.(z)} be an algebraic basic set of polynomials satisfying condition (3) and be such that the zeros of p,.(z) all lie in the circle Jz/ ~ kfnA where A>O, then {p,.(z)} is effective in Jz/ ~ R for all R ~ l if ~X> l, and is effective in every circle /z/ ~ R if IX= l 3 ), provided that the numbers k,. satisfy the condition that lim Ik,. j1'" For A,.(R)

A(R)

=

l.

~

(R + ~tn> ·!k..l

=

R"<"> ( ( l

~

R"'

~

=

+ k~~)"

A u(n)

} lk..l ,.A •

R for R ~ l when ~X>l R for all R when IX= 1.

Hence if IX= l the set is, by theorem (1) of ref. 4, effective in every circle Jz/ ~ R. If ~X> 1 we consider the relation [4, § 5], A(r)

~

(rJR)"' A(R) for all r>R

and put r= l, R= 1-e, then R= l, r= l +e and make e __,.. 0 in each case. Then noting that A(R) is non-decreasing we get the result that A(R) is continuous at R= 1, and the effectiveness of {p,.(z)} in Jz/ ~ 1 is again given by theorem ( l) of ref. 4. We have further the following result. (III) Let {p,.(z)} be an algebraic basic set of polynomials satisfying 8) The corresponding result for non-algebraic simple monic sets is that, when the zeros lie inside JzJ ~ kfn the set is effective in JzJ ~ R for R;;;?: log 2, [2, Th. 2].

74

condition (3). Let c(n) be the number of zeros of Pn(z) lying far from the origin. If the zeros of Pn(z) lie inside a fixed circle and satisfy the condition -.- c(n)

hm -(-)

n--+oo u

n

fJ <

=

IX,

then {Pn(z)} is effective at the origin 4 ), provided that the numbers kn satisfy the condition that lim lknll/n = M < =· For in this case Pn(z)=zu
A(R) :<( R"-fi (R+k)fi M if k or

~

1

A(R) :<( R"-fi M if k< 1 and R is sufficiently small.

In both cases A(O+) = 0 and the required result follows by Th. 3 of ref. 4.

Algebraic basic sets with given coefficients The problem of basic sets of polynomials whose coefficients are of given order of magnitude is equally interestipg and has been considered by many writers [1, 5, 6, 8 and 12]. In fact the two problems of zeros and coefficients are greatly connected. We give here a few results on coefficients. The above results on zeros can be deduced from these results on coefficients, but the whole matter is simple and it rather seems that the deduction of the above results on zeros from the general results of ref. 4 is more rapid than their deduction from the results on coefficients. But examples 1-2 of the above article show the fact that if, even, the more restrictive conditions (on zeros) of article 2 are not satisfied, the results are no more true. The results on eoefficients are: 3.

(I)

If {Pn(z)} is an algebraic basic set of polynomials in which lim

and

n--+oo

= 1

u(n)

n

IPnil :<( Krt
then {Pn(z)} is effective in !z! :<( R for all R ~ (!· For An(e) :<( L;IPn;!ei :<( K{u(n)+1}eu it follows that A (R) :<( R for all R (1) of ref. 4.

~ (!·

(!,

Our result now follows by theorem

4) The corresponding result for non.algebraic simple monic sets is that the set is effective for every integral function of order < 1/{3 (a corollary of Ths. 1&3 of ref. 1). Indeed the algebraic property of a basic set gives it the chance of possessing wide convergence properties.

75 (II)

If {p..(z)} is an algebraic basic set of polynomials in which -

1

lim {u(n)}n

l

=

....... oo

and

K , 0 ~ i ~ u(n)

IP~il ~

then {p,.(z)} is effective in lzl ~ l. For in this case A,.{l) ~ K {u(n)+l} so that A{l)

~ l.

(III) If {p,.(z)} is an algebraic basic set of polynomials satisfying condition (3) and also

IPnil

Mu(n) ' M < 00

~

then {p.. (z)} is effective for every integral function. For here, A .. {l) ~ {u(n)+l}Mu
(IV)

Let i(n) be a sequence of integers such that

lim

i(n)Jn=a>O.

n--+oo

Let {p,.(z)} be an algebraic basic set of polynomials such that

IPnil

~ cp(n,

i)

for 0 ~ i ~ i(n)

where 1

lim {cp (n, i) }"

n-..oo

= 0

,

0 ~ i ~ i(n)

and either (i)

IPnil &

or

~

lim

n-oo

IPnil

(ii) &

KMu
~

u(n) =

n

iX

~

u(n)

< oo

K , i(n)
lim { u(n) }"

n-oo

= £X

< oo

then {p.. (z)} is effective at the origin. In the first case

Hence for R < l A ..(R)

~

{i(n)+ l} cp(n, h.. )+ {u(n)-i(n)}KMu
where cp(n, h..)=max cf>(n, i), 0 ~ i ~ i(n). Hence A(R) ~ M"'R/' so that A(O+ )=0 and the result follows by theorem {3) of ref. 4. In the second case, A,.(R) ~ {i(n)+ l} cp(n, h,.)+{u(n)-i(n)}KRi
gives A(R)

~

tXR!' so that again A(O +) = 0.

76 REFERENCES BoAS Jr., R. P., Basic sets of polynomials. I, Duke Math. Journal, 15, no. 3 (September 1948). 2. EWEIDA, M. T., Order of magnitude of the zeros of polynomials in basic series .• Duke Math. Journal, 14, no. 4 (December 1947). 3. MACROBERT, T. M., Functions of a complex variable (Macmillan, London, 2nd. edit., 1938). 4. RAGY H. MAKAR, On algebraic basic sets of polynomials. I, Proc. Kon. Ned. Akad. v. Wet., A 57, 57-68 (1954). 5. RAGY and BusHRA H. MAKAR, ~ur la base somme de bases de polynomes, Bulletin des Sciences Math. 2e serie, 74 (Juillet-Aout 1950). 6. , On algebraic simple monic sets of polynomials, Proc. of the American Math. Soc., 2, no. 4 (August 1951). 7. RAOUF H. MAKAR, Algebraic and non-algebraic infinite matrices, Proc. of the Royal Netherlands Academy of Sciences, A 13, no. 5 (1951). 8. MURSI M. and RAGY H. MAKAR, Coefficients of basic sets of polynomials and functions represented. Proc. of the Math. and Phys. Soc. of Egypt., 3, no. 1 (1945). 9. NASSIF, M., On the zeros of basic sets of polynomials, Proc. of the Math. and Phys. Soc. of Egypt, 2, no. 4 (1944). 10. , On the product of simple series of polynomials, Journal of London Math. Soc., 22 (1947). 11. , Zeros of simple sets of polynomials, Duke Math. Journal, 19, no. 1 (March 1952). 12. WHITTAKER, J. M., On series of polynomials, Quarterly Journal of Math. (Oxford series), 5 (1934). 13. , Surles series de bases de polynomes quelconques, (Gauthier-Villars, Paris, 1949). 1.