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View of The asymptotic relations for the indefinitely increasing zeros of polynomials with alternate coefficients

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solution b)/

spliì1e ìnctho(Ì

80 S, SALLAM

,l.tthle .tr.J

(n :5, 1t : 0.9f109tì90, /cr: .0250867, h: ¡140) solutiolr Ìtr- ilre

gcn. Avor'. tìtc[ìto(l

l\,IÅ.ttItiI{A].tc.\

_

fìlì\_ulf t),

El.

DE r. Fric) n r

ri nn ì'.ìI#;

ìoT,

^ì.TrlltnÏ,

u,,"

tr,,JTNÁT,YSE NUN' RNTQUN

BT LA TI'ÉORID I}IÌ

L'APPIIO XT]II,,T.J]ION,

l.ouc

.I5, No

l,

19{16,

pp. g[_g3

,TIIE

ÄSY},J

rNoRÐArsrNc Wl PifOlII€ ltu

Á 1.,l.E

T],ELÄîIO}[S zllRos oF Rr\r\ 1lE Cop¡riicUõñäis FOR I-,ûilNOìní¿

,.1.I.IE

ÏI{ DEIìINIiT'ELY

¡-,s

BosKo S. TOI,IIÓ (Bctgladc)

.

-'\s an asyrtrlltof

ic

exJ;ression

fo'

a..polyrromial

wi,r

altc,rrrate coeffi_

cr c^l,s

.l' tlcg'ec

u ( tr,

) iì)'ri..

tol."

-ur"

rò'o"."ì"î^îr,i""îri.l (1)

ur,,e)t'

_

{tr!)n 1

|

r1..,¡.,,_:t

_,,Ë

t _7)u a,,,6n_rrn _-+ _l æ.

rni¿r,rs

t|itlìi""*rï'tl

ro l;his

t.i.or.ial r'"lon*i¿o"

a

finite

scquence of porvnor

(2)

11r(r, ctu).- í.rj .

!ln_1(r:, a,ò

l- (_r)r,

(Lo,,lt

: 8,4,5,.

.

.,

,ru,

arn

f

a,r.

""l'ol1l 2' 1'r ',P) of

the.Po^lyno-

,

r,

"

å'

iü,:;îå r""rn

th

e

trr e or

¡' "'f f''þí;i;: ïï ::Ï;

t

L

:, i

;;

I"{-l)'

Q,,, iv?

v:

0,

((ru}

0)

rx

nl4 -r)

3;t 14 n 5rl4 3r12 7rl4 ,)-

0.3ó25831lì-j 00 .0.31it7.s07E_0ô

-0

352ã83711 1 00

.-0 4990201Jì+00

-

(,.352582611 ¡00 0 tlri(1226311

-0(i

r) . :J5258il1tj -l- 00 0.49902t)llì l o0

soìrr Lion ltr [he gr:n A\.0t'. lnotho<l

0 . 3 547(i 96tt -l_ (x)

0.21391õtiE 02

- 0 .:1521ì97illj*00

- 0 4992.13:lE_i 00

-0 .i15239{ì2It l 00 -0.3,1:ì1366Iì_(,2 0.:ì51 ó110I-t j 00 0 .,t9998641.t 1 00

0.3534291E+00 --0.2067024Ij_04

-

0.35349398 +00

- 0. 500383611 + 00

0 . :15352378 _l_ 00

r) . 261 ô1 55E _ 03 0.3it39488E+00 0.c00803bE F00 'l'ul¡le 4.4.

0.3511344E1_00

-

0 .4748825n_02

-0

.3575117r]+00

-

0 .5013575E+00

-

0 .3560276E+00 0.27404378_-02 0.35471078-l-00 0.5007516rì+00

Diflolcncc

0.8460283E-03 0.2035156ìi _0.1 0.9101629E_03 0.1363575E_02 0,94709778_t):l 0.260ô594E _0:ì 0.1364648E _ 02

0. 178:J490u _ 02

l) iffct'c r rc t:

o.30i9414r, 02

0.6887983I.t_02 0.51143768 02

0 .2709r70F. _O2

0.36iJ1711Èt 02 0.5571804ri_02 0.3199696E__02 0 .7652040Iì_ 03 (nr - 5, l.t -. (t.250tt24, /1, : 6.6164,] B, It _ nl4¡)

3tr 12

r,'nr

l'2

?T

l;¡

7n12

IÌt.ltrtìlìltNctis

1. ll rr r'li o rv s l{ i, Ìr, .J. :rntl C o rv a lt. O l).

. ol _st:r:otttl ordcr dillcrenliul tlilfcrertcc cqurtt

2 ìr I l ¿ì g g â r,.{., ^Sol¿¿/ions lrctrntonirltris ct

.- 0,'l'hc 14tli. l\lrn. OonI. o¡t St¿rt. (ìorlr.

lJ l) ol z ó 1<1, L Lt, , Att r:flicit¡t[ ntunet,icttl n

4. rI orcler no¡t li¡tear

- I lrtcn. Ins. vol., lialcquu-

r. te¡y spline

funeÍiin

(i. ,,,. ti.,rg,rriii:a, 6íi. sorurìons.

c¡., ¡iut0 s.¡:lirtt eJ)J:t,ox.in ttl.r¡(tt) _ 0j.St;r[isiic¡ ìi,,,ä.riL¡,,

i ltc cocfljr,i(Ìtì1, 40 o1, i,he hrgllc,sl jro\\\)i, lcrrlrls

iq

zrr¡¡"

llicir

¡:¡r,

ol

i.ll¿.r 16o1,s-

o1'llrt,r.r¡uiilio.r

113) irrcr,r,lrs<,i jli,l,,iiiri{;r,l.,..irr lrirsrrlirtr,r.;tlrrr,.

L1,r 1hjs

1ì;c,ot,r,r lc

I

t,lìtrs

tli,{i(ìì,1¿ìilr tic

1

tht' qtt<.s1ig¡

ol, ,

ilr Ibsr;lutc r_¿l r

1l (3)

lìccoivccl l7 \l 10f14. ) !, t

I lr nnl ít s I )e l.¡rtI t;tt I

l ¡tit¡tt:'itll ol littttitiI l).(). Jlo.¡. ,i!)t;I .,

/i ¡r ì.ìj-4 / 7'

6-íi.2t7S soluliou b¡-

spline rnctìrorl

(2)

82 B, S. TOMIC

Ilor

ecluation (3) rvith alternate coefficients wc shall ltrove the

follotr- ing

theot'erns :

1

.

Th,e gea,test real root of equatiott, (3), wltose coeJf,icients are

all

altar-

na,te

,

tenil,s

to læ

as ao tends to zero

(4) ro,,

(øo)

* i-cor

ao ---+

!

0.

2. In

cøse q,ll tlte roots

of

equa,tion (3)

toith

alternate coeffic'ients are contpLen, lwo conjugcr,ta compl,er roots witlt, the greatest reu,l ar¿el, þositiue purt llacome req|L und, tlte ¡¡reater one of tltese real rools tenils tl¿en to

!

oo as uo

tentls

to

aet'o.

We

consideÌ'

one rnore finite

sequetlce

of

ecluations

(5)

þ

'No{-f)'tr,,a)P-v :0, p :2,3,4,...,n(a">O).

Fol

equations

(5) we shall prove the following

theoroms :

3. 'Ihe

t'oots t'r,,r(p

:213r4,...,tt), whiclt

tend

to )-æ,

are d,istri-

butecl ts.f'ollotos

(6) 0{?'zz{r++1...<

t.

Betueert,

tlte

roots t"rr,rr, ü1'td, r"zr)+t, ,n*, tlr,ere enists tl¡e asqmptotic t'¿tlut'ion,

(7)

t'zp, zt, @o)

-

't'zt,*t,zr*r(Øo), øo

-+ f

0.

i¡.

I,tt"o¡n (6) and,

(7)'ít

JrilIotos th,at th,e roots, tend,ing

to !æ, of

equ,u-

I,ions

(5)

ctre si.tuatecl,,in the i.nteraa,L ('t,rr,r,"")which tenrÌ,s

lo

zero (rs aoten,ils to zero,

1. lYe

consicler

a larnily of

parabolas

of

degree

(n-12) with

alter,-

nate

coefficients

in its

equation

(8)

U,,+z(t)t"r)

:'å.(-

L)u u,., x;,,+z-u, (øu

>

0)

where øo

is

the parametel

of the

liamily.

For polynomial (8)

we introduce

the asymptotic

relation

(9)

an+z (fi¡ ao)

- n"'

(øon'

- ãtfl *

az), fr

-* +

æ

as

well

as

for the first

ancl

the

secontl derivate

l( i')

&o nz

- (" I ')

ü,

n .r ('i )'',]

t ASYMPTOTIC RELA'IONS .OR ?''E INDEFIN,ITELY IùI.R,EASING ZEROS 83

lVith

regarcl

to fonnula

(2) lve consicler tìre parabolas

(2r)

yr(n,

),) :

tt'l

å¡

a2

-

rcrt:

+

q.z, (r.o

:

#,

À

)

1

cutting

the ,Y-axis. x'or

this reaso'e

introclucc the parameter À

in

place

of rn..Ily r,

_*-e crcnote the

x-axis

of pg,raboia

(2r).-\4i;;ir

r,riis systdin"ái coor<linates

tìre

sysr-em

_!_X):,rf., by izi,

ioã

,Àoliiii., íìJ*, ¡¡ by ,r

aud

lrom this subtract ør,

$;e äbtai'n

(2r)

yr(n, )t)

: ++

4ar),,

tf -

a,rnz

| ørr -

u,r.

Parabola

(2r) is referled to t

putting, in the

system (Xr),

at the

clistance øo. We obtain

the parallel to the

axis

X'

belo.rv

th

corresponcling manner also

for the f formula

(2).

For

the.cal

roots

b.t

trifferent

from

zero orthe equation u+zllz(fr, üo)

: O,

wllet,e

cto: -4-, 4arlt

' ÀÞ

1,

we obtain

(18)

r2r(À)

: -^ (, - f+), n, tt

rrrlx¡ : ?%\

'-+) ,ì.2t

rzr( À)

-, o,

,

À

-+

J,_oo

Q,7

rrr(ì,)

--->

* oo,

À _+

foo

&0, Tzz (øo) ---

a'

ao

-> |

0.

.!'or the equation

u+zUL(fr, úoj

:0 we

obtain

rír(\,rr, - ?',:r\ +# ? -ll

'2

2r¡n_2,

f

__n

(14)

(15)

(16) (17)

(18)

(,19)

rLz(t,,n¡

: ?:rL

(n f 1)¿

À

0'r

1+

'!lia2(r,

u')

-

(10)

Yli-,

, (r,

ø)

-

(11)

frn-

t,

fr

-t

-¡' oo

('*V

nt

J- 2n

tr,2

!

2n

u,t n n!1 +2

I

[(" ;')

uo

*' - (" *rt)

" "*(';) "']

I À>I

1

l

With

regarcl

to relatioti

(9) we considor t,he sequence of functions

(L2) ,,*z!/z(.ürüo): fi"'(*on' -

Q,7s:,

* az)rtt,:lr2r3, ...

(ø*1)'

À 1)t)-]-

(20)

rirQ,,n)-,!' . '

n"

Q'r n+1 , À+ fco

i

(3)

J I\SYMI'TOI-JC ÌìELT\TION^S FOIì T'llE INDE¡'INITELY INCRE.A.SING ZEROS 85

rT in thc

j'olrnul¿¡e

(r4)

arrd

(1iì)

u.e cxparLcl

the

sqrrarg r,oobs

in

¿l

'binomi¿r,l ser,ics,

l'c¡

obtaùr

(21) r"rr()',n)n

-l-

o¡,

À ---+ -l-

æ' Foi' the cquation

u+zlllz' (üt øn)

: 0

rve obtain

v

(22)

r'!r(À,

l) :

,(:13 )

B. S. 'I'OMIÓ J

t'rr(),) --

r!rr(),, n)

-1+t: -);2{ -(''i)(-i l.('l')(I)'* }-

t''"),,

tt)

: 4;:'rtt

;în (, -

nrr,(rr,

I

tt,

f l) --2

L)'

)' t (t";"," )

rifl

1

rt,!2

1- ),>1

,)''ÞI

2

tl2

1

a{

rl2

¡,

2u)'

,IL

In--2 1\

1

-^^ )

1+

(L^ tt -

1,

r;l

()., n)

+ "z , )'+ f

oo

0t 'll' -f L

r:,iQ',

ri') ---+

oo,

À --+

+

oo'

lhc

Tirral conclu.iiotr ott '¡he dis1,r'il>rrtion

of the

coLrsitlclcd ¿bscissas

frt¡m tvhich

I'elatiol

(27). Thc algurnent^bion

fol lclation

(ZB) is

similt¡,to i,hat foi' r'eiation

(27).

2 ilt tlte

poi.nts .

q.= 4; e,, tt)

wtr,rl

n : rl,

(),,

tt) tlte

r.¡rtlhua,tes

"f

ou,rur:

(r2)

i,ncreuse 'i,n'tleÍituc'tely 'ir¡ u,bsohtle ,uulue-iull¿en, ),

->

l-co

; fgr

crrri-

oenier¿ce rt.f' th,e calou,løt,io+t ,¿De orr,rù talce

n:2p -

1 ,¿oitltou,t, d,ln¿i.qíshl¡n¡1 1tp

, ¡¡ ¡i,¡¡rzt' t,l, ¿¡¡g ¡1,

zpt.l/ztr'jr(À, 21t

-

1t), tJ, .=

{lr ll -f á rr, ('rr,

f

1)

(24)

(2r-r )

¡r -s

fe]]6rvs

-¡'^t'

. .+"?-

(pa,

I

V D¡"'

(par-l-u,r-VÐ)

(21t

t

7)2

Irt,z (2.p

!

1) a,i

4p"-r

([t X

t)

< rji

(À, n)

{

r,ir()',

l) i lzr()'){

zn'in

()')'1

I'j!(À, rr')

{r'1"(À'

ir')

{ r"i}')

i!) X

) I

).

'llre

inecltrtllitvra;,.,,,(À)<r'i!(À, tt')e-risl's

Ïot tt'73

a'ticl

Ïor'

À

) P-rr- All otheÌ incqualities ale

v¿Lid

fol'

rt, =.

!, 2,

3,. .

. atitl tol. each

lo¿ll riurnbor ).

>

1-. rnequalities (26) al'g ptol'ccl

by

mczlns

of

elcment"i'r-v c¿t'1-

;iìilîo"|-; iumr"Uy t'', starts Îror^^ thc ïoirnulac t¡ernselvt,s io..

thtr

r¡.entionecl abscis¡ias.

Ämotrg the

u,bscissàs 2erùritr ()'), r'jj(À, tt'),r',r(tr'

rr') atrtl t'"()')

thcrc

orisi,

1,hc

Íollowing asymptotic

rolations

(27)

?'rr(À)

-

t"=r(ì',n')

- ,.:?::,

þt,

!

2)a' )''

->-l

oo

,(28)

r'=r(),,n)

-rü(t',ù - fø,-¡,,', À- +-

ljl¡.,I

t¡,

-

2

.(

'l1r)

,r!,1-(),,

rt) -

"tr'',.,,'

(i') - ,,", '

r¿

-i Z, l'- ' 'l- oo' tt ?

'>

(J6)

,(iì0) 'l'zz

r'r"('t,, rt)

--

r!t!t(),, n.)

ltertce tlicrie tlxisl;s tlto

lnr;:o.nnll (1). 'Ilt'r"

paínt r'":r(t,,

rt)

Li.ts upproni"m'otcl't1

itt

tlt'r: nt'xl'tlltt ,hefween

(tt

't)oitl'ts li'r( i', rr,) and rrr(),)

wlutt

)t

+

-f co'

rllll') zt,+lt¿[rJr(,¡.,2¡t I), r j* -oo, À+ f

co.

illho olclinate

of the culye (8) at the poini, n:

j.'zz(À,

r)

increascs

incìefinitely in

absolute r.a,lue

rvllen

À

-

-Foo.

llhis

orclíñate is

,(:12,\

lln+z {r'rz(À, tt,

)À].:

n+¿!/z {,tóz(À,

z),

À}.

n,. ("r,,,n(.l))r-r -'

¡¡:p2u! ,

flonr which

ct,,,

--

4p'

t(35)

:l

il

))"- t

froln

-whicìr r'rr(ì', n) uor'rzr(ì.,, n

-

errLz()',,

n) | arl

'(33) !lu*2{ri2(\, n),

tr}

* -

oo

,

), --+

f

co

'ln I

rltril":r si¡trilai. lniìnnet n'r: obtùiir

,(3'j)

L,+z'tr'¿'()r, n), Ài

- -

co, ),

+ *-

oo ;

'lllOt'('o\'{rì', l'e lt¿f't' lt,lsó (À)

-

rir(À, 'n)

-|t À* *

oor

uu + l{ 2t,,, r,, ( À) , À} ._ 2fi,r),

0r')

u,r(.--^ I

1)

)'*

+e

2arlt

(lt

'nr('-À l- l) --

__

cor

),

-,

_l_ oc.

(4)

86 B. S. TOMIC

Finally, lhe

ordina,te

of the

curl¡e

(8) at tho point

u

:

rr"(),)

inclefinitely in

altsolutc value

lvhen

)i

- +oo. 'Iho

ordinate ot

(8) at

1,he

point ø:'rsz(À)is

'1

As.!¿lvrpror-rc REr-AT[oNs ¡'oR' TIIE rNDE¡'rNrîEL]. rNcÈEAsrNG

zERos

g7

.and frorn this

(r,,,t.rr(À),

r f :

(r{r.rr(À),

1} * y,

.Àr22(À)

: r(aa)

ar{2(rL

+

2XÀ

+ltñ _-)ì _ 1) +

1} _-_+1- co, ).

+f

oo.

fl'lre tangent to the curvo

,U

:.!/r.,2(fr, À) at the point, r:

?.rs(À)

folnrs

¿¡rL acute a,ngle rvhich

tcntls

to

l

or thi,;

irL,rcrion

";';;;;;;':ï",r,ï:.-

-r

oo' 'r'rrc

rirst

crc'ivate

(r15 )

rr;;t()).rì,rrr(À)

li*trLt'r,

(À), ÀÌ

:

6

Increases.

the

curve'

!1,+ztrrzz (À), ),1

:

r'iz(-t) {uor!, (À)

- ttrrrr()') {

ør\¡

-

&" Ì.,11, (7,)

+ (116) |

u,nr!);z

(À)+...+ (-1)'+z ûu+z: ri;'(),) -e¡ -|

+ -o:--. T

. .

. I (- .[)'*'

1.":1,

,l-, *,

).

--,

-r- oo.

'

i 'rr(

7,) ritt

( À)

l

3. ilre

sh,ct'll represent cet'ttti+t, th,eorents concterninq th,e arc of th,e curue.

U: !/r*r(*,ì,)

i.n th'e interaal (r'i*r,r(À),

*oo),

wh'ere the poi,nt

u:

r'rl*r,r(ìt)' i,s

the

l,ast

ltoi,ttt

oJ inflection.

With

regard 1,o tho asymptotic relation (10) rve cotrsicler 1,he sequence

of

parabolas

(37) ll :'Qz(n,n):

(tr'

|

2)uur2

-

(n

*

L)ørtt

! ttcr,r, tu:I,2,3,...

All

parabolas (37) pass l,hlougJr

the fixed points P(rr'

rYrr)

ancl 80rr,.

Nr;)

rvhosc oldinaters a¡rc

(38)

Nrr.(üo,n)

=

nz {rrr(À),

tt'1¡:2ør(i' -l/it --'- f) < 0, ),.¡

1,

(39) Nrr(ao,n)='4r{rrr(ì'.),tt) :2ør(l'l /l', - ¡- 1) }

0, À

> 1;

it is

easy

to provc

1,ha,1, these

quantitics are not

clcpenclent

on

ru and therefore we

write them without the

argument ra,

for

instance

(40)

J/rr(øo)

= tYrr(ao,n):

N"r(ao,

1) :

Saor!r()')

-

2ctrrrr(À)

J-

nr.

[t

we use the parameter À, thenwe

wlite ]rr(À).X'or À:1

rve har.e

Nzz(l):

:

0.

In virtue of

(15)

antl

(17) we h¿ì,\'e

(41)

^]\[rYrr(r)

: +oo'

With

regartl

to

the asyrnptotic relation (11) we consicler l,he secluence,

of

parabolas E

:

(r{(w, nJ

=

(42)

ütfrI

ü2, Lr2,

À ___+

f

oo,

ftortr u'hich it

follorvs

'(46)

Ui*r't:'rz(Ào), Àoi

:

r1 (Ào)

>

o,

'where

"l(À") is a.llositi'e

¿ir.r ar,'r-¡it'a.iry. grtrir,t rrurnber clepcncli'g

ou

Àor

rvhitc Àn

is a suffiòien'y

sreãl

;;i;;'of

the

para'reter

À.

since

vi*,(.r,

Ào) is a poivnomial, rvc

ü"", ;,;;l;;*o

rheorern,

UL*r@, Ào)

* *

oo, ø __+ _þoo

3;åt",1.,rî::tlli:f,,îï î#:j

in the neighbourhood of rhe ponr n

: {

æ

'$r) r. ?li*, (n',

)ro)

)

u{ (Àn)

(48) 2. n' : rrr(ì,,) > 0,

7.,

)

À0.

In virtue of

(18)

we

have

(49) o {

I'zz (Ào)

<

r.rr(},,)

for

À,

}

À0,

frorn which, in virtue of (4g), it

follows

,(50) o<rrr(t)<n,.

ï'rom (42)

ancl (50)

it

follows

fllrmonru (

tt,

'{;;;::":#;,l,7iu?,,ì, }; ¡: i:ii,liíi"r,,^i.,,n",,::,:;:{{:";:::!!

Fi',,;; "riai" ¡\L' r : r"(l'')' (51)

Ur*z{r.rr(Ào), Ào}

_ _

.E(Ào)

<

O.

i'1ril|:iil',(;îïî

be chosen as bo be the same

in

rerario,rs (a6) and (þr).

'rhr"isr';.rñå:ï,1ËÄ?irl3'.=I';i'li;làht""lr*;""ìlriliî3ä",î;r

I

1- (p-7) ørr!;r(X) -¡

-(-

1) r+r &t, +,

r3;'(À)

. Àtr, (À)

*

-l- oo,

n+2

2

'n-l

L

)"'* - (

2

11,

2 't'ù ü.

rn

virbue of uro

iormura(;:

)-r (,;', ) : (j]

î 1)

.çr.e hå,r.e,

(2(n,

l-

1)

:

(2(r,

n) |

'r¡r(u,n')'

and florn thc

last,

fornttla it

follorvs immetliately

(43)

(2(n,r't' +- 1)

.- -(r(,r,1)'f É'¡r(r,

u)

(5)

S8 E}, S. TOM]Ó

thus tlre

point

of ini,crseotiolrt'.p¡2 p+z( )') oΡthi,s pa:a,bolarvit'h l,hc ¿ùxis rY?+2

is si1,na1,cil to thc rigìrt of tÌre

poirlt

rrr(),), nàìnel¡'

ï,t2)

0

<r'rr(À)

{?ì,+2,r,+z().).

Orving

to (10), frorn

(l-r2) i1,

follot's

(53) r, ,.r*r(

)') ---+

{ oo,

),

+ * æ'

,I'he l'ir,st r'[erir'¿1,e

tf

lJro palabola,

!:]l

t,+z(c'¡ À) zr,t

tho poiltl, ø:"

:

ríz(),r ?r) is

y', * r{t""r(\, p), ?' )

:

{I'1,( ),, 1t)'rt:' - z

-1-

(-1¡'*t' from l'hich it

follows

(t""r(),,

p))'-"

Y'rr*"1,r''.r()', P),

)'I -t -cor

7'

+ *

@r thus

(54)

u'o¡or,r''"(),o,

?),

)'u)t

- -

lJ( )'n)

<

0'

'Ihc

va,lue ?16 cilìr so bo ghoser ¿Ìs

to

lte the s¿uno il.i rela'tiotrs (46), (52) anc'li

lf7j. f.to,u

i+O) anci (ir4)

it

follorvs tha,t

in thc intclva'l

{r'ir(À0,

-?)'lrr(}o)}

|íãiii.*C,tcrì,'ítc

Uioir('r,, Ào) charrges

the sig.

ancl t'akes tire. r':¡luc', 0, 1,hus

(55)

'!tL*r'rr|*r,r*'()',,), )'o]

:

(),

rvhereb¡r t'L*".o*r(Ào)

is the

gre,àtest-

r'oot of the

equiìtiorì

.!lL¡z(r', )'ò.:

--0. O\"iíg ió (¡¿)

anct (46),

fol thc

¡tbsciss¿'

I'i,*r.r*,.(\ò l'hero ¡oltls

the inequa,lit¡r

(56)

0

<

rir(À0,

?) {

t''1,+z,n+t( )'o)

<

r'rr()'o)'

ilhc

sccond rlerivai,e of' 1,ltc pa,l'a,llola

I :11r,*"(t',)') it't' thc

point

n :

rí'"(À, p) is

ll',i*

" It'J'"Q', /t),

)') : (";').,

9 ASYMPTOTIC RDLAI'IONS FOR THE INDEFINITELY INCIìtrASING ZEROS 89

illhc

secont'l rlcrir.¿r,te

oll thc

para,bola, !/

:

ll ,*r(.;tr,

À) a,t lJtc point

¡;

-

t'Lz(À, p) is

8i

\(5E)

'y',1,"{t""r(),,

?), \} -: 2

lt""r(),, p)}1,-2 . (r[?'j.r(]., ?),,11l I

t

2 2

p-2

2

est'Lor-"(I,

?)T

. .

. -(-l¡, )

\ (Lp

1

(tl't'|r()', p),

pl :

(Lz

l"r!-z()t,

p) '

.(r{'r:rr()',p), 1t)

(z

f

1)'

^*v rr pØ+2) I r.-- itrl.r.)-t, \

{

-,,, -

I ),r,

+ (,',:=e;

1

ì

I

I

I

'p+2

ì

I

1ir?)

lt'r,' *"It"l'"(,7',,rr), 7'i

- -@t i' -+

i'-oo' from rvhiclh

o,nsuc*rcrrlr¡, ,,,. ,;ï:;ì11;lt;t ^'t;*Lii:],i,j]*"' ar

i,rre

poi.r, ø:.

.-

rJl(),u.7i) f'o:'

lr

stti'l'icionl,lr- glcztt

ylrlue 7.n. : .

:

t/'í*ztt'!z(

),,p),

).1

r

-F oo,

).->J co,

,.,"rr,,

.rtÏff'

)'+

-'Loo;

t(59)

!'ri *rttt'lr(),0, 'p), i.ul

: -l-

1i-(À0)

>

{).

(lorrseclucltl¡-

thc

parabolzl

ll -,!/t,+z(r" iu) is

conc¿ù\rc ¿rt

tlrtr point r J

== t'3zQ,o, ?).

._ _Fror'

(õ7) anrl (59)

it follo*'s

lh¿r,t

in

t,lrt, i.l,e'r.¿rIrrt,ílr(),,,,p), i,j,().0.,

p)) the

scconcl derir'¿r,te changcs

thc

sigrr

nnd llht¡s

tLrtr 'í'ätüó

"0,'ii,..i

,(

G{l)

utri*"{r,rl ,,r(),u), Àr.}

:

0,

whcrcb¡'r'i*_r,n(Àu_) is

tirc

groatcst

root, oI tho

eqnzrtiott

t/i,*r(t:, Io) :0.

rJn'irrg to (57) and (59), for thc abscissa tii*r,r(Ào) l,ìicrle

hold,i"ilre

ircclualit¡,

(61)

0

<

,ril(À,,

?) {

,r,.} *r.r(),0)

{

r.,rz(),0, p).

Sirrco

l,i:.r,r\À)is

1,hela,sl,

rea,l clo ol flre poì¡,nomal

,4,i,*r(n;, À), t,his

nreâns

that

y'ri

*:(a, À),

1;r.r,ìiirg orrl¡.

posil;ivt r.aliùs. tendí'tf -¡t".r,s

,r

increas<¡s

ïïoltt

lrl¡2.-,1, (À) 1o

fco. ,;,*, (r,

Àj

lenrains

pelnrancntl¡'

posil,ive

as +;;

ánci

l,his rneans Ural, the parabola i,s

c

L¡,

1 ,"i.

Conse-c¡uentl¡' in-

the irrtctr.al

{r'ri

*r.r(,

.,s

on

thc

¿tl'c cr1'

the

para,bol,\ ll.:.1¡

,,-r(i,

),) -a,nt1

thetrfole, r.lìcn

). incrcases,

¡o

nûw zcìr'o can atisr¡

jlt'hirrd the

zelt¡ )'1t+z.u+z( ).). ltìhis

is thc

concepl,

ol

thc lir.sL

zclo. tlhus ir'e

lra,r'r' tlro

.illnnonrilr

(13)._ll'ol a, sttJJíc.icntl .t¡ lrn'{tc ),,

à

).o l,lte unt.:e !/

:

!/r,+z

'( 'r;, ).') c¡tts th,e

urís

x1,+, ttt Ìh,e poitt,t ;t'.= t't,+2,t,+,(i,') uirrutÌ¡ttt.

),'i,nñ)i"t,,ii

[r.t itrct'eu,se, no nelt) zero cfl,lt. ctr,ise be]ti¡¿tl tiis-öert¡"

At thc point

D{t

"r(},),7¡r*rl.t,, tarigcirL

to the cuÌl¡e ? :

.ttr,+z(t,-),)- 1,his curvc

is

conca¡.e

ilt

1,hc-inten,ai cur'\ro

lics

¿lltor-e

thc

tangerrt

ai; tir

the al¡sciss¿r, oli the poin1, o1i iitt,ersecrti we

llat

e

(62)

0

{

r'zz(À)

{

11, 12,1,*, (À)

{ dr*,(},), ). }

).0.

nl'he

poirtt of

interseciic¡rL

ii thc lrcnfionetl

1'auÍìerit r,r'ith 1,hrr u,ris -lu.,,, .has

thc

altsciss¿i,

i(63)

¡: =-

ilt,-,(t) : r-:z(À) .þU]U(ÀI

r:Ì

tt l, *o', r,,"()'), ),t, (ú, +'t

(" ;').'' f'rí'(t"PD'-'{t -

.)

2

0,,þ

1

t)

('; a2

riL()', P)

+ (--t)n p-l I

)

(6)

90 B. S' ToMrc

[n virl,ue of (16) alcl (45) tve

hâ\'e

v ..2{r2r(À), À

10r 11 ASYMPTOTIC R,DLAIIONS IIOR, TIIE INDEFùNITELY INCREASING zEn¡os 91

¿¡ncl l,hcrefore

flom (63) it

follorvs

that

(64) {d,*r(À) -

rrr(À)} ->

0 as

À

-' * æ'

Frorn

(62),

in r.irtuc of

(64),

we

obtain

(65)

?'e2(

),)-

l'tt+z,tt+z (À)

as

À

-' +oo'

'Ihe eqnations

UL*r(n, À)

: 0 antl

(7t

!

2)aon¿

-.

\'tt

i

,\)n.tn

lrr

i

0'

n"toog'io the sa-é finít,'

*equcrrcc

of

equations (5) since

theil first

t'hretr

;.',ñîâ"e

equal. Thcreforo

io their

gleâtest

loots

l,h.ere can

bc

zr,ppliectr' r,he asyrnpl,otic

relalion

({i.5) rvhich now has the following

foml

(66) t {

r''r*r,o*t (À)

- t"rr(\,

P),

\- -f

oo'

In tlte

same

tnânner

1,hc equal,iotls

,rri

+z?,,:;,'),) 0

anrl

(':')

&onz

- (oIt)".*('r) ", :

o

Eirrc

tho

rclation

(67) o<ri*r,r(À)-fj;(À,1), À-'f

co.

By

(33), (34), (35)

antl

(36)

the

absolute Yzr,lues o1 i,ht¡ oÏrlinates of'

the pai'aÈoÉ

u

:

ùor."(,^,,

À) at the

poinl,s -rr

:zffu.Ì', (À).L

r;:',(^,p)

S

<',"'";l\,p) {r;;:^)" itiéi"is" inclcfinitcl¡' rvith ]'

a1c1 zrccolcling

to

the

aforeïaíciihe

arìc

of l,his

parabola,

is

conc¿lt't:

in the itt¡r'val

þ"'o*r,n(\),

i', * ,,, + 2( À)) .

''

'"'"if'tnä

isquàr'e rool,s

in folmulae

(14) anti (23) are exptrtrcletl

in

series'

or

torrnulae 121¡ a,ntt (2,3)

alc

atlclccl

tìpr

-\\re obi,¿r,in

r',,()')

t'!,1,(\, n')

- Tffi;,

^

"-

foo

alrcl

in

vierv

of

1,ht', as¡rrppNoi,ic rolations (65) ancl (67) tve have

(68)

t't'r¡2,e+z ().)

-

1'!,i

*".,

()'))

-+ f oo'

7,

- *æ'

llhe arc oI the

palaboln,

'!J

!lr,+z(n, À)

iu the

inte,rval{r^í-tr,r(À), t'o*",r..r(),)\

is

callecl

tlie ,,last a,ic" of lhis

parabola'

thus,

1'he Îollorving theorcm has 'becn Plovecl :

llnnonnlr (4)t:

!l'h'e ,,Icr,st tr,r'a"

oJ

the

parabolú y-- ttr+z(3t.!)

I'ies'

he¡otu Lltø anis

X'r*,

ctnd,

tiíe

qbsoh,u,te, acLluc

ol

crclt, -ord,inate

oJ tlt'is

cw!

cts wel,l øs

the

tri"'¡¡ín

{lr*r'*r(i') -- t",l*r,r(T)) of theinterual {ri*r,r(\),

r o *2,e*z(^)1t increase'i ndeJirui,telg

willt'

)''

" ''-fn'íiôw

of ret¿i,tiooÑ 1OS¡,"100) anct (6?),

flom the

asymptotic

rt{ation

(30)

we

obtain

{69l !l-".r-rtÙ - ,'r.r,r, - 1,

}. ---+ ì-oo.

t"tt+2,t¡+r(1.)

- t'jl*r,,

(À)

4. ||he inequality rrr.zr(\)

{,t,zrt+t,2t¡t-'

(À)

antl

the

asyrnptotic lela-

Liotr't'2r, ,2o (ìt)

-

t'zn *t, ,,

*,

( À).

In virtue of

teorem (4)

it

follows

: l'he ,,last ârc" of the

parabola

.Jl

:

Uzt,

(r,

À) touches

the

axis

,Y* from the

sicle

of

the positive ordinates

for the value

),.

:

o'zp

of

the parameter, À. Ir'rom

this

rve d.ecluce the

_ 'lÌrr¡onnrr (!)

o.f euen d,egree

decrect,ses untl

if all

th¿n ¿¡r¡t Ttolyno-

m,ial has ot least

ott,

d, ,ualue

i :

qz,

of the parameter

À

can at'i,se zoh,eñ,

L

cotttimtes

to

increo,se incl,eJinitelu.

ft

is knormr from thc theory of algebraic equatioirs [5 | :

thc

presence

.of a positive

minimum or of

a negative maximum

of a

polynornial mea,ns

tho

existence

of trvo

conjugate compìex

roots.

Consequently

two'

coniu- ,grì,te complex

roots

are bouncl

to the ,,last

a,rc"

iÎ tÌris arc

cloes

not

cut

bhe axi-q

Xrr. Ileirce we

har.e

lnnonnlt (6). Tlte

truo cortluçlute atntple:t' t'oots wlticlt ut'e bounil, to lh,e

í{

the posil,i,ac put'ameter rni,n,í,rnum oJ th,e

t'

ta,lres llt,e ,,Iast a,rc)' t:q,ltte

},,:

oJ tlte ltcwa,bol,u dz, u,ncl tltese

tuo ll -

cott¡ttgate compler!l¿t, (u, ],.) aanislt, t'oots change

to

one ilou,hlc rectl t'oot.

Äs long as the

palameter

)' has a vA,lue rvhicrh is

infinitely little

cliffe-

rent

(9

-*

frorrr and

is

arbitralill'closc srnaller than to

ancl

arrrthe

above cur\rc

thc

axis

! : Xroírt

llzt, (n, theneighSou'hoodì,)r

l, :

o.r,

-

e,

of the last

clouble '-eal zerut. 'Llhus,

for'

À

:

oro

I e the

complex zeros

belonging

to the

positivc

minirnum

are situatecl-in

ttre

neighbourhood of l,he

last

double real zero, n'hile the other complex zeros,

if they

exist, have ,smallcl real

palts.

IIor ), )

o.r, th.e

,,last Alctt of the

curve

(70) ll:!lrr,*r(n,

À)

: *.yzn(fr,\) -ürr,*r,

À

)øru

påi,sses

through the

poinl;s 1'ro,rr,_r(\) at;id.

rrr,rr(À) of the

axis

Xr,

arrc{ lies belorv

this axis in

1;he

interval

between these

two poinl,s. The arc

of

the

culve (70) is

in

the

interval

lrr"r,rr(l'),

f

oorì cor cave ancl nronotonously ìncreasing. Owing

to this, the culve

(70) cuts the

axis Xro*r,lying

above

the

axis X22,7 a.'t, the poinl, rzp+r,zt,+t(7,) situatecl

to the rig[t

of the straight

\inen:!'zr,zr(À),

À

)

arr,, i.e.

{71)

o

{ rrr,rr(},) 1r'rr,*r,rr*r(l),

), )> d-zn.

'Ilre

point

of intelscctiorr of the tangent to the

cutve

U

:

!yp1,a1(n, Ì'),

at tho point

1'{rrn,ro()'),

-

(Lzt,+t} of the s}.stern

(Irr*r),

wiLh thc axis Xgr,+r

,tas the

abscissa

'(72) úrr*r(À)

_-'t'2t,.2t,

(^)

- ¡:1"^¡^l^ t t' Þ

a2r'

'We wish to pror.e th:lt

.rJi,,*r{rro,2r,(),),

À]

incleases

inclefinitel¡' with

?t.

Ay

(53) +vc have

(73) I'ro.rr(\)**oo, À*tæ

+0r )'---++oo

Y i, *rt

rrr(ì'),

)'\

(7)

92 B, S. TO1VIIE 12:

nnd L)y (52)

$+)

0

{

?'r, (À)

< rrr,rr,(}'),,

'i,

}

a-¡t,.

llro

va,lue oT

thc lilsl,

r'lclir.ale

y'"r*r(r,

À)

at

the

poiltt ,u:

r'rr,.',,( 7) is

(?5)

3l'rn*r',rzr,zo( ?'), ),1 =-

r\i,ri,$) '

'qr{r'rr,2r,(}'),27t

-

!1,'

( ì.

.{r t' - t

'Jlhe palrl:olir,

(?6) y :'q"(rr'12p -

1)

=

(2p

f [)c'0,r2

27ta,t

+ \2p -

])4.,

crtt¡

thc axis

,lo

at 1ùc ¡ioirtl's.lrr

-

?"r( ),,27t

- 1)alrtl

';r2

--

rt2(À,27r

-

-l-)-

In viltue

oT 1,he inequa,Iit¡' (26) rve ha,r'c

0 <

r'1,r()',

2p - l) { t'rr().),

).

> I,

¿urd

n'ith

lee'ltt'd 1,o 1hc¡ inrrqttalil,¡. (74)

it

lcrllol's

\77) 0 q ljr(r',

2p

- t) 1lrr(

).)

{ i'r,,,or(),), it

-). 'r.zr¡.

P¿¡r'a,bol¿1, (76),

for'

,t,

,-

t'zz(À), hrts 1,hc ol'dinatcr

(?s) r,-ir'rr(À).:Ì7r ll -- 2ør(I

F |/7,'z

-

- l)+

-i-

oo,

).

+ |

co.

lll-tt', olclinatcs ol'pzr,rtbolil (76) alcr 1;osii'ive

fol

rrr

i

t'',r(),r2'p

--

1) a,rld tìre¡' i¡cleasg rnolototrr)ttsly as ,u

- l-

co.

lhus,

olt'jttg Lo (74)r l'hc¡ro <¡rists tlÌe' irrequalil,y

'Qzttl'rr.rr\ì,)r 27t

- lI >

'rt"',1'zr(7,),2p

-

l

ì > 0. )'

)>' tr",r,

rtrul, ou'itrg

to

(?8), l't'orn 1.his

it

follorvs

(79)

r,,[t'2;,.2r,()'),2]t

- 1]

-1-

oo,

)' ---+

f

co.

l-i:orr¡ (7õ)

u,ntl

(7fì)

i1,

l'ollotr'¡: .

. ,i

(E0) ,

;U|p+: it'2r,2r,(À), 7')

-' -l

co, À

-' + co.

.':

lhc fonirulile (7:ì)

:tncl

(80) git'c :.

i

(81)

',trrnt(},)

-

t'2r,21,(À)l

+ 0,

).

-+

co

a,rrtl, sinccr thc, ctn'r't'

ll -.

i!/z.t)+,( ,', À) is croltciìt't' i1 tþe ilt1,e¡r.al 1r"lrtr,"r-t(\)r.

-l-"oÌ, ilt

vir'l,rLt'oT t,his concai-it¡-

rn(ì in virl,uc

ol'(71

)we

obtir,in

(S2) (l

.1 t'zn.rn(7.)

-r ,':,,.,.rr,,(À) {

/z,,nr

(À),

),

}

o.2r,.

thus, florn

(82),

in

virl.nu

tif

(81), rvc

ob1'ain

'

(8ll) t'21,.21,\)r) - t'21t+t rrn,(),), ), -> ..l- oo,

l'or'

1,hc r-rcga,1,i

rt'.

r'oots

of tlìo

g'rcfttost

lt'bsolute

\/a,ltto /'r, *r,r( À) rìn{l 1'r,,,,

(}.)

of' r'o¿rl ocÌrt¿¡tions

u'itlt

posii,ivc

ooolificictrts

:

I'

\

tt,,,;'tt'-'-

: 0, p :

21tt2n

!

1,

)-=0

13 ÂSYI\{PTOTIC R,IILAîJONS T'OR TIIE INDEFINITEËY INCN,EASING ZEROS [ì(i

ib car bc plovctl by

ureans

of

i,ìro s¿r,rne rnt¡thocl

[a]

th¿r,t

fol

thern l,helc erisl,s

tho

as-vrnpt,oi;ic lola,l,ion

1'2,+t,t(À)

-

?'2,,,r(

7r),

).

+ f

co.

Sincc tlre proofs for the inctlurr,Iities

)'2e,21,(1,)

.i

r2r,¡,.2p+z( )') :l,ìr(f i"2?)+3,2,

*r(À) {

I'2.p¡1,21t+1(À)

ale

r-er¡'

lcngthyr

\\'(', shir,ll

gil'c thetrt

itt sell,ìr'alrc par)er.

ft

shoulcl l¡t-r

lnentiont'tl thnt

1,he proofs

for'thcse

iut:r¡ut-

iifil¡s

l1¿¡'1r¡r i,heir' ìrasis

in

'Jìhoolenr (4).

lil l.'Ii r,. ti N o ti s

1. a n i ó, I , I¡iia alael¡ra lt iztlarr.lc. Narrðna l<rrjiga, Booglart 1954, pp irlr ó(i.

2. l, a rr t'e rr 1, FI., 'frttilt| d'AIgi:br¿. lonrcr ì ll, Gzrr.rthicr \¡iÌlnt's ot Fils, lmplinrenls Li lrr':rilos, I)n liulcau clcs Ìolg-iLuclos, dc l'(rcole poì¡-[c.chÌriquc, I)alis 189.1, pp. 112-11.1

ll. l\I:r l cl c r. \i., 'l'|rc Cìeonrclry ol llrc Zcros of a PoL¡¡trontittl in tt Oom¡tlc;t Yarinl¡lc. -\Iatht'ltr¿r

ticll SrLlvors. Nurnbel III, r\nrelicurr II¿'Lthcrrr¡Lical Socic[r', Nol'Yolk 1949, pp ](19- 110

4. 'l- o rrt i ó ll. li., Les rclnlions usgntplolír¡trcs ¡xtur lcs :ét os ct oÌssunl irtdéfinirtuttL des polt¡ti)

¡ncs tt cocll'iri<'nls posilils. Acad. jìo¡'. l:lclÍ1., llull. Cl. Sci.. 5(! S(r'ic, 'l'onrc Xt,lX, [JL'uxtl]cs

1 9(i3 8, lr p. 799 81 8

:i. Zr\!,rtsltitt,\1.L,.,Spraooòttilipoüislcttitn¡nt'lotltttttrt:sr:n.ji.jualgthruíct:sl:ilti l¡¡tttscendcn!¡tilti ut rtntt.jcu.ii.i. lìizrììrìtg..iz, Nloskvn 1960, p. 5¡J.

lìct:cir,ccì J \ t I983 Li tt i uc r s i I ¡¡ Bel¡¡ rarLc I:rttttllt¡ t'¡ )[rtlnutirrrl Iittgirt,, r itt,¡

ri,

\', '.

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