Expansion of functions in a system of polynomials biorthogonal on a closed contour with a system of functions regular at infinitely remote point
We study properties of the systems of polynomials constructed according to the schemes similar to the schemes used for the Bernoulli and Euler polynomials, formulate conditions for the existence of functions associated with polynomials and conditions of representation of polynomials by contour integ...
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| Sprache: | Ukrainisch Englisch |
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2010
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508848387784704 |
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| author | Sukhorolskyi, M. A. Сухорольський, М. А. |
| author_facet | Sukhorolskyi, M. A. Сухорольський, М. А. |
| author_sort | Sukhorolskyi, M. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:39:03Z |
| description | We study properties of the systems of polynomials constructed according to the schemes similar to the schemes used for the Bernoulli and Euler polynomials, formulate conditions for the existence of functions associated with polynomials and conditions of representation of polynomials by contour integrals, and present the classes of analytic functions expandable in series in the systems of polynomials. The expansions of functions are illustrated by examples. |
| first_indexed | 2026-03-24T02:31:44Z |
| format | Article |
| fulltext |
UDK 517.53.57
M. A. Suxorol\s\kyj (Nac. un-t „L\viv. politexnika”)
ROZVYNENNQ FUNKCIJ ZA SYSTEMOG POLINOMIV,
BIORTOHONAL|NYX NA ZAMKNENOMU KONTURI
Z SYSTEMOG REHULQRNYX U NESKINÇENNO
VIDDALENIJ TOÇCI FUNKCIJ
We investigate properties of systems of polynomials constructed according to schemes similar to the
Bernoulli and Euler polynomials. We formulate conditions for the existence of functions associated with
polynomials and conditions under which the polynomials are representable by contour integrals, and
present classes of analytic functions, that are expanded in series in the systems of polynomials. The
expansion of functions is illustrated by examples.
Yssledugtsq svojstva system polynomov, postroenn¥x po analohyçn¥m s polynomamy Bernul-
ly y ∏jlera sxemam. Sformulyrovan¥ uslovyq suwestvovanyq assocyyrovann¥x s polynomamy
funkcyj, uslovyq predstavlenyq polynomov konturn¥my yntehralamy y pryveden¥ klass¥
analytyçeskyx funkcyj, razlahaem¥x v rqd¥ po systemam polynomov. RazloΩenyq funkcyj
proyllgstryrovan¥ prymeramy.
Vstup. Metody rozvynennq analityçnyx funkcij u rqdy za systemamy polino-
miv u kompleksnij oblasti z vykorystannqm systemy asocijovanyx funkcij vy-
kladeno v roboti [1]. Vstanovleno umovy, za qkyx dlq zadano] systemy funkcij
isnu[ vidpovidna systema asocijovanyx funkcij (linijnyx funkcionaliv), biorto-
honal\nyx na zamknenomu konturi, ta umovy rozvynennq analityçnyx funkcij za
cymy systemamy. Vlastyvosti system polinomiv (harmoniçnyx, interpolqcijnyx,
Fabera, çastynnyx sum stepenevyx rqdiv ta inßyx), a takoΩ rozvynennq ana-
lityçnyx funkcij za cymy systemamy rozhlqnuto takoΩ v robotax [2 – 4]. U da-
nij roboti doslidΩeno vlastyvosti system polinomiv, pobudovanyx za analohiç-
nog z polinomamy Bernulli ta Ejlera sxemog
P z C a b zn n
k
k n k
k
k
n
n
( ) =
−
= =
∞
∑
0 0
, (1)
de z — kompleksna zminna, Cn
k
— binomial\ni koefici[nty, ak , bk — posli-
dovnosti dijsnyx çysel. Qkwo ak = 1 i b Bk k= — çysla Bernulli, to
P z B zn n( ) ( )= — polinomy Bernulli; qkwo Ω ak = 1, b Ek k= — çysla Ejlera,
to P zn ( ) = E zn ( ) — polinomy Ejlera.
Znajdeno ocinky dlq polinomiv i asocijovanyx z nymy funkcij, na osnovi
qkyx vstanovleno dostatni umovy rozvynennq analityçnyx funkcij za systema-
my polinomiv.
1. Vlastyvosti polinomiv. Doslidymo vlastyvosti systemy polinomiv (1),
koefici[nty qkyx zadovol\nqgt\ umovy
b0 0≠ , ak ≠ 0 , k = 0, 1, … ; (2)
lim
!k
kk
a
k
A
→∞
= , 0 ≤ A < ∞, lim
!k
kk
b
k
B
→∞
= , 0 ≤ B < ∞, (3)
i qkwo A = 0 abo B = 0, to vidpovidno
lim
k
k
k a a
→∞
= , 0 < a < ∞, lim
k
k
k b b
→∞
= , 0 ≤ b < ∞. (4)
U roboti [5] rozhlqnuto systemu polinomiv (1) za umovy b bk
k= .
© M. A. SUXOROL|S|KYJ, 2010
238 ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
ROZVYNENNQ FUNKCIJ ZA SYSTEMOG POLINOMIV … 239
Vlastyvist\ 1. Tvirna funkciq F z t( , ) =
t
n
P z
n
n n
!
( )
∞∑ systemy (1) ma[
vyhlqd
F z t F zt F z( , ) ( ) ( )= 1 2 ,
de
F z
a
n
zn
n
n
1
0
( )
!
=
=
∞
∑ , F z
b
n
zn
n
n
2
0
( )
!
=
=
∞
∑ (5)
— funkci], analityçni vidpovidno v oblastqx z
A
<
1
, z
B
<
1
.
Dovedennq. Analityçnist\ funkcij F z1( ) , F z2( ) vyplyva[ z umov (2), (3).
Zminyvßy porqdok pidsumovuvannq rqdiv u vyrazi dobutku funkcij (5), oderΩy-
mo tvirnu
F zt F t1 2( ) ( ) =
a
k
zt
b
m
t
a b
k n k
k
k
k m
m
m k n k
n k!
( )
! !( )!=
∞
=
∞
−
=
∑ ∑ =
−0 0
∞∞
=
∞
∑∑
k
n kt z
0
=
=
t
n
n a b
k n k
z
n
k n k
k
n
n
k
!
!
!( )!
−
==
∞
−∑∑
00
=
t
n
P z
n
n
n
!
( )
=
∞
∑
0
.
Vlastyvist\ 2. Nexaj vykonugt\sq umovy (2), (4). Todi magt\ misce inteh-
ral\ni zobraΩennq
P z
i
b zt t dtn
n( ) ( ) ( )= +∫
1
2
0 1
1
π
γ
Γ
, (6)
qkwo b bk
k= 0 , i
P z
i
a z t t dtn
n( ) ( ) ( )= +∫
1
2
0 2
2
π
γ
Γ
, (7)
qkwo a ak
k= 0 , de γ1( )t =
a
t
k
kk +=
∞∑ 10
, γ 2( )t =
b
t
k
kk +=
∞∑ 10
— analityçni
funkci] vidpovidno v oblastqx t a> , t b> ; Γ1 , Γ2 — kola t R= 1, a <
< R1 < ∞, t R= 2 , b < R2 < ∞.
Dovedennq. Analityçnist\ funkcij γ1( )t i γ 2( )t vyplyva[ z umov (4). U
spravedlyvosti formul (6) i (7) perekonu[mosq bezposeredn\og perevirkog.
Pislq pidstanovky vyraziv cyx funkcij u pravi çastyny formul (6), (7) i obçys-
lennq intehraliv oderΩymo polinomy u formi (1).
Vlastyvist\ 3. SpravdΩugt\sq nastupni ocinky:
P z m a z bn
n( ) ( )≤ +0 ε ε , (8)
qkwo vykonugt\sq umovy (2), (4);
P z M n A B zn
k n k k
k
n
( ) !≤ −
=
∑0
0
ε ε = M n
A z B
A z B
n n
0
1 1
! ε ε
ε ε
( ) −
−
+ +
, (9)
qkwo vykonugt\sq umovy (2), (3) i A ≠ 0 , B ≠ 0 ;
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
240 M. A. SUXOROL|S|KYJ
P z m n B
a z
B
n
n( ) ! exp≤
0 ε
ε
ε
, (10)
qkwo vykonugt\sq umovy (2) i lim
k
k
k a
→∞
= a, 0 < a < ∞, lim
!k
kk
b
k→∞
= B, 0 <
< B < ∞;
P z m n A z
b
A z
n
n
( ) ! exp≤ ( )
0 ε
ε
ε
, (11)
qkwo vykonugt\sq umovy (2) i lim
!k
kk
a
k→∞
= A, 0 < A < ∞, lim
k
k
k b
→∞
= b, 0 ≤
≤ b < ∞, de a aε ε= + , b bε ε= + , A Aε ε= + , B Bε ε= + , M 0 , m0 —
stali velyçyny, ε — qk zavhodno male dodatne çyslo.
Dovedennq. Zhidno z umovamy (4) ma[mo nerivnosti a m ak
k≤ 1 ε , bk ≤
≤ m bk
2 ε , mi = const. Ocinyvßy polinomy (1) z uraxuvannqm cyx nerivnostej,
oderΩymo
P zn ( ) ≤ C a b zn
k
k n k
k
k
n
−
=
∑
0
≤ m m C a b zn
k k n k k
k
n
1 2
0
ε ε
−
=
∑ =
= m m a z b
n
1 2 ε ε+( ) .
Zvidsy vyplyva[ nerivnist\ (8).
Nerivnist\ (9) oderΩymo, ocinyvßy vyraz polinomiv (1) z uraxuvannqm spiv-
vidnoßen\ (3), zapysanyx u vyhlqdi
a
k
M Ak k
!
≤ 1 ε ,
b
k
M Bk k
!
≤ 2 ε , Mi = const.
OtΩe,
P zn ( ) ≤ C a b zn
k
k n k
k
k
n
−
=
∑
0
≤ M M n A B zk n k k
k
n
1 2
0
! ε ε
−
=
∑ =
= M M n
A z B
A z B
n n
1 2
1 1
! ε ε
ε ε
( ) −
−
+ +
.
Nerivnist\ (10) oderΩymo, ocinyvßy vyraz polinomiv (1) z uraxuvannqm neriv-
nostej a m ak
k≤ 1 ε ,
b
k
M Bk k
!
≤ 2 ε .
Dijsno,
P zn ( ) ≤
n a b
k n k
z m M n
a B
k
z
k n k k
k
n k n k
k
k
!
!( )!
!
!
−
=
−
−
≤∑
0
1 2
ε ε
==
∑
0
n
≤
≤ m M n B
k
a z
B
n
k
k
1 2
0
1
!
!
ε
ε
ε
=
∞
∑ = m M n B
a z
B
n
1 2 ! expε
ε
ε
.
Analohiçno oderΩymo nerivnist\ (11). Ocinyvßy polinomy (1) z uraxuvannqm
nerivnostej
a
k
M Ak k
!
≤ 1 ε , b m bk
k≤ 2 ε , matymemo
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
ROZVYNENNQ FUNKCIJ ZA SYSTEMOG POLINOMIV … 241
P zn ( ) ≤ C a b zn
k
k n k
k
k
n
−
=
∑
0
= n
a b z
k n k
k n k
k
k
n
!
!( )!
−
= −∑
0
≤
≤ M m n
b A z
k
k n k
k
n
1 2
0
!
!
ε ε( ) −
=
∑ ≤ M m n A z
k
b
A z
n
k
k
1 2
0
1
!
!
ε
ε
ε
( )
=
∞
∑ =
= M m n A z
b
A z
n
1 2 ! expε
ε
ε
( )
.
Vlastyvist\ 4. Nexaj vykonugt\sq umovy (2) i ER — prostir funkcij
f z( ) , odnoznaçnyx i analityçnyx u kruzi z R< , 0 < R ≤ ∞. Todi systema po-
linomiv (1) [ povnog i nezaleΩnog u prostori ER .
Dovedennq. Systema stepeniv nezaleΩna i povna u prostori ER . Za vyko-
nannq umov (2) koefici[nty bilq starßyx stepeniv polinomiv ne dorivnggt\ nu-
levi i funkci] zk , k = 0, 1, … , odnoznaçno vyraΩagt\sq çerez polinomy P zn ( ) .
Tomu [1, s. 137] systema (1) povna i nezaleΩna u prostori ER .
2. Asocijovani systemy funkcij. Zapyßemo zaleΩnosti stepeniv zk vid
polinomiv systemy (1) u vyhlqdi
z
C g
a
P zk k
n
k n
kn
k
n= −
=
∑
0
( ) , (12)
de gk , k = 0, 1, … , — koefici[nty, wo vyznaçagt\sq z systemy rivnqn\
C C b g
n k
n k
k
r
r n
k
r
n
k r r n
=
− −∑ =
=
≠
1
0
, ,
, .
(13)
Nexaj
f z
f
n
z
n
n
n
( )
( )
!
=
( )
=
∞
∑ 0
0
(14)
— stepenevyj rqd analityçno] funkci] f z ER( ) ∈ . Znajdemo formal\ne rozvy-
nennq ci[] funkci] za systemog polinomiv (1). Pidstavyvßy vyrazy stepeniv (12)
u formulu (14) i zminyvßy porqdok pidsumovuvannq, oderΩymo
f z( ) =
f
n
C g
a
P z
f
n
n
n
k
n k
nk
n
k
n
n( ) ( )( )
!
( ) ~
( )
!
0 0
00
−
==
∞
∑∑
CC g
a
P z
n
k
n k
nn kk
k
−
=
∞
=
∞
∑∑
0
( ) .
(15)
Vvivßy poznaçennq
L f
f
n
C g
a
k
n
n
k
n k
nn k
( )
( )
!
( )
= −
=
∞
∑ 0
, (16)
zapyßemo formulu (15) u vyhlqdi
f z L f P zk
k
k( ) ~ ( ) ( )
=
∞
∑
0
. (17)
Qkwo dlq bud\-qkyx skinçennyx znaçen\ k isnu[ hranycq
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
242 M. A. SUXOROL|S|KYJ
lim
r
r
k r k
r
r C
g
a
S
→∞
− = , 0 ≤ S < ∞, (18)
to funkci]
ω k
r
k
r k
rr k
r
z
C g
a z
( ) = −
=
∞
+∑ 1
1 (19)
analityçni v oblasti z S> i spravedlyvi ekvivalentni rivnqnnqm (13) spivvid-
noßennq
1
2π
ω
i
P z z dzn k
Γ
∫ ( ) ( ) =
1
0
, ,
, ,
n k
n k
=
≠
(20)
de Γ — dodatno ori[ntovane kolo z R= 1 , S < R1 < R , wo oxoplg[ osoblyvi
toçky funkcij ω k z( ) .
Zaznaçymo [1], wo qkwo vykonugt\sq umovy (20), to systema polinomiv (1) i
systema asocijovanyx (pry[dnanyx) funkcij ω k kz( ){ } =
∞
0 nazyvagt\sq biortoho-
nal\nymy na zamknenomu konturi.
Takym çynom, koΩnij funkci] f z ER( ) ∈ , S < R ≤ ∞, moΩna postavyty u vid-
povidnist\ rqd (17), koefici[nty qkoho vyznaçagt\sq za formulog (16) abo za
formulog
L f
i
f z z dzk k( ) ( ) ( )= ∫
1
2π
ω
Γ
. (21)
Nastupna teorema vstanovlg[ we odyn sposib vyznaçennq koefici[ntiv aso-
cijovanyx funkcij.
Teorema 1. Nexaj poslidovnist\ çysel bk , k = 0, 1, … , zadovol\nq[ umo-
vyR(2), (3). Todi funkciq G z( ) =
g
k
k
k !=
∞∑ 0
, koefici[nty stepenevoho rqdu
qko] zadovol\nqgt\ rivnqnnq systemy (13), [ obernenog do funkci] F z2( ) i
analityçnog v deqkomu okoli poçatku koordynat,
G z
F z
( )
( )
=
1
2
. (22)
Dovedennq. PeremnoΩyvßy stepenevi rqdy funkcij F z2( ) i G z( ) ,
znajdemo
F z G z2( ) ( ) =
b g z
k n
n k
k n
nk
+
=
∞
=
∞
∑∑
! !00
=
b g z
k m k
m k k
m
m kk
−
=
∞
=
∞
−∑∑
!( )!0
=
= z
b g
k m k
m m k k
k
m
m
−
==
∞
−∑∑
!( )!00
.
Za vykonannq rivnosti (22) i nezaleΩnosti funkcij zm spravdΩugt\sq spivvid-
noßennq
b g
r m r
m r r
r
m
−
= −∑
!( )!0
=
1 0
0 0
, ,
, .
m
m
=
≠
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
ROZVYNENNQ FUNKCIJ ZA SYSTEMOG POLINOMIV … 243
Peretvoryvßy livi çastyny cyx spivvidnoßen\ z uraxuvannqm zaminy m = n – k,
b g
r m r
m r r
r
m
−
= −∑
!( )!0
=
b g
r n k r
n k r r
r
n k
− −
=
−
− −∑
!( )!0
=
=
k
n
C C b gn
k r
k r
k
n k r r
r
n k!
!
+
+ − −
=
−
∑
0
=
k
n
C C b gn
r
r
k
n r r k
r k
n!
!
− −
=
∑ ,
oderΩymo rivnqnnq (13). Analityçnist\ funkci] G z( ) v deqkomu okoli nul\o-
vo] toçky vyplyva[ z analityçnosti funkci] F z2( ) i umovy F2 0( ) = b0 ≠ 0.
Rozhlqnemo vlastyvosti asocijovanyx funkcij (19).
Vlastyvist\ 5. Nexaj vykonugt\sq umovy (2) i funkci] G z( ) ta F z1( )
taki, wo
lim
k
k
k g g
→∞
= , 0 ≤ g < ∞, lim
k
k
k a a
→∞
= , 0 < a < ∞. (23)
Todi pry[dnani funkci] ω k z( ) analityçni v oblasti z s> , s
g
a
= , i spravd-
Ωu[t\sq ocinka
ω
ε ε
k k
z
m
a z g
( ) ≤
−( )−
+
0
1 , (24)
de m0 = const, a−ε = a – ε, gε = g + ε.
Dovedennq. Za vykonannq umov (23) ma[mo a m ak
k≥ −1 ε , g m gk
k≤ 3 ε ,
mi = const. Ocinyvßy funkci] (19) z uraxuvannqm cyx nerivnostej, oderΩymo
nerivnist\
ω k z( ) =
C g
a z
r k
k
r
r kr
r k
+
+=
∞
+ +∑
0
1
1
≤
m
m
C g
a z
g r k
k r
r k
r
r k
1 0
1
1+
−
+
=
∞
+ +∑ ε
ε
=
=
a m
m a z g
g
k
−
−
+−
ε
ε ε1
1
1
( )
,
qka vykonu[t\sq za umovy
g
a z
ε
ε−
< 1 i, vidpovidno, z
g
a
>
−
ε
ε
. Z ci[] nerivnosti
vyplyvagt\ ocinka (24), a takoΩ, vnaslidok dovil\no] malosti velyçyny ε, ana-
lityçnist\ funkcij ω k z( ) v oblasti z s> .
Vlastyvist\ 6. Nexaj vykonugt\sq umovy (2) i funkci] G z( ) , F z1( ) taki,
wo
lim
!k
kk
g
k
G
→∞
= , 0 ≤ G < ∞, lim
!k
kk
a
k
A
→∞
= , 0 < A < ∞. (25)
Todi pry[dnani funkci] ω k z( ) analityçni v oblasti z S> , S
G
A
= , i
spravdΩu[t\sq ocinka
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
244 M. A. SUXOROL|S|KYJ
ω
ε ε
k k
z
M
k A z G z
( )
!
≤
−( )−
0 1
, (26)
de M 0 = const, A−ε = A – ε, Gε = G + ε.
Dovedennq. Ocinyvßy funkci] (19) z uraxuvannqm nerivnostej
a
k
k
!
≥
≥ M Ak
1 −ε ,
g
k
M Gk k
!
≤ 3 ε , Mi = const, qki vyplyvagt\ z umov (25), oderΩymo
ω k z( ) =
C g
a z
r k
k
r
r kr
r k
+
+=
∞
+ +∑
0
1
1
≤
M
k M A z
G
A zk k
r
r
3
1
1
0! −
+
−=
∞
∑
ε
ε
ε
=
=
M
k M A z A z Gk k
3
1
1
1
! −
− − −ε ε ε
.
Zvidsy vyplyvagt\ ocinka (26) i, vnaslidok dovil\no] malosti velyçyny ε, anali-
tyçnist\ funkcij ω k z( ) v oblasti z S> .
Vlastyvist\ 7. Nexaj vykonugt\sq umovy (2) i funkci] G z( ) , F z1( ) taki,
wo
lim
k
k
k g g
→∞
= , 0 ≤ g < ∞, lim
!k
kk
a
k
A
→∞
= , 0 < A < ∞. (27)
Todi pry[dnani funkci] ω k z( ) analityçni v oblasti z > 0 i spravdΩu[t\sq
ocinka
ω
ε
ε
ε
k k k
z
m
k A z
g
A z
( )
!
exp≤
−
+
−
0
1 , (28)
de m0 = const, a−ε = a – ε, gε = g + ε.
Dovedennq. Za vykonannq umov (27) ma[mo
a
k
M Ak k
!
≥ −1 ε , g m gk
k≤ 3 ε ,
M1 , m3 = const. Ocinyvßy funkci] (19) z uraxuvannqm cyx nerivnostej, oder-
Ωymo
ω k z( ) =
1 1
1
0k z
g
r
r k
a zk
r
r r k
r! !
( )!
+
=
∞
+
∑ +
≤
≤
m
k M A z r
g
A z
g
k k
r
r! !1
1
0
1
−
+
−=
∞
∑
ε
ε
ε
=
m
k M A z
g
A zk k
3
1
1!
exp
−
+
−
ε
ε
ε
.
Z ci[] nerivnosti vyplyvagt\ ocinka (28) i analityçnist\ funkci] ω k z( ) v ob-
lasti z > 0 .
3. Rozvynennq analityçnyx funkcij. Rozhlqnemo dostatni umovy rozvy-
nennq funkcij f z ER( ) ∈ , 0 < R ≤ ∞, za systemog polinomiv (1).
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
ROZVYNENNQ FUNKCIJ ZA SYSTEMOG POLINOMIV … 245
Teorema 2. Nexaj vykonugt\sq umovy (2), (3), lim
!k
kk
g
k
G
→∞
= , 0 < G < ∞,
i f z ER( ) ∈ , S2 < S < R ≤ ∞, de S
G
A
= , S
B
A
2 = .
Todi u kruzi z R< spravdΩu[t\sq formula
f z L f P zk
k
k( ) ( ) ( )=
=
∞
∑
0
, (29)
de L fk ( ) — koefici[nty, wo vyznaçagt\sq za formulog (16) abo (21).
Dovedennq. Stepenevyj rqd (14) rivnomirno zbiha[t\sq v oblasti z R≤ −ε ,
0 < R−ε = R – ε. Peretvorymo joho z uraxuvannqm formuly (12):
f z
f
n
C g
a
P z
n
n
k
n k
nk
n
k
n
( )
( )
!
( )
( )
= −
==
∞
∑∑ 0
00
. (30)
Za umov teoremy spravedlyvog [ ocinka (26) i, vidpovidno, funkci] ωn z( ) anali-
tyçni v oblasti z S> . OtΩe, magt\ misce formuly (16) i (21). Vykorystavßy
nerivnosti M Ak
1 −ε ≤
a
k
k
!
≤ M Ak
1 ε ,
g
k
M Gk k
!
≤ 3 ε ,
f
k
k( )( )
!
0
≤
M
Rk
4
−ε
, Mi =
= const, ocinymo sumu rqdu (30):
f
n
C g
a
P z
n
n
k
n k
nk
n
k
n
( )( )
!
( )
0
00
−
==
∞
∑∑ ≤
≤
M M
M A R
G P z
kn
n
n k
k
k
n
3 4
1 0 0
1
( )
( )
!− −=
∞ −
=
∑ ∑
ε ε
ε =
=
M M
M
P z
k G
G
A R
k
k
k
n
n k
3 4
1 0
( )
! ε
ε
ε ε=
∞
− −=
∞
∑ ∑
≤
≤
M M
M
P z
k G
G
A R
k
k
k
n
n
3 4
1 0 0
( )
! ε
ε
ε ε=
∞
− −=
∞
∑ ∑
=
=
M M
M
G
A R
P z
k G
n
n
k
k
k
3 4
1 0 0
ε
ε ε ε− −=
∞
=
∞
∑ ∑ ( )
!
≤
≤ M
G
A R
B
G
A z
B
m
m
k
k
ε
ε ε
ε
ε
ε
ε− −=
∞
=
∞
∑ ∑
0 0
=
∑
n
n
k
0
=
= M
G
A R
A z
B
B
G
m
m
n
n
ε
ε ε
ε
ε
ε
ε− −=
∞
=
∞
∑ ∑
0 0
=
∞
∑
k
k n
≤
≤ M
G
A R
A z
B
B
G
m
m
n
n
ε
ε ε
ε
ε
ε
ε− −=
∞
=
∞
∑ ∑
0 0
=
∞
∑
k
k 0
=
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
246 M. A. SUXOROL|S|KYJ
= M
G
A R
A z
B
B
G
1 1 1
1 1
−
−
−
− −
− −
ε
ε ε
ε
ε
ε
ε
−1
, M
M M M
M
= 3 4 0
1
.
Za vykonannq umov
G
A R
ε
ε ε− −
< 1,
A z
B
ε
ε
< 1 ,
B
G
ε
ε
< 1 oderΩani rqdy zbihagt\sq.
Vraxovugçy tut dovil\nu malist\ velyçyny ε, moΩna stverdΩuvaty, wo rqd
(27) zbiha[t\sq rivnomirno u kruzi z ≤ R0 < R za umov S2 < S < R ≤ ∞.
Naslidok. Za umov teoremyR2 spravedlyvym [ rozvynennq
1
0t z
t P zk k
k−
=
=
∞
∑ ω ( ) ( ) , (31)
qke rivnomirno zbiha[t\sq pry z r≤ i t ≥ ρ , de r i ρ — bud\-qki çysla,
wo zadovol\nqgt\ umovy 0 < r < R, ρ > max ( , )r S .
Dovedennq. Rivnomirna zbiΩnist\ rqdu (31) vyplyva[ z ocinok (9) i (26). Jo-
ho sumu znajdemo, vykorystavßy formuly (12) i (19):
ω k k
k
t P z( ) ( )
=
∞
∑
0
=
C g
a t
P z
r
k
r k
rr k
r k
k
−
=
∞
+
=
∞
∑∑ 1
1
0
( ) =
=
1
1
00 t
C g
a
P z
r
r
k
r k
rk
r
k
r
+
−
==
∞
∑∑ ( ) =
z
t
r
r
r
+
=
∞
∑ 1
0
=
1
t z−
.
Teorema 3. Nexaj vykonugt\sq umovy (2),
lim
!k
kk
a
k
A
→∞
= , 0 < A < ∞, lim
k
k
k b b
→∞
= , 0 ≤ b < ∞,
lim
!k
kk
g
k
G
→∞
= , 0 ≤ G < ∞, i f z ER( ) ∈ , S < R ≤ ∞, S
G
A
= .
Todi u kruzi z R< ma[ misce rozvynennq (29), koefici[nty qkoho vyzna-
çagt\sq za formulog (16) abo (21).
Dovedennq. Spravedlyvist\ formul (16) i (21) zabezpeçu[t\sq (qk i u pope-
rednij teoremi) vykonannqm nerivnosti (26). Ocinymo sumu rqdu (30) z uraxuvan-
nqm nerivnosti (11) i nerivnostej M Ak
1 −ε ≤ Ak ≤ M Ak
1 ε ,
g
k
k
!
≤ M Gk
3 ε ,
f
k
k( )( )
!
0
≤
M
Rk
4
−ε
, mi , M j = const, qki vyplyvagt\ z umov teoremy. Takym çy-
nom, ma[mo ocinku
f
n
C g
a
P z
n
n
k
n k
nk
n
k
n
( )( )
!
( )
0
00
−
==
∞
∑∑ ≤
M M
M R A
G
k
P z
n n
n
n k
k
n
k
3 4
1 0 0
1
ε ε
ε
−=
∞ −
=
∑ ∑
!
( ) ≤
≤ m
b
A z R A
G A z
n
n k k
k
n
n
exp
( )
ε
ε ε ε
ε ε
( )
− −
−
==
∑1
000
∞
∑ =
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
ROZVYNENNQ FUNKCIJ ZA SYSTEMOG POLINOMIV … 247
= m
b
A z
A z
G
G
R A
k
k
exp ε
ε
ε
ε
ε
ε ε
=
∞
− −
∑
0
=
∞
∑
n
n k
≤
≤ m
b
A z
A z
G
G
R A
k
k
exp ε
ε
ε
ε
ε
ε ε
=
∞
− −
∑
0
=
∞
∑
n
n 0
, m
M M m
M
= 3 4 0
1
,
qka spravdΩu[t\sq za umov z
G
A
< ε
ε
,
G
A
Rε
ε
ε
−
−< . Vnaslidok dovil\nosti çysla
ε > 0 rqd (29) rivnomirno zbiha[t\sq do funkci] f z( ) v oblasti z ≤ R0 < R .
Teorema 4. Nexaj vykonugt\sq umovy (2), (4), lim
k
k
k g
→∞
= g, 0 ≤ g < ∞, i
f z ER( ) ∈ , s0 < R ≤ ∞, de s0 =
b g
a
+
. Todi u kruzi z < R – s0 ma[ misce
rozvynennq (29), koefici[nty qkoho vyznaçagt\sq za formulog (16) abo (21).
Dovedennq. Oskil\ky vykonu[t\sq nerivnist\ (24), funkci] ω k z( ) anali-
tyçni v oblasti z
g
a
> i, vidpovidno, spravedlyvymy [ formuly (16) i (21).
Rozhlqnemo rqd (30), qkyj oderΩano z stepenevoho rqdu (14), rivnomirno zbiΩ-
noho v oblasti z R≤ −ε , 0 < R−ε = R – ε. Peretvorymo joho z uraxuvannqm ne-
rivnosti (8) i nerivnostej m ak
1 −ε ≤ ak ≤ m ak
1 ε , gk ≤ m gk
3 ε ,
f
k
k( )( )
!
0
≤
M
Rk
4
−ε
,
mi , M 4 = const, wo vyplyvagt\ z umov teoremy:
f z( ) =
f
n
C g
a
P z
n
n
k
n k
nk
n
k
n
( )( )
!
( )
0
00
−
==
∞
∑∑ ≤
≤
M m
m R a
C g P z
n
n
n
k n k
k
n
k
4 3
1 0 0
1
( )
( )
− −=
∞
−
=
∑ ∑
ε ε
ε =
= m
R a
C g a z b
n
k
n
k n k k
k
n1
0 0( )− −=
∞
−
=
∑ ∑ +( )
ε ε
ε ε ε =
= m
a z b g
R an
n
ε ε ε
ε ε
+ +
− −=
∞
∑
0
, m
M m m
m
= 4 3 0
1
.
Suma oderΩanoho maΩorantnoho rqdu isnu[ za umovy z <
a
a
−ε
ε
R−
ε –
–
b g
a
ε ε
ε
+
−
. Vnaslidok dovil\no] malosti velyçyny ε rqd (30) i, vidpovidno, rqd
(29) rivnomirno zbihagt\sq v oblasti z ≤ R0 < R – s0 .
Teorema 5. Nexaj vykonugt\sq umovy (2), (4), lim
!k
kk
g
k→∞
= G , 0 < G < ∞, i
f z( ) — funkciq eksponencial\noho typu, lim ( )( )
k
kk f
→∞
0 = σ, 0 ≤ σ <
a
G
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
248 M. A. SUXOROL|S|KYJ
Todi v oblasti z < ∞ ma[ misce rozvynennq (29), koefici[nty qkoho vy-
znaçagt\sq za formulog (16).
Dovedennq. Za umovog 0 < a < ∞ i g = ∞, tomu rqdy (19) rozbihagt\sq.
Ocinymo bezposeredn\o sumu rqdu (30), oderΩanoho z rivnomirno zbiΩnoho v ob-
lasti z R≤ −ε , 0 < R−ε = R – ε, stepenevoho rqdu (14) z uraxuvannqm formu-
lyR(12). Vraxovugçy nerivnist\ (8) i nerivnosti m ak
1 −ε ≤ ak ≤ m ak
1 ε ,
g
k
k
!
≤
≤ M Gk
3 ε , f k( )( )0 ≤ m k
4σε , mi , M j = const, qki vyplyvagt\ z umov teoremy,
oderΩu[mo
f z( ) =
f
n
C g
a
P z
n
n
k
n k
nk
n
k
n
( )( )
!
( )
0
00
−
==
∞
∑∑ ≤
≤ M
G
a
a z b
k G
n n
n
n
k
k
k
nσε ε
ε
ε ε
ε−=
∞
=
∑ ∑ +( )
0 0 !
≤
≤ M
G
a
a z b
k G
n n
n
n
k
k
k
σε ε
ε
ε ε
ε−=
∞
=
∞
∑ ∑ +( )
0 0 !
=
= M
G
a
a z b
G
1
1
−
+
−
−
σε ε
ε
ε ε
ε
exp , M
M m m
m
= 3 0 4
1
.
Cq ocinka spravdΩu[t\sq za umovy σε
ε
ε
< −a
G
. OtΩe, rqd (30) dlq cilo] funkci]
zbiha[t\sq rivnomirno v bud\-qkij skinçennij oblasti z R≤ 0 < ∞, qkwo til\-
ky σ <
a
G
. Za cyx umov v oblasti z R≤ 0 < ∞ takoΩ rivnomirno zbiha[t\sq
rqd (29).
Teorema 6. Nexaj vykonugt\sq umovy (2), lim
k
k
k a
→∞
= a , 0 < a < ∞,
lim
!k
kk
b
k→∞
= B, 0 ≤ B < ∞, lim
k
k
k g
→∞
= g, 0 ≤ g < ∞, i f z( ) — funkciq eks-
ponencial\noho typu, lim ( )( )
k
kk f
→∞
0 = σ, 0 ≤ σ <
a
B
.
Todi v oblasti z < ∞ ma[ misce rozvynennq (29), koefici[nty qkoho vy-
znaçagt\sq za formulamy (16) abo (21).
Dovedennq. Za vykonannq umov teoremy spravdΩugt\sq ocinky (10), (24) i
pry[dnani funkci] ω k z( ) analityçni v oblasti z
g
a
> . Tomu spravedlyvymy [
formuly (16) i (21). Ocinymo sumu rqdu (30) z uraxuvannqm nerivnosti (10) i ne-
rivnostej m ak
1 −ε ≤ ak ≤ m ak
1 ε , gk ≤ m gk
3 ε , f k( )( )0 ≤ m k
4σε , mi , M j =
= const, wo vyplyvagt\ z umov teoremy:
f
n
C g
a
P z
n
n
k
n k
nk
n
k
n
( )( )
!
( )
0
00
−
==
∞
∑∑ ≤
f
n
C g
a
P z
n
n
k
n k
nk
n
k
n
( )( )
!
( )
0
00
−
==
∞
∑∑ ≤
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
ROZVYNENNQ FUNKCIJ ZA SYSTEMOG POLINOMIV … 249
≤ M
a z
B a
g B
n
n
n
n k k
exp
(
ε
ε
ε
ε
ε εσ
−−=
∞ −
∑
0 kkk
n
)!=
∑
0
≤
≤ M
a z
B
B
a k
g
n
n k
exp
!
ε
ε
ε ε
ε
σ
−=
∞
=
∞
∑ ∑
0 0
1 εε
εB
=
= M
a z g
B
B
a
n
n
exp ε ε
ε
ε ε
ε
σ+
−=
∞
∑
0
= M
a z g
B
B
a
exp ε ε
ε
ε ε
ε
σ+
−
−
−
1
1
,
M
m m m
m
= 0 3 4
1
.
Takym çynom, za umovy σ <
a
B
rqd (30) zbiha[t\sq i, vidpovidno, rqd (29) zbi-
ha[t\sq rivnomirno do funkci] f z( ) u bud\-qkomu kruzi skinçennoho radiusa.
4. Rozvynennq funkcij za systemamy polinomiv typu Bernulli — Ejle-
ra. Rozhlqnemo systemu polinomiv
P z C a b zn n
k k
n k
k
k
n
n
( ) =
−
= =
∞
∑ 0
0 0
, (32)
qka [ çastynnym vypadkom systemy (1) za umovy a ak
k= 0 . Tvirna funkciq sys-
temy (32) nabere vyhlqdu
t
n
P z
a
k
zt
b
m
t
n
n
n
k
k m
mk
m
!
( )
!
( )
!=
∞
=
∞
=
∞
∑ ∑∑=
0
0
00
.
Zvidsy F zt1( ) = ea zt0 , F t2( ) =
b
m
tm m
m !=
∞∑ 0
.
A. Nexaj vykonugt\sq umovy teoremyR4. Todi polinomy systemy (29) moΩna
zobrazyty u vyhlqdi konturnoho intehrala (7).
Dlq pry[dnanyx funkcij oderΩymo z formuly (19) taki vyrazy:
ω k z( ) =
C g
a z
r k
k
r
r k r k
r
+
+ + +
=
∞
∑
0
1
0
1
=
( )
!
−1
0
0
k
k
k
kk a
d
dz
ω
=
( ) ( )
( )
−
− +∫
1 1
20
0
1
2
k
k ka i
t dt
t zπ
ω
Γ
,
(33)
de ω0( )z =
g
a
r
rr
0
0=
∞∑
1
1zr + , z > s =
g
a0
.
Koefici[nty rozvynennq (29) funkci] f z ER( ) ∈ , s < R ≤ ∞, otryma[mo z (19)
u vyhlqdi
L fk ( ) =
1 0
0k
f g
n k a
n
n k
n
n k!
( )
( )!
( )
−
=
∞
−
∑ =
1 1
20
0
2
k a i
d f t
dt
t dt
k
k
k!
( )
( )
π
ω
Γ
∫ . (34)
B. Vykonannq umov teoremyR5 takoΩ zabezpeçu[ intehral\ne zobraΩennq po-
linomiv u vyhlqdi (7), odnak rqdy, wo zobraΩugt\ pry[dnani funkci], rozbi-
hagt\sq. Tomu dlq koefici[ntiv rozvynennq funkci] eksponencial\noho typu
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
250 M. A. SUXOROL|S|KYJ
f z( ) , 0 ≤ σ <
a
G
0 , za systemog polinomiv (29) moΩna vykorystaty til\ky for-
mulu (16), qka nabere vyhlqdu
L f
k
f g
n k a
k
n
n k
n
n k
( )
!
( )
( )!
( )
=
−
−
=
∞
∑1 0
0
.
V. Nexaj vykonugt\sq umovy teoremyR6. Todi polinomy ne moΩna zobrazyty
u vyhlqdi konturnoho intehrala (7). Dlq pry[dnanyx funkcij ta koefici[ntiv
rozvynennq funkci] eksponencial\noho typu f z( ) , 0 ≤ σ <
a
B
0 , magt\ misce
formuly (33) i (34).
Pryklad 1. Nexaj b zk
k= −( )0 i a0 1= . Todi
P zn ( ) = C z zn
k
k
n
n k k
=
−∑ −
0
0( ) = ( )z z n− 0 , F zt t e ezt z t( , ) = − 0 ,
F t
t
k
e
k
k
t
1
0
( )
!
= =
=
∞
∑ , F t2( ) =
( )
!
−
=
∞
∑ z
k
t
k
k
k0
0
= e z t− 0 ,
G t( ) = ez t0 =
z
k
t
k
k
k0
0 !=
∞
∑ .
Zvidsy g zk
k= 0 , a = 1, b z= 0 , g z= 0 . Za formulog (33) znajdemo
ω0( )t =
z
t
k
kk
0
10 +=
∞∑ =
1
0t z−
. Dlq vyznaçennq koefici[ntiv rozvynennq anali-
tyçno] funkci] f z ER( ) ∈ , R > 2 0z , za systemog polinomiv (32) oderΩymo z
(34) formulu
L fn ( ) =
1 1
2
1
2
0n i
d f t
dt t z
dt
n
n!
( )
π Γ
∫ −
=
1 0
n
d f z
dt
n
n!
( )
i, vidpovidno, rqd (29), qkyj zhidno z teoremogR4 zbiha[t\sq u kruzi z R< –
– 2 0z . OtΩe, koefici[nty rozvynennq funkci] f z( ) za polinomamy P zn ( ) =
= ( )z z n− 0 [ koefici[ntamy Tejlora ci[] funkci],
f z
n
d f z
dt
z z
n
n
n
n( )
!
( )
( )= −
=
∞
∑ 1 0
0
0 .
ZauvaΩymo, wo teoremaR4 vstanovlg[ kruh z centrom u poçatku koordynat,
vseredyni qkoho vidpovidnyj rqd rivnomirno zbiha[t\sq.
Pryklad 2 . Rozhlqnemo systemu polinomiv Ejlera E zn ( ) =
= Cn
k
k
n
=∑ 0
E zn k
k
− , qki [ koefici[ntamy rozvynennq vidpovidno] tvirno] funk-
ci]
2
1
e
e
zt
t+
= E z
t
n
n
n
n
( )
!=
∞∑ 0
. Za formulamy (5) i (22) znajdemo
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
ROZVYNENNQ FUNKCIJ ZA SYSTEMOG POLINOMIV … 251
F t et
1( ) = , F t
et2
2
1
( ) =
+
, G t( ) =
1
2F t( )
=
1
2
1
2
1
0
+
=
∞
∑
k
t
k
k
!
.
Zvidsy ma[mo a0 1= , g0 1= , gk =
1
2
, k ≥ 1, i za formulog (30) znajdemo
ω0( )t =
1
2
1 1
1t t
+
−
. Oskil\ky vykonugt\sq umovy teoremyR6, ma[mo hranyci
vidpovidnyx poslidovnostej a = 1, B =
2
π
, g = 1. Koefici[nty rozvynennq
funkci] eksponencial\noho typu f z( ) , σ
π
<
2
, znaxodymo za formulog (34):
L fn ( ) =
1 1
4
1 1
1
2
n i
d f z
dz z z
dz
k
k!
( )
π Γ
∫ +
−
=
1 0 1
2n
f fn n
!
( ) ( )( ) ( )+
.
Rozvynennq funkci] eksponencial\noho typu za systemog polinomiv Ejlera zbi-
ha[t\sq u bud\-qkomu kruzi skinçennoho radiusa i ma[ vyhlqd
f z
f f
n
E z
n n
n
n
( )
( ) ( )
!
( )
( ) ( )
=
+
=
∞
∑1
2
0 1
0
.
Pryklad 3. Rozvynennq funkcij za systemog polinomiv Bernulli B zn ( ) =
= C B zn
k
n kk
n k
−=∑ 0
. Tvirna funkciq systemy polinomiv [ takog:
te
e
B z
t
n
zt
t n
n
n−
=
=
∞
∑
1 0
( )
!
.
Zvidsy ma[mo F t et
1( ) = , F t2( ) =
t
et − 1
, G t( ) =
e
t
t − 1
=
t
k
k
k ( )!+=
∞∑
10
i, vid-
povidno, ω0( )t =
1
10 kk +=
∞∑
1
1t k + = ln
t
t − 1
. Znaxodymo hranyci poslidov-
nostej koefici[ntiv vidpovidnyx rqdiv a = 1, g = 1, B =
2
π
. Oskil\ky spravd-
Ωugt\sq umovy teoremyR6, pidstavyvßy vyraz funkci] ω0( )t u formulu (34),
znajdemo koefici[nty rozvynennq eksponencial\no] funkci] f z( ) , σ
π
<
2
:
L f f z dz0
0
1
( ) ( )= ∫ , L fn ( ) =
1 1
4 1
2
n i
d f z
dz
z
z
dz
n
n!
( )
ln
π Γ
∫ −
=
=
1 1
4
1
1
11
1
2
n i
d f z
dz t t
dz
n
n!
( )
π
−
−∫ −
−
Γ
=
1
1 01 1
n
f fn n
!
( ) ( )( ) ( )− −− , n ≥ 1.
Ma[mo takyj rqd funkci] f z( ) za systemog polinomiv Bernulli:
f z( ) = f t dt( )
0
1
∫ +
f f
n
B z
n n
n
n
( ) ( )( ) ( )
!
( )
− −
=
∞ −∑
1 1
1
1 0
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
252 M. A. SUXOROL|S|KYJ
5. Rozvynennq funkcij za systemog polinomiv typu Mellina. Rozhlqne-
mo systemu polinomiv
P z C a b zn n
k
k
n k k
k
n
n
( ) =
−
= =
∞
∑ 0
0 0
, (35)
qku oderΩymo z (1) za umovy b bk
k= 0 , b0 0≠ . Dlq tvirno] funkci] systemy po-
linomiv (35) oderΩymo vyraz
t
n
P z
n
n
n
!
( )
=
∞
∑
0
=
a zt
k
b t
m
k
k
k
m
m
( )
!
( )
!=
∞
=
∞
∑ ∑
0
0
0
.
Zvidsy F z1( ) =
a z
k
k
k
k !=
∞∑ 0
, F z2( ) = ebz
, G z( ) = e b z− 0 i, vidpovidno, gk =
= ( )−b k
0 , b = b0 , g = b0 .
Pry[dnani funkci] znajdemo za formulog (19) u vyhlqdi ω k z( ) =
=
C b
a
n
k n k
n
n k
( )− −
=
∞∑ 0
1
1zn + abo
ω k z( ) =
1
0
0
k b
d
dz
z z
k
k
k
k
!
( )ω( ) =
1 1
20
0
1b i
t t dt
t zk
k
kπ
ω ( )
( )− +∫
Γ
, (36)
de ω0( )t =
( )−
+=
∞∑ b
a t
n
n
nn
0
10
.
Koefici[nty rozvynennq analityçno] funkci] f z( ) za systemog polinomiv
(35) ßuka[mo za formulog (21) abo z uraxuvannqm (36) — za formulog
L fk ( ) =
1
2π
ω
i
f z z dzk( ) ( )
Γ
∫ =
=
1 1
20
0
k b i
f z
d
dz
z z dz
k
k
k
k
!
( ) ( )
π
ω
Γ
∫ ( ) =
1
2πi
d f z
dz
z dz
k
k k
( )
( )Φ
Γ
∫ , (37)
de Φk t( ) =
( )−
− +=
∞∑ b
a t
n
n
n kn k
0
1 .
A. Nexaj spravdΩugt\sq umovy teoremyR3 i, vidpovidno, lim
!k
kk
a
k→∞
= A,
0 < A < ∞. Dlq pry[dnanyx funkcij magt\ misce formuly (36) v oblasti
z > 0 , oskil\ky
lim
k k
k
a→∞
1
= lim
!
!k k
k
k
a k→∞
1
=
1 1
A kk
klim
!→∞
= 0.
B. Nexaj vykonugt\sq umovy teoremyR4, lim
k
k
k a
→∞
= a, 0 < a < ∞. Todi po-
linomy (35) moΩna zapysaty v intehral\nij formi (6).
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
ROZVYNENNQ FUNKCIJ ZA SYSTEMOG POLINOMIV … 253
Dlq funkci] f z ER( ) ∈ , s < R ≤ ∞, s
b
a
=
2 0 , spravedlyvog [ formula (29)
v oblasti z < R –
2 0b
a
. Koefici[nty rozvynennq ci[] funkci] za systemog
polinomiv (35) ßuka[mo za formulamy (37).
Dlq funkci] f z( ) , analityçno] v oblasti z < R , 0 < R < ∞, ma[ misce
formula (29), a koefici[nty rozvynennq za systemog polinomiv (35)
vyznaçagt\sq za formulamy (37).
Pryklad 4. Rozhlqnemo funkcig f z( ) , analityçnu v oblasti z < R, zi
stepenevym rqdom
f z A zk
k
k( ) =
=
∞
∑
0
, (38)
koefici[nty qkoho zadovol\nqgt\ umovy Ak ≠ 0 , k = 0, 1, … , lim
k
k
k A
→∞
= A,
0 < A < ∞, i, vidpovidno, R =
1
A
. Znajdemo rozvynennq funkci] f z( ) za syste-
mog polinomiv (35), koefici[nty qkyx [ koefici[ntamy stepenevoho rqdu (38),
a Ak k= , tobto spravdΩugt\sq umovy teoremy 4, b = g = b0 , a = A.
Pry[dnani funkci] ω k z( ) , zhidno z formulog (36), magt\ vyhlqd
ω k z( ) =
( )− −
=
∞
∑ b C
A
r k
r
k
rr k
0
1
1zr + .
Pidstavyvßy ]x u formulu (37) i vraxuvavßy (38), oderΩymo
L fn ( ) =
( ) ( )
!
( )− −
=
∞
∑ b f
n
C
A
n k n
n
k
nn k
0 0
= ( )− −
=
∞
∑ b Cn k
n
k
n k
0 =
1
1 0
1+( ) +b k
.
OtΩe, ma[mo rozvynennq
f z
b
C b A z
m
m
n
k n k
k
k
k
n
( )
( )
=
+ +
=
∞
−
=
∑ ∑1
1 0
1
0
0
0
,
qke [ peretvorennqm Ejlera [6] stepenevoho rqdu funkci] f z( ) .
Vysnovky. Rozklad funkcij za systemog polinomiv, biortohonal\nog z de-
qkog inßog systemog (asocijovanyx) funkcij, [ rozvynennqm metodu rozkladu
funkcij u stepenevi rqdy (za dodatnymy stepenqmy zminno]) v kompleksnij ob-
lasti. Pry c\omu systema asocijovanyx funkcij dlq systemy stepeniv, qk vydno
z prykladu 1, [ systemog vid’[mnyx stepeniv ci[] zminno]. MoΩna takoΩ rozvy-
vaty funkci] v rqdy za systemog asocijovanyx z polinomamy funkcij, dlq qko]
asocijovanymy [ polinomy.
ZauvaΩymo, wo dlq bud\-qko] nezaleΩno] i povno] systemy funkcij za pev-
nyx umov [1] moΩna pobuduvaty vidpovidnu systemu asocijovanyx funkcij i
konstrugvaty rqdy za ci[g systemog funkcij.
Intehral\ne zobraΩennq (6) abo (7) polinomiv i, vidpovidno, intehral\ne zob-
raΩennq sum rqdiv za systemog polinomiv leΩyt\ v osnovi [7] pobudovy rozv’qz-
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
254 M. A. SUXOROL|S|KYJ
kiv (u vyhlqdi konturnyx intehraliv) zvyçajnyx dyferencial\nyx rivnqn\ ta riv-
nqn\ z çastynnymy poxidnymy.
1. Markußevyç A. Y. Yzbrann¥e hlav¥ teoryy analytyçeskyx funkcyj. – M.: Nauka,1976. –
192 s.
2. Hajer D. Lekcyy po teoryy approksymacyy v kompleksnoj oblasty. – M.: Myr, 1986. –
216Rs.
3. Dzqd¥k V. K. Vvedenye v teoryg ravnomernoho pryblyΩenyq funkcyj polynomamy. – M.:
Nauka, 1977. – 512 s.
4. Uolß DΩ. Ynterpolqcyq y approksymacyq racyonal\n¥my funkcyqmy v kompleksnoj ob-
lasty. – M.: Yzd-vo ynost. lyt., 1961. – 508 s.
5. Suxorol\s\kyj M. A. Rozvynennq analityçnyx funkcij za systemamy polinomiv typu Melli-
na // Visn. un-tu „L\viv. politexnika”. Ser. fiz.-mat. nauky. – 2005. – # 346. – S. 111 – 115.
6. Suxorol\s\kyj M. A. Oblast\ zbiΩnosti peretvorennq Ejlera stepenevoho rqdu analityçno]
funkci] // Ukr. mat. Ωurn. – 2008. – 60, # 8. – S. 1145 – 1152.
7. Kamke ∏. Spravoçnyk po ob¥knovenn¥m dyfferencyal\n¥m uravnenyqm. – M.: Nauka,
1974. – 576 s.
OderΩano 02.07.07,
pislq doopracgvannq — 30.11.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 2
|
| id | umjimathkievua-article-2860 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:31:44Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d6/5220cb3c1e8be2277aa234f24925c0d6.pdf |
| spelling | umjimathkievua-article-28602020-03-18T19:39:03Z Expansion of functions in a system of polynomials biorthogonal on a closed contour with a system of functions regular at infinitely remote point Розвинення функцій за системою поліномів, біортогональних на замкненому контурі з системою регулярних у нескінченно віддаленій точці функцій Sukhorolskyi, M. A. Сухорольський, М. А. We study properties of the systems of polynomials constructed according to the schemes similar to the schemes used for the Bernoulli and Euler polynomials, formulate conditions for the existence of functions associated with polynomials and conditions of representation of polynomials by contour integrals, and present the classes of analytic functions expandable in series in the systems of polynomials. The expansions of functions are illustrated by examples. Исследуются свойства систем полиномов, построенных по аналогичным с полиномами Бериулли и Эйлера схемам. Сформулированы условия существования ассоциированных с полиномами функций, условия представления полиномов контурными интегралами и приведены классы аналитических функций, разлагаемых в ряды по системам полиномов. Разложения функций проиллюстрированы примерами. Institute of Mathematics, NAS of Ukraine 2010-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2860 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 2 (2010); 238–254 Український математичний журнал; Том 62 № 2 (2010); 238–254 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2860/2472 https://umj.imath.kiev.ua/index.php/umj/article/view/2860/2473 Copyright (c) 2010 Sukhorolskyi M. A. |
| spellingShingle | Sukhorolskyi, M. A. Сухорольський, М. А. Expansion of functions in a system of polynomials biorthogonal on a closed contour with a system of functions regular at infinitely remote point |
| title | Expansion of functions in a system of polynomials biorthogonal on a closed contour with a system of functions regular at infinitely remote point |
| title_alt | Розвинення функцій за системою поліномів, біортогональних на замкненому контурі з системою регулярних у нескінченно віддаленій точці функцій |
| title_full | Expansion of functions in a system of polynomials biorthogonal on a closed contour with a system of functions regular at infinitely remote point |
| title_fullStr | Expansion of functions in a system of polynomials biorthogonal on a closed contour with a system of functions regular at infinitely remote point |
| title_full_unstemmed | Expansion of functions in a system of polynomials biorthogonal on a closed contour with a system of functions regular at infinitely remote point |
| title_short | Expansion of functions in a system of polynomials biorthogonal on a closed contour with a system of functions regular at infinitely remote point |
| title_sort | expansion of functions in a system of polynomials biorthogonal on a closed contour with a system of functions regular at infinitely remote point |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2860 |
| work_keys_str_mv | AT sukhorolskyima expansionoffunctionsinasystemofpolynomialsbiorthogonalonaclosedcontourwithasystemoffunctionsregularatinfinitelyremotepoint AT suhorolʹsʹkijma expansionoffunctionsinasystemofpolynomialsbiorthogonalonaclosedcontourwithasystemoffunctionsregularatinfinitelyremotepoint AT sukhorolskyima rozvinennâfunkcíjzasistemoûpolínomívbíortogonalʹnihnazamknenomukonturízsistemoûregulârnihuneskínčennovíddaleníjtočcífunkcíj AT suhorolʹsʹkijma rozvinennâfunkcíjzasistemoûpolínomívbíortogonalʹnihnazamknenomukonturízsistemoûregulârnihuneskínčennovíddaleníjtočcífunkcíj |