Construction of the Padé approximants for some Appell hypergeometric series by means of method of generalized moment representations
By means of extension of V.K. Dzyadyk's method of generalized moment representations to the case of two--dimensional number sequences Pad\'e approximants for some Appell hypergeometric series are constructed
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Інститут математики НАН України
2013
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| Назва журналу: | Transactions of Institute of Mathematics of NAS of Ukraine |
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872552655798140928 |
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| author | Golub, A. P. Chernets’ka, L. O. Голуб, А. П. Чернецька, Л. О. |
| author_facet | Golub, A. P. Chernets’ka, L. O. Голуб, А. П. Чернецька, Л. О. |
| author_institution_txt_mv | [
{
"author": "А. П. Голуб",
"institution": "Інститут математики НАН України"
},
{
"author": "Л. О. Чернецька",
"institution": "Інститут математики НАН України"
}
] |
| author_sort | Golub, A. P. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-01-29T14:44:27Z |
| description | By means of extension of V.K. Dzyadyk's method of generalized moment representations to the case of two--dimensional number sequences Pad\'e approximants for some Appell hypergeometric series are constructed |
| first_indexed | 2026-08-04T01:02:53Z |
| format | Article |
| fulltext |
Çáiðíèê ïðàöü Ií-òó ìàòåìàòèêè ÍÀÍ Óêðà¨íè 2013, Ò.10, �1, 69�94
ÓÄÊ 517.53
À.Ï. Ãîëóá, Ë.Î. ×åðíåöüêà (Ií�ò ìàòåìàòèêè ÍÀÍ Óêðà¨íè, Êè¨â)
ÏÎÁÓÄÎÂÀ ÀÏÐÎÊÑÈÌÀÍÒ ÏÀÄÅ ÄËß ÄÅßÊÈÕ ÃI-
ÏÅÐÃÅÎÌÅÒÐÈ×ÍÈÕ ÐßÄI ÀÏÏÅËß ÇÀ ÄÎÏÎÌÎ-
ÃÎÞ ÌÅÒÎÄÓ ÓÇÀÃÀËÜÍÅÍÈÕ ÌÎÌÅÍÒÍÈÕ ÇÎÁ-
ÐÀÆÅÍÜ
By means of extension of V.K. Dzyadyk's method of generalized moment
representations to the case of two�dimensional number sequences Pad�e
approximants for some Appell hypergeometric series are constructed.
Çà äîïîìîãîþ ïîøèðåííÿ ìåòîäó óçàãàëüíåíèõ ìîìåíòíèõ çîáðàæåíü
Â.Ê. Äçÿäèêà íà âèïàäîê äâîâèìiðíèõ ÷èñëîâèõ ïîñëiäîâíîñòåé ïîáóäîâà-
íî àïðîêñèìàíòè Ïàäå äëÿ äåÿêèõ ãiïåðãåîìåòðè÷íèõ ðÿäiâ Àïïåëÿ.
 ðîáîòàõ [1], [2] ìåòîä óçàãàëüíåíèõ ìîìåíòíèõ çîáðàæåíü
Â.Ê. Äçÿäèêà ïîøèðåíî íà âèïàäîê äâîâèìiðíèõ ÷èñëîâèõ ïîñëiäîâ-
íîñòåé i ïîáóäîâàíî àïðîêñèìàíòè Ïàäå äëÿ äåÿêèõ ãiïåðãåîìåòðè÷-
íèõ ðÿäiâ Àïïåëÿ òà Ãóìáåðòà. Íàâåäåìî ùå îäèí ïðèêëàä çàñòîñó-
âàííÿ öüîãî ìåòîäó.
Çàóâàæèìî, ùî ðiçíîìàíiòíi ìîäèôiêàöi¨ áàãàòîâèìiðíèõ i, çî-
êðåìà, äâîâèìiðíèõ àïðîêñèìàöié Ïàäå âèâ÷àëèñÿ â ðîáîòàõ [3]�[10],
çîêðåìà â [8]�[10] äîñëiäæóâàëèñÿ àïðîêñèìàíòè Ïàäå äëÿ äåÿêèõ
ôóíêöié Àïïåëÿ.
Íàâåäåìî íåîáõiäíi îçíà÷åííÿ.
Îçíà÷åííÿ 1. Áóäåìî ãîâîðèòè, ùî äëÿ äâîâèìiðíî¨ ÷èñëîâî¨ ïî-
ñëiäîâíîñòi {sk,m}∞k,m=0 ì๠ìiñöå óçàãàëüíåíå ìîìåíòíå çîáðàæåííÿ
íà äîáóòêó ëiíiéíèõ ïðîñòîðiâ X òà Y çà îçíà÷åíîþ íà öüîìó äîáóò-
êó áiëiíiéíîþ ôîðìîþ 〈., .〉, ÿêùî â ïðîñòîði X âêàçàíî äâîâèìiðíó
ïîñëiäîâíiñòü åëåìåíòiâ {xk,m}∞k,m=0, à â ïðîñòîði Y � äâîâèìiðíó
ïîñëiäîâíiñòü åëåìåíòiâ {yj,n}∞j,n=0 òàêi, ùî
sk+j,m+n = 〈xk,m, yj,n〉, k, j,m, n ∈ Z+. (1)
Äâîâèìiðíié ÷èñëîâié ïîñëiäîâíîñòi {sk,m}∞k,m=0 ìîæíà ïîñòà-
âèòè ó âiäïîâiäíiñòü ôîðìàëüíèé ñòåïåíåâèé ðÿä âiä äâîõ çìiííèõ
âèãëÿäó
c© À.Ï. Ãîëóá, Ë.Î. ×åðíåöüêà, 2013
70 À.Ï. Ãîëóá, Ë.Î. ×åðíåöüêà
f(z, w) =
∞∑
k,m=0
sk,mzkwm. (2)
Çàäà÷à ïðî äâîâèìiðíi óçàãàëüíåíi ìîìåíòíi çîáðàæåííÿ ìîæå
áóòè ñôîðìóëüîâàíà â îïåðàòîðíîìó âèãëÿäi. À ñàìå, ïðèïóñòèìî,
ùî ïðîñòîðè X òà Y ¹ íîðìîâàíèìè, i â ïðîñòîði X iñíóþòü êî-
ìóòóþ÷i ìiæ ñîáîþ îáìåæåíi îïåðàòîðè A,B : X → X , òàêi ùî
âèêîíóþòüñÿ ðiâíîñòi
Axk,m = xk+1,m,
Bxk,m = xk,m+1
äëÿ ∀k,m ∈ Z+. Íåõàé òàêîæ â ïðîñòîði Y iñíóþòü îáìåæåíi îïå-
ðàòîðè A?, B? : Y → Y , ñïðÿæåíi äî îïåðàòîðiâ A òà B âiäíîñíî
áiëiíiéíî¨ ôîðìè 〈., .〉 â òîìó ðîçóìiííi, ùî ∀x ∈ X ,∀y ∈ Y :
〈Ax, y〉 = 〈x,A?y〉,
〈Bx, y〉 = 〈x,B?y〉.
Òîäi çîáðàæåííÿ (1) ìîæå áóòè çàïèñàíèì ó âèãëÿäi
sk,m = 〈xk,m, y0,0〉 =
〈
AkBmx0,0, y0,0
〉
, k, m ∈ Z+, (3)
i ðÿä (2) áóäå çáiæíèì â îêîëi ïî÷àòêó êîîðäèíàò äî àíàëiòè÷íî¨
ôóíêöi¨, ùî ì๠çîáðàæåííÿ
f(z, w) =
〈
R̂z(A)R̂w(B)x0,0, y0,0
〉
, (4)
äå ðåçîëüâåíòíà ôóíêöiÿ R̂z(A) âèçíà÷à¹òüñÿ ðiâíiñòþ
R̂z(A) = (I − zA)−1.
 [1] ïîêàçàíî, ÿê äëÿ ôóíêöié âèãëÿäó (4) ìîæíà áóäóâàòè äâî-
âèìiðíi àïðîêñèìàíòè Ïàäå.
Íåõàé X = Y = L2 ([0, 1], dµ) äëÿ äåÿêî¨ ìiðè, ùî âèçíà÷à¹òüñÿ
íåñïàäíîþ ôóíêöi¹þ µ(t), ùî ì๠íåñêií÷åííó êiëüêiñòü òî÷îê çðîñ-
òàííÿ íà [0, 1]. Ðîçãëÿíåìî â ïðîñòîði X êîìóòóþ÷i îáìåæåíi ëiíiéíi
îïåðàòîðè
(Aϕ) (t) = tϕ(t),
Ïîáóäîâà àïðîêñèìàíò Ïàäå . . . 71
(Bϕ) (t) = (1− t)ϕ(t).
�õ ðåçîëüâåíòíi ôóíêöi¨ ìàþòü âèãëÿä
(
R̂z(A)ϕ
)
(t) =
ϕ(t)
1− zt
,
(
R̂w(B)ϕ
)
(t) =
ϕ(t)
1− w(1− t)
.
Òàêèì ÷èíîì, íà îñíîâi (4) ïðè x0,0(t) = y0,0(t) ≡ 1 ôóíêöiþ
f(z, w) ìîæíà çàïèñàòè ó âèãëÿäi
f(z, w) =
〈
R̂z(A)R̂w(B)x0,0, y0,0
〉
=
1∫
0
dµ(t)
(1− zt)(1− w(1− t))
=
=
1
w + z − zw
w
1∫
0
dµ(t)
1− w(1− t)
+ z
1∫
0
dµ(t)
1− zt
. (5)
Êîåôiöi¹íòè sk,m â ðîçâèíåííi ôóíêöi¨ f(z, w) â ðÿä (2), çãiäíî ç
(3) áóäóòü ìàòè âèãëÿä
sk,m =
1∫
0
xk,m(t)y0,0(t)dµ(t) =
1∫
0
(AkBmx0,0)(t)y0,0(t)dµ(t) =
=
1∫
0
tk(1− t)mdµ(t), k, m = 0,∞. (6)
Çà òåîðåìîþ 1 ç [2] ùîá ïîáóäóâàòè àïðîêñèìàíòè Ïàäå ôóíêöi¨
âèãëÿäó (5), íàì ïîòðiáíî ïîáóäóâàòè íåòðèâiàëüíèé óçàãàëüíåíèé
ìíîãî÷ëåí âèãëÿäó
XN1,N2 =
N1∑
k=0
N2∑
m=0
c
(N1,N2)
k,m xk,m,
72 À.Ï. Ãîëóá, Ë.Î. ×åðíåöüêà
òàêèé ùî âèêîíóþòüñÿ óìîâè áiîðòîãîíàëüíîñòi
〈XN1,N2 , yj,n〉 = 0
ïðè (j, n) ∈ ([0, N1]× [0, N2]) \ {(N1, N2)}. Îñêiëüêè XN1,N2(t) â äàíî-
ìó âèïàäêó áóäå àëãåáðà¨÷íèì ìíîãî÷ëåíîì ñòåïåíÿ N1 + N2, ÿêèé
îðòîãîíàëüíèé äî ìíîãî÷ëåíiâ ñòåïåíÿ 6 N1+N2−1, òî âií ñïiâïàäà-
òèìå ç òî÷íiñòþ äî ñòàëîãî ìíîæíèêà ç àëãåáðà¨÷íèì ìíîãî÷ëåíîì
PN1+N2(t) ñòåïåíÿ N1 + N2, îðòîíîðìîâàíèì íà [0, 1] ç âàãîþ dµ(t)
(äèâ. [11, c. 116]):
XN1,N2(t) = PN1+N2(t). (7)
Çàóâàæèìî, ùî ïîëiíîì (7) ïðè öüîìó áóäå îðòîãîíàëüíèì íå
ëèøå äî xk,m(t), (k, m) ∈ ([0, N1]×[0, N2])\{(N1, N2)}, àëå i äî xk,m(t)
ïðè (k,m) ∈ {(k, m), k, m ∈ Z+, k + m 6 N1 + N2 − 1}. Òîìó iíäåêñè
êîåôiöi¹íòiâ ÷èñåëüíèêà àïðîêñèìàíòè Ïàäå ôóíêöi¨ f âèãëÿäó (5)
áóäåìî áðàòè íå ç ìíîæèíè N ? = ([0, 2N1] × [0, 2N2]) \ ([N1, 2N1] ×
[N2, 2N2]) (ÿê ïðîïîíó¹òüñÿ â òåîðåìi 1 ç [1]), à ç ìíîæèíè
N = {(k, m), k + m 6 2N1 + 2N2 − 1}\{(k, m), k > N1,m > N2},
à iíäåêñè êîåôiöi¹íòiâ çíàìåííèêà � ç îáëàñòi
D = [0, N1] × [0, N2].
Äëÿ îáðàíî¨ íàìè îáëàñòi N â òåîðåìi 1 ç [2] ìè ïîâèííi ïîêëàñòè
x(k) = 2N1 + 2N2 − 1− k, y(m) = 2N1 + 2N2 − 1−m.
Çàïèøåìî ìíîãî÷ëåí PN1+N2(t) ó âèãëÿäi:
PN1+N2(t) =
N1+N2∑
j=0
p
(N1+N2)
j tj .
Ç (7) îòðèìà¹ìî:
N1∑
k=0
N2∑
m=0
c
(N1,N2)
k,m xk,m(t)=
N1∑
k=0
N2∑
m=0
c
(N1,N2)
k,m tk(1−t)m=
N1+N2∑
j=0
p
(N1+N2)
j tj . (8)
Ç ðiâíîñòi (8) êîåôiöi¹íòè c
(N1,N2)
k,m , k = 0, N1,m = 0, N2 ìîæíà
âèçíà÷èòè áåçëi÷÷þ ñïîñîáiâ. Ôóíêöi¨ âèãëÿäó (5) ¹ ñèìåòðè÷íèìè
Ïîáóäîâà àïðîêñèìàíò Ïàäå . . . 73
çà ñâî¨ìè çìiííèìè òîäi i òiëüêè òîäi, êîëè dµ(t) ≡ dµ(1 − t). Òîìó
äîðå÷íî ðîçãëÿíóòè äâà âèïàäêè.
Âèïàäîê (I).  íåñèìåòðè÷íîìó âèïàäêó ÿê îäèí ç âàðiàíòiâ
çíàõîäæåííÿ êîåôiöi¹íòiâ c
(N1,N2)
k,m , k = 0, N1, m = 0, N2 ç ðiâíîñòi (8)
ðîçãëÿíåìî òàêèé (äèâ. ìàëþíîê íèæ÷å).
Ïîêëàäåìî:
N1+N2∑
j=0
p
(N1+N2)
j tj =
N1−1∑
k=0
c
(N1,N2)
k,0 tk + tN1
N2∑
m=0
c
(N1,N2)
N1,m (1− t)m.
k
m
N2
N1
-
6
Ïðè k = 0, N1 − 1,m = 0 îòðèìà¹ìî:
c
(N1,N2)
k,m = p
(N1+N2)
k .
Äàëi áóäåìî ìàòè:
N1+N2∑
j=N1
p
(N1+N2)
j tj−N1 =
N2∑
m=0
c
(N1,N2)
N1,m (1− t)m,
çâiäêè
N2∑
k=0
(−t)k
N2∑
j=k
p
(N1+N2)
j+N1
(
j
k
)
=
N2∑
m=0
c
(N1,N2)
N1,m tm.
Îòæå, ïðè k = N1,m = 0, N2 îòðèìà¹ìî:
c
(N1,N2)
N1,m = (−1)m
N2∑
j=m
p
(N1+N2)
j+N1
(
j
m
)
.
74 À.Ï. Ãîëóá, Ë.Î. ×åðíåöüêà
Òàêèì ÷èíîì, ìà¹ìî:
c
(N1,N2)
k,m =
p
(N1+N2)
k ïðè k = 0, N1 − 1,m = 0,
(−1)m
N2∑
j=m
(
j
m
)
p
(N1+N2)
j+N1
ïðè k = N1,m = 0, N2,
0, äëÿ iíøèõ (k, m) ∈ D .
(9)
Âèïàäîê (II).  ñèìåòðè÷íîìó âèïàäêó ì๠ñåíñ íàáëèæàòè ôóíê-
öiþ f ëèøå ñèìåòðè÷íèìè ðàöiîíàëüíèìè ïîëiíîìàìè, à òîìó ïî-
êëàäåìî N1 = N2 = N i âñi íåäiàãîíàëüíi êîåôiöi¹íòè ðiâíèìè íóëþ:
c
(N,N)
k,m = 0 ïðè k 6= m. Äëÿ çðó÷íîñòi äiàãîíàëüíi êîåôiöi¹íòè c
(N,N)
k,k
áóäåìî íàäàëi ïîçíà÷àòè ÷åðåç c
(N)
k .
Ðiâíiñòü (8) íàáóäå âèãëÿäó
N∑
k=0
c
(N)
k tk(1− t)k =
2N∑
j=0
p
(2N)
j tj .
Äëÿ âèçíà÷åííÿ êîåôiöi¹íòiâ c
(N)
k âñòàíîâèìî íàñòóïíèé äîïî-
ìiæíèé ðåçóëüòàò.
Ëåìà. Íåõàé äåÿêèé àëãåáðà¨÷íèé ìíîãî÷ëåí
P2N (t) =
2N∑
m=0
p(2N)
m tm (10)
ñòåïåíÿ 2N ¹ òàêèì, ùî ∀t ∈ [0, 1] âèêîíó¹òüñÿ ðiâíiñòü
P2N (t) = P2N (1− t). (11)
Òîäi âií ìîæå áóòè ¹äèíèì ÷èíîì çàïèñàíèì ó âèãëÿäi
P2N (t) =
N∑
j=0
c
(N)
j tj(1− t)j , (12)
i ïðè öüîìó äëÿ êîåôiöi¹íòiâ c
(N)
j , j = 0, N , ñïðàâäæóþòüñÿ ðiâíî-
ñòi
c
(N)
j =
p
(2N)
0 ïðè j = 0,
j∑
m=1
(2j −m− 1)!m
j!(j −m)!
p(2N)
m ïðè j > 1.
(13)
Ïîáóäîâà àïðîêñèìàíò Ïàäå . . . 75
Äîâåäåííÿ. Ç ðiâíîñòåé (10) òà (12) áóäåìî ìàòè
p
(2N)
k =
c
(N)
0 ïðè k = 0,
k∑
j=[ k+1
2 ]
(−1)k−j
(
j
k − j
)
c
(N)
j ïðè k = 1, 2N.
Íà îñíîâi äàíîãî çîáðàæåííÿ íåâàæêî ïåðåêîíàòèñÿ â ñïðàâåä-
ëèâîñòi ðiâíîñòåé (13) äëÿ íåâåëèêèõ çíà÷åíü N = 0, 1, 2, 3, . . . .
Ïðèïóñòèìî çà iíäóêöi¹þ, ùî ðiâíîñòi (13) âèêîíóþòüñÿ ïðè äå-
ÿêîìó öiëîìó N − 1 ∈ Z+. Ðîçãëÿíåìî òîäi àëãåáðà¨÷íèé ìíîãî÷ëåí
P2N (t) ñòåïåíÿ 2N , òàêèé ùî äëÿ íüîãî ì๠ìiñöå âëàñòèâiñòü (11).
Òîäi àëãåáðà¨÷íèé ìíîãî÷ëåí
P̃2N−2(t) = P2N (t)− p
(2N)
2N (−1)N+1tN+1(1− t)N+1
ìàòèìå ñòåïiíü 2(N −1) i äëÿ íüîãî òàêîæ ñïðàâäæóâàòèìåòüñÿ âëà-
ñòèâiñòü (11). Òîìó çà ïðèïóùåííÿì iíäóêöi¨ éîãî ìîæíà çàïèñàòè
¹äèíèì ÷èíîì ó âèãëÿäi
P̃2N−2(t) =
N−1∑
j=0
c̃
(N−1)
j tj(1− t)j ,
äå
c̃
(N−1)
j =
p̃
(2N−2)
0 ïðè j = 0,
j∑
m=1
(2j −m− 1)!m
j!(j −m)!
p̃(2N−2)
m ïðè j > 1.
Àëå òîäi
P2N (t) = P̃2N−2(t) + p
(2N)
2N (−1)N+1tN+1(1− t)N+1 =
=
N−1∑
j=0
c̃
(N−1)
j tj(1− t)j + p
(2N)
2N (−1)N+1tN+1(1− t)N+1.
Âðàõîâóþ÷è, ùî
p̃(2N−2)
m = p(2N)
m ïðè m = 0, N − 1,
76 À.Ï. Ãîëóá, Ë.Î. ×åðíåöüêà
äëÿ äîâåäåííÿ ëåìè çàëèøèëîñü âñòàíîâèòè, ùî çà óìîâè (11) äëÿ
êîåôiöi¹íòiâ ìíîãî÷ëåíà P2N (t) ì๠ìiñöå ñïiââiäíîøåííÿ
(−1)Np
(2N)
2N =
N∑
m=1
(2N −m− 1)!m
N !(N −m)!
p(2N)
m . (14)
Íåâàæêî ïåðåêîíàòèñÿ, ùî óìîâà (11) òÿãíå çà ñîáîþ íåîáõiäíiñòü
âèêîíàííÿ ñïiââiäíîøåíü
p(2N)
m = (−1)m
2N∑
k=m
p
(2N)
k
(
k
m
)
, m = 0, 2N. (15)
Ïåðøi N ñïiââiäíîøåíü (15) ìîæíà ðîçãëÿäàòè ÿê ñèñòåìó ëiíié-
íèõ àëãåáðà¨÷íèõ ðiâíÿíü âiäíîñíî N íåâiäîìèõ
p
(2N)
N+1, p
(2N)
N+2 , ..., p
(2N)
2N .
Âèçíà÷íèê öi¹¨ ñèñòåìè ì๠âèãëÿä
∆N =
∣∣∣∣∣∣∣∣∣∣
1 1 . . . 1
N + 1 N + 2 . . . 2N(
N+1
2
) (
N+2
2
)
. . .
(
2N
2
)
. . . . . . . . . . . .(
N+1
N−1
) (
N+2
N−1
)
. . .
(
2N
N−1
)
∣∣∣∣∣∣∣∣∣∣
,
à âåêòîð ïðàâèõ ÷àñòèí äîðiâíþ¹
(
−p
(2N)
1 − p
(2N)
2 − . . .− p
(2N)
N ,−2p
(2N)
1 − 2p
(2N)
2 − . . .−Np
(2N)
N ,
−
(
3
2
)
p
(2N)
3 −
(
4
2
)
p
(2N)
4 − . . .−
(
N
2
)
p
(2N)
N , . . . ,
(−1 + (−1)m)p(2N)
m −
(
m + 1
m
)
p
(2N)
m+1 − . . .−
(
N
m
)
p
(2N)
N , . . . ,
(−1 + (−1)N−1)p(2N)
N−1 −
(
N
N − 1
)
p
(2N)
N
)T
. (16)
Ïîáóäîâà àïðîêñèìàíò Ïàäå . . . 77
Îòîæ, ùîá çíàéòè çà ôîðìóëàìè Êðàìåðà çi ñïiââiäíîøåíü (15)
êîåôiöi¹íò p
(2N)
2N , ïîòðiáíî ïiäðàõóâàòè âèçíà÷íèê ∆N òà âèçíà÷íèê,
ùî îòðèìó¹òüñÿ ç ∆N çàìiíîþ îñòàííüîãî ñòîâï÷èêà íà âåêòîð ïðà-
âèõ ÷àñòèí (16).
Î÷åâèäíî,
∆N =
(N+1)!(N+2)! · . . . · (2N)!
0!1! · . . . · (N − 1)!
∣∣∣∣∣∣∣∣∣∣∣∣∣
1
(N+1)!
1
(N+2)!
. . .
1
(2N)!
1
N !
1
(N+1)!
. . .
1
(2N−1)!
. . . . . . . . . . . .
1
2!
1
3!
. . .
1
(N+1)!
∣∣∣∣∣∣∣∣∣∣∣∣∣
=
= (−1)
(N−1)N
2
N∏
k=1
(N+k)!
N−1∏
k=1
k!
∣∣∣∣∣∣∣∣∣∣∣∣∣
1
2!
1
3!
. . .
1
(N + 1)!
1
3!
1
4!
. . .
1
(N + 2)!
. . . . . . . . . . . .
1
(N + 1)!
1
(N + 2)!
. . .
1
(2N)!
∣∣∣∣∣∣∣∣∣∣∣∣∣
. (17)
Ïîçíà÷èìî îñòàííié âèçíà÷íèê ÷åðåç ∆̃N . Öåé âèçíà÷íèê ¹
âèçíà÷íèêîì Ãàíêåëÿ ïîñëiäîâíîñòi
{
1
(k + 2)!
}∞
k=0
. Äëÿ öi¹¨ ïîñëi-
äîâíîñòi ì๠ìiñöå óçàãàëüíåíå ìîìåíòíå çîáðàæåííÿ (äèâ. [12, c. 37])
1
(k + j + 2)!
=
1∫
0
xk(t)yj(t)dt, k, j = 0,∞,
äå
xk(t) =
tk
k!
, k = 0,∞,
yj(t) =
(1− t)j+1
(j + 1)!
, j = 0,∞.
78 À.Ï. Ãîëóá, Ë.Î. ×åðíåöüêà
Ìíîãî÷ëåí XN−1(t) =
N−1∑
k=0
γ
(N−1)
k xk(t), äëÿ ÿêîãî âèêîíóþòüñÿ
óìîâè áiîðòîãîíàëüíîñòi
1∫
0
XN−1(t)yj(t)dt = 0, j = 0, N − 2,
î÷åâèäíî, ç òî÷íiñòþ äî ìóëüòèïëiêàòèâíî¨ êîíñòàíòè ñïiâïàä๠ç îð-
òîãîíàëüíèì çñóíóòèì íà [0, 1] ìíîãî÷ëåíîì ßêîái P
(0,1)
N−1(t) (äèâ. [13,
c. 580-581]) . Íåõàé êîíñòàíòó âèáðàíî òàê, ùîá ñòàðøèé êîåôiöi¹íò
XN−1(t) äîðiâíþâàâ 1. Òîäi
XN−1(t) = P
(0,1)
N−1 (t) =
=
(N − 1)!
(2N − 1)!
N−1∑
m=0
(−1)m
(
N − 1
m
)
(2N −m− 1)!m
(N −m− 1)!
tN−m−1. (18)
Àëå, ÿê âiäçíà÷åíî â [14], öåé ìíîãî÷ëåí ìîæíà çîáðàçèòè òàêîæ
ó âèãëÿäi
XN−1(t) = ξN−1
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
1
2!
1
3!
. . .
1
(N + 1)!
1
3!
1
4!
. . .
1
(N + 2)!
. . . . . . . . . . . .
1
N !
1
(N + 1)!
. . .
1
(2N − 1)!
1 t . . .
tN−1
(N − 1)!
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
, (19)
äå ξN−1 � êîíñòàíòà, ùî âèçíà÷à¹òüñÿ óìîâàìè íîðìóâàííÿ. Òîäi,
ëåãêî áà÷èòè, ùî
1∫
0
XN−1(t)yN−1(t)dt = ξN−1∆̃N .
Ïîáóäîâà àïðîêñèìàíò Ïàäå . . . 79
Îñêiëüêè
yN−1(t) =
(1− t)N
N !
=
(1− t)
N !
(
(−1)N−1tN−1 + . . .
)
=
=
(1− t)
N !
(
(−1)N−1XN−1(t) + . . .
)
,
à ñòàðøèé êîåôiöi¹íò ìíîãî÷ëåíà 1
ξN−1
XN−1(t), î÷åâèäíî, äîðiâíþ¹
∆̃N−1
(N − 1)!
, òî ç iíøîãî áîêó
ξN−1
(N − 1)!
∆̃N−1
(−1)N−1
N !
hN−1 = ξN−1∆̃N ,
äå (äèâ. [13, c. 580])
hN−1 =
1∫
0
[XN−1(t)]2(1− t)dt =
(N − 1)!(N − 1)!N !N !
2N(2N − 1)!(2N − 1)!
.
Îòæå, ìà¹ìî
∆̃N
∆̃N−1
=
(−1)N−1(N − 1)!N !
2N(2N − 1)!(2N − 1)!
,
çâiäêè
∆̃N =
N∏
k=1
(−1)k−1(k − 1)!k!
(2k)!(2k − 1)!
,
à òîìó íà îñíîâi (17) ìà¹ìî
∆N = (−1)
(N−1)N
2 N !
N∏
k=1
(−1)k−1(k − 1)!(N + k)!
(2k)!(2k − 1)!
. (20)
Âñòàíîâèìî, ùî íàñïðàâäi ïðè êîæíîìó íàòóðàëüíîìó N âèçíà÷-
íèê ∆N = 1. Î÷åâèäíî, ùî ∆1 = 1. Íåõàé ïðè äåÿêîìó N ∈ N:
∆N = 1. Òîäi çãiäíî ç (20)
∆N+1 = (N + 1)!
N+1∏
k=1
(k − 1)!(N + k + 1)!
(2k)!(2k − 1)!
=
80 À.Ï. Ãîëóá, Ë.Î. ×åðíåöüêà
= ∆N
N !(N + 1)
(2N + 2)!(2N + 1)!
N+1∏
k=1
(N + 1 + k)!
N∏
k=1
(N + k)!
=
=
(N + 1)!
(2N + 1)!
N∏
k=1
(N + 1 + k) =
(N + 1)!
(2N + 1)!
· (2N + 1)!
(N + 1)!
= 1.
Ïåðåéäåìî òåïåð äî ïiäðàõóíêó âèçíà÷íèêà
∆(N)
N =
∣∣∣∣∣∣∣∣∣∣
1 1 . . . 1 β0
N + 1 N + 2 . . . 2N − 1 β1(
N+1
2
) (
N+2
2
)
. . .
(
2N−1
2
)
β2
. . . . . . . . . . . . . . .(
N+1
N−1
) (
N+2
N−1
)
. . .
(
2N−1
N−1
)
βN−1
∣∣∣∣∣∣∣∣∣∣
,
äå
βm=(−1 + (−1)m) p(2N)
m −
(
m+1
m
)
p
(2N)
m+1−
(
m+2
m
)
p
(2N)
m+2−. . .−
(
N
m
)
p
(2N)
N .
Î÷åâèäíî, ÿê i ðàíiøå
∆(N)
N =
N−1∏
k=1
(N+k)!
N−1∏
k=1
k!
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
1
(N+1)!
1
(N+2)!
. . .
1
(2N−1)!
β0 · 0!
1
N !
1
(N+1)!
. . .
1
(2N−2)!
β1 · 1!
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
1
2!
1
3!
. . .
1
N !
βN−1 ·(N−1)!
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
=
= (−1)
(N−1)N
2
N−1∏
k=1
(N + k)!
N−1∏
k=1
k!
×
Ïîáóäîâà àïðîêñèìàíò Ïàäå . . . 81
×
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
1
2!
1
3!
. . .
1
N !
βN−1 · (N−1)!
1
3!
1
4!
. . .
1
(N + 1)!
βN−2 · (N−2)!
. . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
1
(N + 1)!
1
(N + 2)!
. . .
1
(2N − 1)!
β0 · 0!
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
.
Ïîçíà÷èìî îñòàííié âèçíà÷íèê ÷åðåç ∆̃(N)
N i ðîçêëàäåìî éîãî çà
åëåìåíòàìè îñòàííüîãî ñòîâï÷èêà. Îòðèìà¹ìî
∆̃(N)
N =
N−1∑
k=0
(−1)kβk k!A(N)
k ,
äå
A
(N)
k =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
1
2!
1
3!
. . .
1
N !
1
3!
1
4!
. . .
1
(N + 1)!
. . . . . . . . . . . .
1
(N − k)!
1
(N − k + 1)!
. . .
1
(2N − k − 2)!
1
(N − k + 2)!
1
(N − k + 3)!
. . .
1
(2N − k)!
. . . . . . . . . . . .
1
(N + 1)!
1
(N + 2)!
. . .
1
(2N − 1)!
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
�
öå âèçíà÷íèê, ùî îòðèìó¹òüñÿ ç âèçíà÷íèêà ∆̃N âèêèäàííÿì îñòàí-
íüîãî ñòîâï÷èêà i ðÿäêà ç íîìåðîì N−k. Ïðè öüîìó ëåãêî ïîìiòèòè,
ùî íà îñíîâi (18) òà (19) ìîæíà çàïèñàòè
1
ξN−1
XN−1(t) =
N−1∑
k=0
tN−k−1
(N − k − 1)!
(−1)kA
(N)
k =
=
1
ξN−1
(N − 1)!
(2N − 1)!
N−1∑
m=0
(−1)m
(
N − 1
m
)
(2N −m− 1)!
(N −m− 1)!
tN−1−m.
82 À.Ï. Ãîëóá, Ë.Î. ×åðíåöüêà
Çâiäñè îòðèìó¹ìî
A
(N)
k =
1
ξN−1
(N − 1)!
(2N − 1)!
(
N − 1
k
)
(2N − k − 1)!.
Îòæå,
∆̃(N)
N =
1
ξN−1
(N − 1)!
(2N − 1)!
N−1∑
k=0
(−1)kβkk!
(
N − 1
k
)
(2N − k − 1)!.
Îñêiëüêè
ξN−1 =
(N − 1)!
∆̃N−1
,
òî
∆̃(N)
N =
∆̃N−1
(2N − 1)!
N−1∑
k=0
(−1)kβk k!
(
N − 1
k
)
(2N − k − 1)!.
Ïiäñòàâëÿþ÷è ñþäè çíà÷åííÿ êîåôiöi¹íòiâ βk, k = 0, N − 1, îò-
ðèìà¹ìî
∆̃(N)
N =
∆̃N−1
(2N−1)!
N−1∑
k=0
(−1)kk!
(
N−1
k
)
(2N −k−1)!
{
(−1+(−1)k)p(2N)
k −
−
(
k + 1
k
)
p
(2N)
k+1 −
(
k + 2
k
)
p
(2N)
k+2 − . . .−
(
N
k
)
p
(2N)
N
}
.
Âðàõîâóþ÷è, ùî íà ïiäñòàâi (17)
∆N−1 = (−1)
(N−2)(N−1)
2
N−1∏
p=1
(N + p− 1)!
N−2∏
p=1
p!
∆̃N−1 = 1,
ìà¹ìî
∆̃N−1 = (−1)
(N−2)(N−1)
2
N−2∏
p=1
p!
N−1∏
p=1
(N + p− 1)!
.
Ïîáóäîâà àïðîêñèìàíò Ïàäå . . . 83
Îòæå,
∆̃(N)
N = (−1)
(N−2)(N−1)
2
N−2∏
p=1
p!
N−1∏
p=1
(N + p− 1)!
1
(2N − 1)!
N−1∑
k=0
(−1)kk!×
×
(
N − 1
k
)
(2N − k − 1)!
{
(−1 + (−1)k)p(2N)
k −
N∑
r=k+1
(
r
k
)
p(2N)
r
}
,
à òîìó
∆(N)
N =
(−1)N
N !(N − 1)!
N−1∑
k=0
(−1)kk!
(
N − 1
k
)
(2N − k − 1)!×
×
{
(1− (−1)k)p(2N)
k +
N∑
r=k+1
(
r
k
)
p(2N)
r
}
=
=
2(−1)N−1
N !
[ N
2 ]−1∑
m=0
(2N − 2m− 2)!
(N − 2m− 2)!
p
(2N)
2m+1+
+
(−1)N
N !
N∑
k=1
(−1)k−1 (2N − k)!
(N − k)!
N∑
r=k
(
r
k − 1
)
p(2N)
r =
=
(−1)N
N !
{
N∑
r=1
p(2N)
r
r∑
k=1
(−1)k−1 (2N − k)!
(N − k)!
(
r
k − 1
)
−
−2
[ N
2 ]−1∑
m=0
(2N − 2m− 2)!
(N − 2m− 2)!
p
(2N)
2m+1
}
,
çâiäêè i âèïëèâàþòü ñïiââiäíîøåííÿ (14), à ç íèìè é òâåðäæåííÿ
ëåìè.
Íà îñíîâi âèùåíàâåäåíèõ ìiðêóâàíü ñôîðìóëþ¹ìî òàêi ðåçóëüòà-
òè.
84 À.Ï. Ãîëóá, Ë.Î. ×åðíåöüêà
Òåîðåìà 1. Äëÿ àíàëiòè÷íî¨ ôóíêöi¨, ùî ì๠iíòåãðàëüíå çîá-
ðàæåííÿ (5), ó âèïàäêó íåñèìåòðè÷íî¨ ìiðè dµ(t) 6≡ dµ(1 − t) ïðè
äîâiëüíèõ N1, N2 ∈ N ðàöiîíàëüíi ôóíêöi¨
[N /D ]f (z, w) =
PN (z, w)
QN1,N2(z, w)
,
òàêi ùî
QN1,N2(z, w) =
N1∑
j=0
N2∑
n=0
c
(N1,N2)
N1−j,N2−nzjwn,
PN (z, w) =
N1−1∑
k=0
N2−1∑
m=0
zkwm
k∑
j=0
m∑
n=0
c
(N1,N2)
N1−j,N2−nsk−j,m−n+
+zN1
N2−1∑
m=0
N1+2N2−1−m∑
k=0
zkwm
N1∑
j=0
m∑
n=0
c
(N1,N2)
j,N2−n sk+j,m−n+
+wN2
N1−1∑
k=0
2N1+N2−1−k∑
m=0
zkwm
k∑
j=0
N2∑
n=0
c
(N1,N2)
N1−j,n sk−j,m+n,
äå êîåôiöi¹íòè c
(N1,N2)
k,m , k = 0, N1, m = 0, N2 çàäîâîëüíÿþòü ðiâíîñòi
(8), ìàòèìóòü ðîçêëàäè â ñòåïåíåâi ðÿäè, êîåôiöi¹íòè ÿêèõ ñïiâ-
ïàäàòèìóòü ç êîåôiöi¹íòàìè (6) äëÿ ôóíêöi¨ (5) äëÿ âñiõ (j, n) ∈
E = {(j, n) ∈ Z2
+ : j + n 6 2N1 + 2N2 − 1}.
Çîêðåìà, öå ñïðàâåäëèâî äëÿ ðàöiîíàëüíî¨ ôóíêöi¨:
[N /D ](I)
f (z, w) =
P
(I)
N (z, w)
Q
(I)
N1,N2
(z, w)
,
äå
Q
(I)
N1,N2
(z, w) = wN2
N1∑
j=1
p
(N1+N2)
N1−j zj+
+
N2∑
n=0
(−1)N2−nwn
n∑
k=0
p
(N1+N2)
N1+N2−k
(
N2 − k
N2 − n
)
,
Ïîáóäîâà àïðîêñèìàíò Ïàäå . . . 85
P
(I)
N (z, w) =
N1−1∑
k=0
N2−1∑
m=0
zkwm
m∑
n=0
(−1)N2−n
N2∑
r=N2−n
p
(N1+N2)
r+N1
(
r
N2−n
)
sk,m−n+
+zN1
N2−1∑
m=0
N1+2N2−1−m∑
k=0
zkwm
m∑
n=0
(−1)N2−n
N2∑
r=N2−n
p
(N1+N2)
r+N1
(
r
N2−n
)
sk+N1,m−n+
+wN2
N1−1∑
k=0
2N1+N2−1−k∑
m=0
zkwm
k∑
j=0
p
(N1+N2)
N1−j sk−j,m+
+wN2
N1−1∑
k=0
2N1+N2−1−k∑
m=0
zkwm
N2∑
n=0
(−1)n
N2∑
r=n
p
(N1+N2)
r+N1
(
r
n
)
sk,m+n,
äå p
(N1+N2)
j � êîåôiöi¹íòè àëãåáðà¨÷íîãî ìíîãî÷ëåíà PN1+N2(t), îð-
òîíîðìîâàíîãî íà [0, 1] ç âàãîþ dµ(t).
Òåîðåìà 2. Äëÿ àíàëiòè÷íî¨ ôóíêöi¨, ùî ì๠iíòåãðàëüíå çîá-
ðàæåííÿ (5), ó âèïàäêó ñèìåòðè÷íî¨ ìiðè dµ(t) ≡ dµ(1− t) ïðè äî-
âiëüíîìó N ∈ N ðàöiîíàëüíi ôóíêöi¨
[N /D ]f (z, w) =
PN (z, w)
QN,N (z, w)
,
òàêi ùî
QN,N (z, w) =
N1∑
j=0
N∑
n=0
c
(N,N)
N−j,N−nzjwn,
PN (z, w) =
N−1∑
k=0
N−1∑
m=0
zkwm
k∑
j=0
m∑
n=0
c
(N1,N2)
N−j,N−nsk−j,m−n+
+zN
N−1∑
m=0
3N−1−m∑
k=0
zkwm
N∑
j=0
m∑
n=0
c
(N,N)
j,N−nsk+j,m−n+
+wN
N−1∑
k=0
3N−1−k∑
m=0
zkwm
k∑
j=0
N∑
n=0
c
(N,N)
N−j,nsk−j,m+n,
86 À.Ï. Ãîëóá, Ë.Î. ×åðíåöüêà
äå êîåôiöi¹íòè c
(N,N)
k,m , k, m = 0, N çàäîâîëüíÿþòü ðiâíîñòi
N∑
k=0
N∑
m=0
c
(N,N)
k,m tk(1− t)m =
2N∑
j=0
p
(2N)
j tj ,
ìàòèìóòü ðîçêëàäè â ñòåïåíåâi ðÿäè, êîåôiöi¹íòè ÿêèõ ñïiâ-
ïàäàòèìóòü ç êîåôiöi¹íòàìè (6) äëÿ ôóíêöi¨ (5) äëÿ âñiõ
(j, n) ∈ E = {(j, n) ∈ Z2
+ : j + n 6 4N − 1}.
Çîêðåìà, öå ñïðàâåäëèâî äëÿ ðàöiîíàëüíî¨ ôóíêöi¨:
[N /D ](II)
f (z, w) =
P
(II)
N (z, w)
Q
(II)
N,N (z, w)
,
äå
Q
(II)
N,N (z, w) = p
(2N)
0 zNwN +
N∑
k=1
k∑
j=1
(2k − j − 1)!j
k!(k − j)!
p
(2N)
j zN−kwN−k,
P
(II)
N (z, w) =
N−1∑
r=0
N−1∑
m=0
zrwm
N∑
k=N−r
k∑
j=1
(2k − j − 1)!j
k!(k − j)!
p
(2N)
j sr−N+k,m−N+k+
+zN
N−1∑
m=0
3N−1−m∑
r=0
zrwm
N∑
k=1
k∑
j=1
(2k − j − 1)!j
k!(k − j)!
p
(2N)
j sr+k,m−N+k+
+wN
N−1∑
r=0
3N−1−r∑
m=0
zrwm
N∑
k=1
k∑
j=1
(2k − j − 1)!j
k!(k − j)!
p
(2N)
j sr−N+k,m+k,
äå p
(2N)
j , j = 0, N� êîåôiöi¹íòè àëãåáðà¨÷íîãî ìíîãî÷ëåíà P2N (t),
îðòîíîðìîâàíîãî íà [0, 1] ç âàãîþ dµ(t).
Ó âèïàäêó ìiðè
dµ(t) = tν(1− t)σ
dt, ν, σ > −1
êîåôiöi¹íòè ñòåïåíåâîãî ðîçâèíåííÿ ôóíêöi¨ âèãëÿäó (6) áóäóòü
äîðiâíþâàòè
sk,m =
1∫
0
tk+ν(1− t)m+σ
dt =
Γ(k + ν + 1)Γ(m + σ + 1)
Γ(k + m + ν + σ + 2)
, k, m = 0,∞.
Ïîáóäîâà àïðîêñèìàíò Ïàäå . . . 87
Îòæå, ïîáóäîâàíà ôóíêöiÿ
f(z, w) =
∞∑
k=0
∞∑
m=0
Γ(k + ν + 1)Γ(m + σ + 1)
Γ(k + m + ν + σ + 2)
zkwm (21)
áóäå ÷àñòèííèì âèïàäêîì ãiïåðãåîìåòðè÷íîãî ðÿäó Àïïåëÿ
F3(α, α′, β, β′, γ, z, w) =
∞∑
k,m=0
(α)k(α′)m(β)k(β′)m
(γ)k+mk!m!
zkwm
(äèâ. [15, c. 220, ôîðìóëà (8)]) ïðè α = ν + 1, α′ = σ + 1, β = β′ = 1,
γ = ν + σ + 2.
Ó öüîìó âèïàäêó ìíîãî÷ëåí XN1,N2(t) áóäå ñïiâïàäàòè ç òî÷íiñòþ
äî ñòàëîãî ìíîæíèêà ç îðòîíîðìîâàíèì çñóíóòèì íà [0, 1] ìíîãî÷ëå-
íîì ßêîái P
(ν,σ)∗
N1+N2
(t) ñòåïåíÿ N1 + N2.
Âðàõîâóþ÷è ÿâíèé âèðàç äëÿ êîåôiöi¹íòiâ îðòîãîíàëüíèõ ìíîãî-
÷ëåíiâ ßêîái (äèâ. [13, ñ. 581, ï.(22.3.3)]) (êîíñòàíòó äëÿ çðó÷íîñòi
ïîêëàäåìî ðiâíîþ 1)
P
(ν,σ)∗
N1+N2
(t)=
N1+N2∑
m=0
(−1)m
(
N1+N2
m
)
Γ(N1 + N2 + ν + σ +1 + m)
Γ(ν + 1 + m)
tm, (22)
pN1+N2
j = (−1)j
(
N1 + N2
j
)
Γ(N1 + N2 + ν + σ + 1 + j)
Γ(ν + 1 + j)
,
äëÿ ðÿäiâ âèãëÿäó (21) ïðè ν 6= σ ì๠ìiñöå òàêèé ðåçóëüòàò.
Òåîðåìà 3. Äëÿ ãiïåðãåîìåòðè÷íîãî ðÿäó Àïïåëÿ
f(z, w) = F3(α, α′, 1, 1, γ, z, w) =
∞∑
k,m=0
(α)k(α′)m
(γ)k+m
zkwm
ïðè α = ν + 1, α′ = σ + 1, γ = ν + σ + 2, ν, σ > −1, ν 6= σ äëÿ áóäü-
ÿêèõ N1, N2 ∈ N ðàöiîíàëüíà ôóíêöiÿ
[N /D ](I)
f (z, w) =
P
(I)
N (z, w)
Q
(I)
N1,N2
(z, w)
,
88 À.Ï. Ãîëóá, Ë.Î. ×åðíåöüêà
äå
Q
(I)
N1,N2
(z, w) = wN2
N1∑
j=1
(−1)N1−j
(
N1 + N2
N1 − j
)
×
×Γ(2N1 + N2 + ν + σ + 1− j)
Γ(N1 + ν + 1− j)
zj +
N2∑
n=0
wn
n∑
k=0
(−1)N1−k−n
(
N2 − k
N2 − n
)
×
×
(
N1 + N2
N1 + N2 − k
)
Γ(2N1 + 2N2 + ν + σ + 1− k)
Γ(N1 + N2 + ν + 1− k)
,
P
(I)
N (z, w) =
N1−1∑
k=0
N2−1∑
m=0
zkwm
m∑
n=0
(−1)N2−n
N2∑
r=N2−n
(−1)N1+r
(
N1 + N2
N1 + r
)
×
×
(
r
N2−n
)
Γ(2N1+N2 + ν + σ + 1 + r)
Γ(N1 + ν + 1 + r)
Γ(k + ν + 1)Γ(m− n + σ + 1)
Γ(k + m− n + ν + σ + 2)
+
+zN1
N2−1∑
m=0
N1+2N2−1−m∑
k=0
zkwm
m∑
n=0
(−1)N2−n
N2∑
r=N2−n
(−1)N1+r
(
N1 + N2
N1 + r
)
×
×
(
r
N2 − n
)
Γ(2N1+N2+ν+σ+1+r)
Γ(N1 + ν + 1 + r)
Γ(N1+k+ν+1)Γ(m−n+σ+1)
Γ(N1 + k + m− n + ν + σ + 2)
+
+wN2
N1−1∑
k=0
2N1+N2−1−k∑
m=0
zkwm
k∑
j=0
(−1)N1−j
(
N1 + N2
N1 − j
)
×
×Γ(2N1 + N2 + ν + σ + 1− j)
Γ(N1 + ν + 1− j)
Γ(k − j + ν + 1)Γ(m + σ + 1)
Γ(k + m− j + ν + σ + 2)
+
+wN2
N1−1∑
k=0
2N1+N2−1−k∑
m=0
zkwm
N2∑
n=0
(−1)n
N2∑
r=n
(−1)N1+r
(
N1 + N2
N1 + r
)
×
×Γ(2N1 + N2 + ν + σ + 1 + r)
Γ(N1 + ν + 1 + r)
(
r
n
)
Γ(k + ν + 1)Γ(m + n + σ + 1)
Γ(k + m + n + ν + σ + 2)
,
ìàòèìå ðîçêëàä â ñòåïåíåâèé ðÿä, êîåôiöi¹íòè ÿêî-
ãî ñïiâïàäàòèìóòü ç êîåôiöi¹íòàìè ðÿäó (21) äëÿ âñiõ
(j, n) ∈ E = {(j, n) ∈ Z2
+ : j + n 6 2N1 + 2N2 − 1}.
Ïîáóäîâà àïðîêñèìàíò Ïàäå . . . 89
Ó âèïàäêó, êîëè ν = σ ìíîãî÷ëåí XN,N (t) áóäå ñïiâïàäàòè ç
òî÷íiñòþ äî ñòàëîãî ìíîæíèêà ç îðòîíîðìîâàíèì çñóíóòèì íà [0, 1]
ìíîãî÷ëåíîì Ãåãåíáàóåðà C
(ν+ 1
2 )∗
2N (t). Êîåôiöi¹íòè öüîãî ìíîãî÷ëå-
íà ìîæíà îá÷èñëèòè çi ñïiââiäíîøåííÿ çâ'ÿçêó ç ìíîãî÷ëåíîì ßêîái
(äèâ. [13, ñ. 584, ï. (22.5.27)]):
C
(ν)
N (t) =
(2ν)N(
ν + 1
2
)
N
P
(ν− 1
2 ,ν− 1
2 )
N (t).
Âðàõîâóþ÷è (22), çàïèøåìî
P
(ν,ν)∗
2N (t) =
2N∑
m=0
(−1)m
(
2N
m
)
Γ(2N + 2ν + 1 + m)
Γ(ν + 1 + m)
tm.
Îòæå,
C
(ν+ 1
2 )∗
2N (t) =
(2ν + 1)2N
(ν + 1)2N
2N∑
m=0
(−1)m
(
2N
m
)
Γ(2N + 2ν + 1 + m)
Γ(ν + 1 + m)
tm.
Îòðèìà¹ìî â òàêîìó ðàçi
p2N
j = (−1)j (2ν + 1)2N
(ν + 1)2N
(
2N
j
)
Γ(2N + 2ν + 1 + j)
Γ(ν + 1 + j)
. (23)
Ïiäñòàâëÿþ÷è (23) â (13), çàïèøåìî êîåôiöi¹íòè c
(N)
k â òàêîìó
âèãëÿäi
c
(N)
k =
Γ2(2N + 2ν + 1)
Γ(2N + ν + 1)
, k=0,
k∑
j=1
(−1)j
(
2N
j
)
(2ν+1)2N
(ν+1)2N
(2k−j−1)!j
k!(k−j)!
Γ(2N+2ν+1+j)
Γ(ν+1+j)
, k>1.
(24)
Îòæå, äëÿ ðÿäiâ âèãëÿäó
f(z, w) =
∞∑
k=0
∞∑
m=0
Γ(k + ν + 1)Γ(m + ν + 1)
Γ(k + m + 2ν + 2)
zkwm (25)
90 À.Ï. Ãîëóá, Ë.Î. ×åðíåöüêà
íà îñíîâi òåîðåìè 2 ìîæíà ïîáóäóâàòè ðàöiîíàëüíi àïðîêñèìàíòè
Ïàäå, à ñàìå ì๠ìiñöå òàêèé ðåçóëüòàò.
Òåîðåìà 4. Äëÿ ãiïåðãåîìåòðè÷íîãî ðÿäó Àïïåëÿ
f(z, w) = F3(α, α′, 1, 1, γ, z, w) =
∞∑
k,m=0
(α)k(α′)m
(γ)k+m
zkwm
ïðè α = α′ = ν + 1, γ = 2ν + 2, ν > −1 äëÿ áóäü-ÿêîãî N ∈ N ðàöiî-
íàëüíà ôóíêöiÿ
[N /D ](II)
f (z, w) =
P
(II)
N (z, w)
Q
(II)
N,N (z, w)
,
äå
Q
(II)
N,N (z, w) =
Γ2(2N + 2ν + 1)
Γ(2N + ν + 1)
zNwN +
N∑
k=1
c
(N)
k zN−kwN−k.
P
(II)
N (z, w)=
N−1∑
r=0
N−1∑
m=0
zrwm
N∑
k=N−r
c
(N)
k
Γ(r−N+k+ν+1)Γ(m−N+k+ν+1)
Γ(r + m− 2N + 2k + 2ν + 2)
+
+zN
N−1∑
m=0
3N−1−m∑
r=0
zrwm
N∑
k=1
c
(N)
k
Γ(r + k + ν + 1)Γ(m−N + k + ν + 1)
Γ(r + m−N + 2k + 2ν + 2)
+
+wN
N−1∑
r=0
3N−1−r∑
m=0
zrwm
N∑
k=1
c
(N)
k
Γ(r −N + k + ν + 1)Γ(m + k + ν + 1)
Γ(r + m−N + 2k + 2ν + 2)
,
äå c
(N)
k ìàþòü âèãëÿä (24), ìàòèìå ðîçêëàä â ñòåïåíåâèé ðÿä, êîå-
ôiöi¹íòè ÿêîãî ñïiâïàäàòèìóòü ç êîåôiöi¹íòàìè ðÿäó (25) äëÿ âñiõ
(j, n) ∈ E = {(j, n) ∈ Z2
+ : j + n 6 4N − 1}.
Ùîá ïðîiëþñòðóâàòè ðåçóëüòàò òåîðåìè 4, ðîçãëÿíåìî ÷àñòèííèé
âèïàäîê ðÿäó (25) ïðè ν = 0. Òîäi ôóíêöiÿ f ìàòèìå âèãëÿä
f(z, w) =
ln((1− z)(1− w))
zw − w − z
. (26)
Ïîáóäîâà àïðîêñèìàíò Ïàäå . . . 91
Ïîêëàäåìî N = 2. Çà òåîðåìîþ 4 îòðèìà¹ìî ðàöiîíàëüíó ôóíê-
öiþ
PN (z, w)
Q2,2(z, w)
=
(
1680 + 840(z + w) + 560(z2 + w2)− 200zw+
+420(z3 + w3)− 100(z2w + zw2) + 336(z4 + w4)− 76(z3w + zw3)+
+280(z5 + w5)− 64(z4w + zw4) + 240(z6 + w6)− 56(z5w + zw5)+
+ 210(z7 + w7)− 50(z6w + zw6)
) · (24z2w2 − 480zw + 1680)−1.
Íàâåäåìî ïîðiâíÿëüíó òàáëèöþ, â ÿêó äëÿ êîæíîãî çíà÷åííÿ
çìiííî¨ â ïåðøèé ðÿäîê çàïèøåìî çíà÷åííÿ íàáëèæóâàíî¨ ôóíê-
öi¨ (26), â äðóãèé � ÷àñòèííî¨ ñóìè ñòåïåíåâîãî ðÿäó
P7(z, w) = 1+
1
2
(z+w)+
1
3
(z2+w2)+
1
6
zw+
1
4
(z3+w3)+
1
12
(z2w+zw2)+
+
1
5
(z4+w4)+
1
20
(z3w + zw3)+
1
30
z2w2+
1
6
(z5+w5)+
1
30
(z4w+zw4)+
+
1
60
(z3w2+z2w3)+
1
7
(z6+w6)+
1
42
(z5w+zw5)+
1
105
(z4w2+z2w4)+
+
1
140
z3w3 +
1
8
(z7 + w7) +
1
56
(z6w + zw6) +
1
168
(z5w2 + z2w5)+
+
1
280
(z4w3 + z3w4), â òðåòié � ïîáóäîâàíî¨ àïðîêñèìàíòè.
w\z 0.0 0.2 0.4 0.6 0.8
1 1.115717756 1.277064060 1.527151220 2.011797390
0.0 1 1.115717409 1.276949943 1.523044343 1.941530209
1 1.115717410 1.276949943 1.523044343 1.941530210
1.115717756 1.239686396 1.411479183 1.675638651 2.181644599
0.2 1.115717409 1.239685579 1.411356601 1.671363261 2.109533485
1.115717410 1.239685865 1.411376479 1.671735409 2.112634759
1.277064060 1.411479183 1.596330075 1.877784679 2.409390381
0.4 1.276949943 1.411356601 1.596062597 1.873147658 2.334971366
1.276949943 1.411376479 1.596149340 1.874023685 2.341741096
1.527151220 1.675638651 1.877784679 2.181644599 2.745357222
0.6 1.523044343 1.671363261 1.873147658 2.172121327 2.663451627
1.523044343 1.671735409 1.874023685 2.174752740 2.676199310
2.011797390 2.181644599 2.409390381 2.745357222 3.352995652
0.8 1.941530209 2.109533485 2.334971366 2.663451627 3.193274881
1.941530210 2.112634759 2.341741096 2.676199310 3.224728471
92 À.Ï. Ãîëóá, Ë.Î. ×åðíåöüêà
Âiäïîâiäíèé ãðàôiê ìàòèìå âèãëÿä
Ïîêëàäåìî òåïåð N = 3. Îòðèìà¹ìî ðàöiîíàëüíó ôóíêöiþ
PN (z, w)
Q3,3(z, w)
=
(−665280− 332640(z + w)−221760(z2+w2) + 161280zw −
−166320(z3 +w3)+ 80640(z2w + zw2)− 133056(z4 +w4)+ 57456(z3w+
+zw3)−7056z2w2−110880(z5 +w5)+45864(z4w + zw4)−3528(z3w2+
+z2w3)− 95040(z6 + w6) + 38592(z5w + zw5)− 2808(z4w2 + z2w4)−
−83160(z7 + w7) + 33480(z6w + zw6)− 2448(z5w2 + z2w5)− 73920(z8+
+w8) + 29640(z7w + zw7)− 2208(z6w2 + z2w6)− 66528(z9 + w9)+
+26628(z8w+zw8)−2028(z7w2+z2w7)−60480(z10+w10) + 24192(z9w+
+zw9)− 1884(z8w2 + z2w8)− 55440(z11 + w11) + 22176(z10w + zw10)−
− 1764(z9w2 + z2w9)
) · (720z3w3−30240z2w2 +272160zw−665280)−1.
Ïîáóäîâà àïðîêñèìàíò Ïàäå . . . 93
Íàâåäåìî òàáëèöþ, â ïåðøèé ðÿäîê ÿêî¨ çàïèñàíî çíà÷åííÿ íàá-
ëèæóâàíî¨ ôóíêöi¨ (26), â äðóãèé � ÷àñòèííî¨ ñóìè ñòåïåíåâîãî ðÿäó
P11(z, w) = 1+
1
2
(z+w)+
1
3
(z2+w2)+
1
6
zw+
1
4
(z3+w3)+
1
12
(z2w+zw2)+
+
1
5
(z4 +w4)+
1
20
(z3w+zw3)+
1
30
z2w2 +
1
6
(z5 +w5)+
1
30
(z4w+zw4)+
+
1
60
(z3w2 + z2w3)+
1
7
(z6 +w6)+
1
42
(z5w + zw5)+
1
105
(z4w2 + z2w4)+
+
1
140
z3w3+
1
8
(z7+w7)+
1
56
(z6w+zw6)+
1
168
(z5w2+z2w5)+
1
280
(z4w3+
+z3w4)+
1
9
(z8 +w8)+
1
72
(z7w+zw7)+
1
252
(z6w2 +z2w6)+
1
504
(z5w3+
+z3w5)+
1
630
z4w4 +
1
10
(z9 +w9)+
1
90
(z8w+zw8)+
1
360
(z7w2 +z2w7)+
+
1
840
(z6w3 + z3w6)+
1
1260
(z5w4 + z4w5)+
1
11
(z10 + w10)+
1
110
(z9w+
+zw9)+
1
495
(z8w2 + z2w8)+
1
1320
(z7w3 +z3w7)+
1
2310
(z6w4 +z4w6)+
+
1
2772
z5w5 +
1
12
(z11 + w11) +
1
132
(z10w + zw10) +
1
660
(z9w2 + z2w9)+
+
1
1980
(z8w3 + z3w8) +
1
3960
(z7w4 + z4w7) +
1
5544
(z6w5 + z5w6),
â òðåòié � ïîáóäîâàíî¨ àïðîêñèìàíòè.
w\z 0.0 0.2 0.4 0.6 0.8
1 1.115717756 1.277064060 1.527151220 2.011797390
0.0 1 1.115717755 1.277062003 1.526770377 1.990512901
1 1.115717756 1.277062003 1.526770376 1.990512903
1.115717756 1.239686396 1.411479183 1.675638651 2.181644599
0.2 1.115717755 1.239686395 1.411477036 1.675247520 2.159977487
1.115717756 1.239686395 1.411477464 1.675290370 2.161129002
1.277064060 1.411479183 1.596330075 1.877784679 2.409390381
0.4 1.277062003 1.411477036 1.596325545 1.873147658 2.387302302
1.277062003 1.411477464 1.596327293 1.877468720 2.389704511
1.527151220 1.675638651 1.877784679 2.181644599 2.745357222
0.6 1.526770377 1.675247520 1.873147658 2.180810211 2.722370778
1.526770376 1.675290370 1.877468720 2.181083266 2.726325075
2.011797390 2.181644599 2.409390381 2.745357222 3.352995652
0.8 1.990512901 2.159977487 2.387302302 2.722370778 3.306828505
1.990512903 2.161129002 2.389704511 2.726325075 3.317370936
94 À.Ï. Ãîëóá, Ë.Î. ×åðíåöüêà
Íàâåäåíi ïðèêëàäè ïîêàçóþòü, ùî ïîáóäîâàíi íà îñíîâi òåîðå-
ìè 4 ðàöiîíàëüíi àïðîêñèìàíòè íàáëèæàþòü ôóíêöiþ (26) êðàùå çà
÷àñòèííó ñóìó ñòåïåíåâîãî ðÿäó ç òàêîþ æ êiëüêiñòþ âiëüíèõ êîåôi-
öi¹íòiâ.
1. Ãîëóá À.Ï., ×åðíåöüêà Ë.Î. Äâîâèìiðíi óçàãàëüíåíi ìîìåíòíi çîáðàæåí-
íÿ òà ðàöiîíàëüíi àïðîêñèìàöi¨ ôóíêöié äâîõ çìiííèõ // Óêð. ìàò. æóðí.
� Ïðèéíÿòî äî äðóêó.
2. Ãîëóá À.Ï., ×åðíåöüêà Ë.Î. Äâîâèìiðíi óçàãàëüíåíi ìîìåíòíi çîáðàæåí-
íÿ òà àïðîêñèìàöi¨ Ïàäå äåÿêèõ ðÿäiâ Ãóìáåðòà // Óêð. ìàò. æóðí. �
Ïðèéíÿòî äî äðóêó.
3. Alabiso C., Butera P. N-variable rational approximants and method of
moments // J. Math. Phys. � 1975. � 16, �4. � P. 840 � 845.
4. Cuyt A. How well can the concept of Pad�e approximant be generalized to
the multivariate case? // J. Comput. Appl. Math. � 1999. � 105, �1 � 2. �
P. 25 � 50.
5. Hughes Jones R. General rational approximants in N variables // J. Approx.
Theory. � 1976. � 16. � P. 201 � 233.
6. Lutterodt C. A two-dimensional analogue of Pad�e approximant theory // J.
Phys. A. : Math. � 1974. � 7. � P. 1027 � 1037.
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11. Ñóåòèí Ï.Ê. Êëàññè÷åñêèå îðòîãîíàëüíûå ìíîãî÷ëåíû. � Ì.: Íàóêà,
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|
| id | oai:trim.imath.kiev.ua:article-140 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | Ukrainian |
| last_indexed | 2026-08-04T01:02:53Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/a2/1f3a62ea633a7dc4e6508c335b9d5aa2.pdf |
| spelling | oai:trim.imath.kiev.ua:article-1402018-01-29T14:44:27Z Construction of the Padé approximants for some Appell hypergeometric series by means of method of generalized moment representations Побудова апроксимант Паде для деяких гіпергеометричних рядів Аппеля за допомогою методу узагальнених моментних зображень Golub, A. P. Chernets’ka, L. O. Голуб, А. П. Чернецька, Л. О. By means of extension of V.K. Dzyadyk's method of generalized moment representations to the case of two--dimensional number sequences Pad\'e approximants for some Appell hypergeometric series are constructed За допомогою поширення методу узагальнених моментних зображень В.К. Дзядика на випадок двовимірних числових послідовностей побудовано апроксиманти Паде для деяких гіпергеометричних рядів Аппеля Інститут математики НАН України 2013-07-15 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/140 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 10 No. 1 (2013): Approximation Theory of Functions and Related Problems; 69-94 Сборник Трудов Института математики НАН Украины; Том 10 № 1 (2013): Tеорiя наближення функцiй та сумiжнi питання; 69-94 Збірник Праць Інституту математики НАН України; Том 10 № 1 (2013): Tеорiя наближення функцiй та сумiжнi питання; 69-94 3083-7529 1815-2910 uk https://trim.imath.kiev.ua/index.php/trim/article/view/140/115 Авторське право (c) 2013 Інститут математики НАН України |
| spellingShingle | Golub, A. P. Chernets’ka, L. O. Голуб, А. П. Чернецька, Л. О. Construction of the Padé approximants for some Appell hypergeometric series by means of method of generalized moment representations |
| title | Construction of the Padé approximants for some Appell hypergeometric series by means of method of generalized moment representations |
| title_alt | Побудова апроксимант Паде для деяких гіпергеометричних рядів Аппеля за допомогою методу узагальнених моментних зображень |
| title_full | Construction of the Padé approximants for some Appell hypergeometric series by means of method of generalized moment representations |
| title_fullStr | Construction of the Padé approximants for some Appell hypergeometric series by means of method of generalized moment representations |
| title_full_unstemmed | Construction of the Padé approximants for some Appell hypergeometric series by means of method of generalized moment representations |
| title_short | Construction of the Padé approximants for some Appell hypergeometric series by means of method of generalized moment representations |
| title_sort | construction of the padé approximants for some appell hypergeometric series by means of method of generalized moment representations |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/140 |
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