Equivalence of two methods for construction of regular continued C-fractions
A regular continued C-fraction is associated with a power series. The coefficients of this fraction are determined via either Hankel determinants or inverse derivatives. We prove the equivalence of these approaches to the construction of regular continued C-fractions.
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| Дата: | 2009 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2009
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3076 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509103569240064 |
|---|---|
| author | Katsala, R. A. Pahirya, M. M. Кацала, Р. А. Пагіря, М. М. |
| author_facet | Katsala, R. A. Pahirya, M. M. Кацала, Р. А. Пагіря, М. М. |
| author_sort | Katsala, R. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:44:40Z |
| description | A regular continued C-fraction is associated with a power series. The coefficients of this fraction are determined via either Hankel determinants or inverse derivatives. We prove the equivalence of these approaches to the construction of regular continued C-fractions. |
| first_indexed | 2026-03-24T02:35:47Z |
| format | Article |
| fulltext |
UDK 517.518:519.652
M. M. Pahirq (Mukaçiv. un-t, UΩhorod. nac. un-t),
R. A. Kacala (UΩhorod. nac. un-t)
EKVIVALENTNIST| DVOX METODIV POBUDOVY
PRAVYL|NYX LANCGHOVYX C-DROBIV
A regular continued C-fraction is associated with a power series. The coefficients of this fraction are
determined either via Hankel determinants or via inverse derivatives. We prove the equivalence of
these two approaches to the construction of regular continued C-fractions.
Pravyl\naq cepnaq C-drob\ qvlqetsq sootvetstvugwej stepennomu rqdu. Koπffycyent¥ πtoj
droby opredelqgtsq yly çerez hankelev¥ opredelytely, yly çerez obratn¥e proyzvodn¥e. Do-
kazana πkvyvalentnost\ πtyx podxodov k postroenyg pravyl\n¥x cepn¥x C-drobej.
Vstup. Rozvynennq funkcij odni[] dijsno] zminno] v lancghovyj drib naleΩyt\
do vaΩlyvyx zadaç nablyΩennq funkcij, oskil\ky taki rozvynennq ßyroko vy-
korystovugt\sq u prykladnyx zadaçax porqd iz nablyΩennqmy stepenevymy rq-
damy, ortohonal\nymy bahatoçlenamy, aproksymaciqmy Pade i t.7p. Çasto lan-
cghovyj drib ma[ bil\ß ßyroku oblast\ zbiΩnosti i ne nakopyçu[ poxybku pry
obçyslennqx. Tomu zadaça otrymannq rozvynen\ funkcij u lancghovyj drib ta
vstanovlennq vza[mozv’qzku miΩ riznymy sposobamy takyx rozvynen\ [ aktual\-
nog. Dovedenng ekvivalentnosti dvox pidxodiv do pobudovy pravyl\noho lan-
cghovoho C-drobu prysvqçeno danu robotu.
Rozvynennq funkcij u pravyl\nyj lancghovyj C-drib. Odyn iz sposo-
biv otrymannq rozvynen\ funkcij u lancghovyj drib polqha[ u vykorystanni
rozvynen\ ci[] funkci] u stepenevyj rqd i pobudovi vidpovidnoho danomu stepe-
nevomu rqdu pravyl\noho lancghovoho C-drobu [1 – 4].
Nexaj
f x c x c x c xn
n( ) = + + + … + + …1 1 2
2
(1)
— rozvynennq funkci] u stepenevyj rqd v okoli nulq,
D x
a x a x a x a xn
k
i( ) = +
+ + … + + … = +
=
∞
1
1 1 1
1
1
1 2
1
K (2)
— pravyl\nyj lancghovyj C-drib,
D xn( ) =
P x
Q x
a xn
n k
n
i( )
( )
= +
=
1
11
K (3)
— n-pidxidnyj drib lancghovoho drobu (2).
Oznaçennq. Lancghovyj drib (2) nazyva[t\sq vidpovidnym stepenevomu rq-
du (1), qkwo rozvynennq joho dovil\noho n - pidxidnoho drobu (3) u stepenevyj
rqd zbiha[t\sq iz vyxidnym stepenevym rqdom do çlenu xn
vklgçno.
Vyznaçnykom Hankelq, qkyj pov’qzanyj iz stepenevym rqdom (1), nazyva[t\sq
vyznaçnyk
H n
0
( ) = 1, Hk
n( ) =
c c c
c c c
c c c
n n n k
n n n k
n k n k n
+ + −
+ + +
+ − +
1 1
1 2
1
�
�
� � � �
� ++ −2 2k
, k = 1, 2, 3, … .
Teorema 1 [3, s. 220]. 1. Qkwo dlq zadanoho stepenevoho rqdu (1) isnu[ vid-
povidnyj pravyl\nyj lancghovyj C-drib u toçci x = 0, to
© M. M. PAHIRQ, R. A. KACALA, 2009
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 7 1005
1006 M. M. PAHIRQ, R. A. KACALA
Hk
( )1 ≠ 0, Hk
( )2 ≠ 0, k = 1, 2, 3, … , (4)
i
a H1 1
1= ( )
, a
H H
H H
k
k k
k k
2
1
1 2
1
1
2= − −
−
( ) ( )
( ) ( ) , a
H H
H H
k
k k
k k
2 1
1
1
1
2
1 2+
+ −= −
( ) ( )
( ) ( )
, k = 1, 2, 3, … .
(5)
2. Navpaky, qkwo spivvidnoßennq (4) vykonugt\sq, to pravyl\nyj lancg-
hovyj C-drib (2) iz koefici[ntamy an , qki vyznaçagt\sq formulamy (5), bu-
de vidpovidnym stepenevomu rqdu (1).
Formula Tile. Inßym sposobom, za dopomohog qkoho moΩna otrymaty roz-
vynennq funkcij u lancghovyj drib v okoli toçky x x= ∗ , [ formula Tile [5]
f x( ) = f x
x x
f x
x x
f x
( )
` ( ) ` ` ( )∗
∗
∗
∗
∗
+
−
+
−
[ ] +2
+
−
+ … +
−
+
∗
∗
∗
−
∗
x x
f x
x x
n f xn3 2 1` ( ) ` ( )( ) ( ) …… , (6)
de n f xn` ( )( )−
1 = ( ) ( )n f x – ( ) ( )n f x−2
. Oberneni poxidni
( ) ( )n f x obçyslg-
gt\sq za dopomohog rekurentnyx spivvidnoßen\
( ) ( )0 f x = f x( ) , ` ( )
( )
f x
f x
=
′
1
,
( ) ( )n f x = n f xn` ( )( )−
1 + ( ) ( )n f x−2 , n = 2, 3, … . (7)
Lancghovyj drib (6) moΩna podaty u vyhlqdi ekvivalentnoho jomu pravyl\-
noho lancghovoho C-drobu [6]
f x( ) = ω
ω ω ω
0
1 2
1 1 1
+
−
+
−
+ … +
−
+ …
∗ ∗ ∗( ) ( ) ( )x x x x x xn , (8)
de
ω0 = ∗f x( ) , ω1
1
=
∗` ( )f x
,
ωn n nn f x n f x
=
−
−
∗
−
∗
1
11 2` ( ) ( )` ( )( ) ( ) , n ≥ 2. (9)
Osnovna meta ci[] roboty — vstanovlennq ekvivalentnosti miΩ koefici[nta-
my lancghovyx drobiv (2) i (8).
Podannq koefici[ntiv formuly Tile çerez poxidni funkci]. Nexaj ck =
= f xk( ) ( ) / k!, k = 0, 1, 2, … . Todi oberneni poxidni
( ) ( )k f x vyraΩagt\sq çerez
zvyçajni poxidni funkci] qk vidnoßennq dvox vyznaçnykiv Hankelq [7, 8]
( )
( )
( )( )2 1 1
3
1
k k
k
f x
H
H
− −= , ( )
( )
( )( )2 1
0
2
k k
k
f x
H
H
= + , k ≥ 1, (10)
de
H n
0 1( ) = , H
c c c
c c c
c c
k
n
n n n k
n n n k
n k
( ) =
+ + −
+ + +
+ −
1 1
1 2
1
�
�
� � � �
nn k n kc+ + −� 2 2
, k = 1, 2, 3, … .
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 7
EKVIVALENTNIST| DVOX METODIV POBUDOVY PRAVYL|NYX … 1007
ZauvaΩennq 1. Dovedennq formul (10) u roboti [8] vidriznq[t\sq vid dove-
dennq v [7]. U knyzi N. E. N\orlunda [7, s. 419 – 427] formuly otrymano z vyko-
rystannqm vlastyvostej koefici[ntiv interpolqcijnoho bahatoçlena u formi
N\gtona.
Vyrazymo vsi n f xn` ( )( )−
1
, n ≥ 2, çerez hankelevi vyznaçnyky. Z formul
(7) i (10) pry n = 2 oderΩymo
2` ` ( )f x∗[ ] = `̀ ( ) ( )f x f x∗ ∗− =
c c
c c
c
c
0 1
1 2
2
0− =
=
c c
c c
c
c c
c
c c
c
c
c
c
0 1
1 2
0
1 2
2
0 1
1
2
1
2
2
0
0
−
= = − .
Pry n = 3 otryma[mo
3` `̀ ( )f x∗[ ] = `̀̀ ( ) ` ( )f x f x∗ ∗− =
c
c c
c c
c
c
c c
c c
c c
c c
c c
c
3
1 2
2 3
1
1
2 3
1 2
2 3
1 2
2 3
1
1
0
− =
−
=
=
c c
c
c c
c c
c
c
c c
c c
c
1 2
2
1 2
2 3
1
2
2
1 2
2 3
1
0
−
= .
Analohiçno pry n = 4 ta n = 5
4
1 2
2 3
2
2 3
3 4
2
` `̀` ( )f x
c c
c c
c c
c c
c
[ ] = − , 5 4
2 3
3 4
2
1 2 3
2 3 4
3 4 5
1
` ( )( ) f x
c c
c c
c c c
c c c
c c c
c
=
cc
c c
2
2 3
.
OtΩe, vykonugt\sq spivvidnoßennq
2
1
1 2
1
2
0
2` ` ( )
( )
( ) ( )f x
H
H H
[ ] = −
( )
, 3
1
2 2
2
1
1
1` `̀ ( )
( )
( ) ( )f x
H
H H
[ ] =
( )
,
4
2
1 2
2
2
1
2` `̀ ( )
( )
( ) ( )f x
H
H H
[ ] = −
( )
, 5 4 2
2 2
3
1
2
1` ( )( )
( )
( ) ( )f x
H
H H
=
( )
.
Nam potribna nastupna lema.
Lema [3, s. 225]. Dlq vyznaçnykiv Hankelq Hk
s( )
ma[ misce totoΩnist\
Qkobi
H H H H Hk
s
k
s
k
s
k
s
k
s( ) ( ) ( ) ( ) ( )+
+ −
+ +− = ( )2
1 1
2 1 2
, k = 1, 2, … , s = 1, 2, … . (11)
Teorema 2. Koefici[nty lancghovoho drobu (6) vyznaçagt\sq çerez vyznaç-
nyky Hankelq Hk
n( )
takym çynom:
( ) ` ( )( )
( )
( ) ( )2 2 1
1 2
2
1
2k f x
H
H H
k k
k k
−
−
= −
( )
, ( ) ` ( )( )
( )
( ) ( )2 1 2
2 2
1
1
1k f x
H
H H
k k
k k
+ =
( )
+
. (12)
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 7
1008 M. M. PAHIRQ, R. A. KACALA
Dovedennq. Oskil\ky
( ) ` ( ) ( ) ( )( ) ( ) ( )2 2 1 2 2 2k f x f x f xk k k− − = − ,
to z (10) ta (11) ma[mo
( ) ` ( )( )2 2 1k f xk− =
H
H
H
H
k
k
k
k
+
−
−1
0
2
0
1
2
( )
( )
( )
( ) =
H H H H
H H
k k k k
k k
+ −
−
−1
0
1
2 0 2
2
1
2
( ) ( ) ( ) ( )
( ) ( )
= −
( )
−
H
H H
k
k k
( )
( ) ( )
1 2
2
1
2 .
Druhe iz spivvidnoßen\ (12) dovodyt\sq analohiçno.
ZauvaΩennq 2. Formuly (12) inßym sposobom bulo dovedeno v roboti [7,
s. 428, 429].
Ekvivalentnist\ lancghovyx drobiv (2) i (8). Zapyßemo koefici[nty ωn
pravyl\noho lancghovoho C-drobu (8) çerez vyznaçnyky Hankelq Hk
n( )
. Iz
formul (9) ta (12) otrymu[mo
ω2
2
1
2
1
1 1
1 2
1
2
k
k k k k
k k
H H H H
H H
= −
( )
− −
−
( ) ( ) ( ) ( )
( ) ( ))
( ) ( )
( ) ( )( )
= − −
−
2
2
1
1
1
1
2
H H
H H
k k
k k
, (13)
ω2 1
1
1
2
1
2 2
1 2 2
k
k k k k
k k
H H H H
H H
+
+ −= −
( )
( ) ( ) ( ) ( )
( ) ( ))
( ) ( )
( ) ( )( )
= − − +
2
1
2
1
1
1 2
H H
H H
k k
k k
, (14)
de k = 1, 2, 3, … .
Nexaj ma[mo stepenevyj rqd
f x( ) = f x
f x
x x
f x
x x( )
( )
!
( )
( )
!
( )∗
∗
∗
∗
∗+
′
− +
′′
−
1 2
2 + …
… +7
f x
n
x x
n
n
( ) ( )
!
( )∗
∗− + … 7.
Podilyvßy livu i pravu çastyny ci[] rivnosti na f x( )∗ , otryma[mo
f x
f x
( )
( )∗
= 1
1
+
′
⋅
−∗
∗
∗
f x
f x
x x
( )
( ) !
( ) +
′′
⋅
−∗
∗
∗
f x
f x
x x
( )
( ) !
( )
2
2 + …
… +7
f x
f x n
x x
n
n
( ) ( )
( ) !
( )∗
∗
∗⋅
− + … 7.
Poznaçymo �ci =
f x
f x i
i( ) ( )
( ) !
∗
∗ ⋅
, i = 1, 2, … , todi
f x
f x
( )
( )∗
= 1 1 2
2+ − + − + … + − + …∗ ∗ ∗� � �c x x c x x c x xn
n( ) ( ) ( ) . (15)
Otrymaly stepenevyj rqd vyhlqdu (1). Todi koefici[nty vidpovidnoho jomu
pravyl\noho lancghovoho C-drobu budut\ takymy:
� �ω1 1
1= H ( )
, �
� �
� �
ω2
1
1 2
1
1
2k
k k
k k
H H
H H
= − −
−
( ) ( )
( ) ( )
, �
� �
� �ω2 1
1
1
1
2
1 2k
k k
k k
H H
H H
+
+ −= −
( ) ( )
( ) ( ) , (16)
de
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 7
�H n
0 1( ) = , �
� � � �
� � � �
� � � �
H
c c c
c c c
k
n
n n n k
n n n k( ) =
+ + −
+ + +
1 1
1 2
�� � � �c c cn k n k n k+ − + + −1 2 2
, k = 1, 2, 3, … .
Oskil\ky �ci =
1
f x
ci( )∗
, i = 1, 2, … , to z formul (13), (14) ta (16) ma[mo
�ω ω0 0
1
=
∗f x( )
, � �ω ω1 1 1 1
1 1
= = =
∗ ∗
c
f x
c
f x( ) ( )
,
�
� �
� �
ω2
1
1 2
1
1
2
1
k
k k
k k
kH H
H H
f x
= − = −−
−
∗
( ) ( )
( ) ( )
( ) −− −
∗
∗ ∗
−
1 1
1 2
1
1
1
1 1
H
f x
H
f x
H
f x
k k k
k k k
( ) ( )
( )
( )
( ) ( )
HHk
k
−
=
1
2
2
( )
ω , (17)
�
� �
� �
ω2 1
1
1
1
2
1 2
1
k
k k
k k
H H
H H
f x
+
+ − ∗= − = −
( ) ( )
( ) ( )
( )) ( )
( ) (
( ) ( )
( )
k k k k
k k
H
f x
H
f x
H
f
+ +
∗
− −
∗
1 1
1
1 1
2
1
1
1 1
xx
H
k k
k
∗
+=
)
( )2
2 1ω , k ≥ 1.
Zapyßemo lancghovyj C-drib, qkyj [ vidpovidnym stepenevomu rqdu (15), i
vykorysta[mo znajdeni rivnosti (17). Todi
f x
f x
( )
( )∗
= �
� � �
ω
ω ω ω
0
1 2
1 1 1
+
−
+
−
+ … +
−
+ … =∗ ∗ ∗( ) ( ) ( )x x x x x xn
=
1
1 1 1
0
1 2
f x
x x x x x xn
( )
( ) ( ) ( )
∗
∗ ∗ ∗+
−
+
−
+ … +
−
+ …
ω
ω ω ω
.
Qkwo livu i pravu çastyny rivnosti pomnoΩyty na f x( )∗ , to otryma[mo rozvy-
nennq (8) funkci] f x( ) , a ce, v svog çerhu, pokazu[, wo pobudovy pravyl\noho
lancghovoho C-drobu çerez vyznaçnyky Hankelq ta oberneni poxidni [ ekviva-
lentnymy.
1. Wall H. S. Analytic theory of continued fractions. – New York: D. Van Nostrand Co., 1948. –
433 p.
2. Perron O. Die Lehre von den Kettenbrüchen. – Stuttgart: Teubner, 1957. – Bd 2. – 315 S.
3. DΩouns U., Tron V. Neprer¥vn¥e droby. Analytyçeskaq teoryq y pryloΩenyq. – M.: Myr,
1985. – 414 s.
4. Skorobohat\ko V. Q. Teoryq vetvqwyxsq cepn¥x drobej y ee prymenenye v v¥çyslytel\noj
matematyke. – M.: Nauka, 1983. – 312 s.
5. Thiele T. N. Interpolationsprechnung. – Leipzig: Commisission von B. G. Teubner, 1909. – XII +
175 S.
6. Pahirq M. M., Kacala R. A. Rozvytky deqkyx funkcij u lancghovi droby // Nauk. visn. UΩ-
horod. un-tu. Ser. matematyka i informatyka. – 2007. – Vyp. 14-15. – S. 107 – 116.
7. Nörlund N. E. Vorlesungen über Differenzenrechnung. – Berlin: I. Springer, 1924. – 551 S.
8. Kacala R. A. Zv’qzok miΩ obernenymy poxidnymy ta poxidnymy funkci] odni[] dijsno] zmin-
no] // Nauk. visn. UΩhorod. un-tu. Ser. matematyka i informatyka. – 2008. – Vyp. 16. – S. 73 –
81.
OderΩano 25.12.08
|
| id | umjimathkievua-article-3076 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:35:47Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c2/f888d4c80eff5c213658b89b71121fc2.pdf |
| spelling | umjimathkievua-article-30762020-03-18T19:44:40Z Equivalence of two methods for construction of regular continued C-fractions Еквівалентність двох методів побудови правильних ланцюгових C-дробів Katsala, R. A. Pahirya, M. M. Кацала, Р. А. Пагіря, М. М. A regular continued C-fraction is associated with a power series. The coefficients of this fraction are determined via either Hankel determinants or inverse derivatives. We prove the equivalence of these approaches to the construction of regular continued C-fractions. Правильная цепная C-дробь является соответствующей степенному ряду. Коэффициенты этой дроби определяются или через ганкелевы определители, или через обратные производные. Доказана эквивалентность этих подходов к построению правильных цепных C-дробей. Institute of Mathematics, NAS of Ukraine 2009-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3076 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 7 (2009); 1005-1009 Український математичний журнал; Том 61 № 7 (2009); 1005-1009 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3076/2901 https://umj.imath.kiev.ua/index.php/umj/article/view/3076/2902 Copyright (c) 2009 Katsala R. A.; Pahirya M. M. |
| spellingShingle | Katsala, R. A. Pahirya, M. M. Кацала, Р. А. Пагіря, М. М. Equivalence of two methods for construction of regular continued C-fractions |
| title | Equivalence of two methods for construction of regular continued C-fractions |
| title_alt | Еквівалентність двох методів побудови правильних ланцюгових C-дробів |
| title_full | Equivalence of two methods for construction of regular continued C-fractions |
| title_fullStr | Equivalence of two methods for construction of regular continued C-fractions |
| title_full_unstemmed | Equivalence of two methods for construction of regular continued C-fractions |
| title_short | Equivalence of two methods for construction of regular continued C-fractions |
| title_sort | equivalence of two methods for construction of regular continued c-fractions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3076 |
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