Jacobi, Stieltjes, Gauss : :

Transcription

1 Jacobi, Stieltjes, Gauss : :

2 894 T. J. Stieltjes [St94a] Recherches sur les fractions continues Stieltjes 0 f(u)du, z + u f(u) > 0, z C z + + a a 2 z + a , a p > 0 (a) Vitali (a) Stieltjes z + b z + b 2 p p 2 p 3 z + b 3 z + b 4..., b k (a k ), p k (a k ) > 0 (b) 903 E. B. Van Vleck [Vl03] Stieltjes Stieltjes 920 Hamburger [Ha20] 990 Pringsheim [Pr98] Vleck [Vl0a] + c c , c p C, (c) i

3 b + b 2 + b , b p C (d) (c) Pringsheim g p [0, ], (p =, 2, 3,... ) c p ( g p )g p Vleck g 0 = 0 + g g 2 g p /( g )( g 2 ) ( g p ) 2 Jacobi a 2 z + b a 2 2 z 2 + b 2 z 3 + b 3..., z p C (e) 2 ( ) 2 n p= n I(b p + z p ) x p 2 I(a p )(x p x p+ + x p x p+ ), x n C, n =, 2, 3,... p= I(z p ) > 0 Jacobi {K p (z)} {K p (z)} r p (z) {( g p )g p } 3 Jacobi {c p } {c p } 4 4 {c p } (k 0 + k u + + k n u n ) dϕ c (u) = k 0 c 0 + k c + + k n c n, k n C : (f) 3 2p (p =, 2, 3,... ) ii

4 Stieltjes [St89] Stieltjes 5 (f) Stieltjes Lebesgue {c p } µ p = 0 u p dµ(u), p = 0,, 2,... µ(u) µ(u) Stieltjes Stieltjes Jacobi Stieltjes Jacobi 6 Gauss Gauss a, b C c {0,, 2, 3, } F (a, b, c; z) = n=0 (a) n (b) n (c) n n! zn, z < F (a, b +, c + ; z)/f (a, b, c; z) [Ga76] (a) n Pochhammer (a) 0 := (a) n+ := (a) n (a + n), (n = 0,, 2,... ) Gauss Gauss a, b, c Gauss Jacobi Gauss iii

5 Worpitzky ( g p )g p x p / Jacobi Stieltjes Stieltjes Jacobi Jacobi Jacobi f /f Stieltjes Jacobi Jacobi Stieltjes Stieltjes Stieltjes iv

6 Gauss Gauss v

7 N N 0 R R >0 0 C Ĉ R R #R R R c R A B B A A B A B A B A B A\B A B a A A a R(z) z I(z) z z z = x + iy i.e., z = x iy θ = arg(z) z r = z z = x + iy i.e., z = x 2 + y 2 z = re iθ r = z θ = arg(z) g f(ω) f, g i.e., g(f(ω)) max{a, b} a, b min{a, b} a, b deg f f vi

8 572 R. Bombelli Lord W. Brouncker 4 π = T. J. Stieltjes [St94a] z + + a a 2 z + a , a p > 0, z C Stieltjes Stieltjes Jacobi Jacobi Stieltjes Jacobi Stieltjes Gauss Gauss Jacobi Stieltjes Gauss

9 3... : a b 0 + a 2 b + b (.) a p C, (p =, 2, 3,... ), b p C, (p = 0,, 2,... ) a p, b p p p a p /b p p b 0 + a b + a 2 b 2 + a 3 b 3 + b 0 + b + a a 2 b a n = b 0 + a b + a 2 b a n b n b n.2. tan ( ) ζ = ζ + 3ζ + 5ζ + for ζ > 2 π. 2 3 Jacobi ( ) ω t 0 (ω) := b 0 + ω, t p (ω) := a p, p =, 2, 3,... b p + ω 2

10 t 0 t t p (0) = b 0 + a b + a 2 b a p b p p t 0 t t p (ω) = A p ω + A p B p ω + B p, p = 0,, 2,... A p, B p p p : A =, B = 0, A 0 = b 0, B 0 = ; A p+ = b p+ A p + a p+ A p, B p+ = b p+ B p + a p+ B p, p = 0,, 2,.... (.2) A n A n B n B n = A n b n A n + a n A n 2 B n b n B n + a n B n 2 A n 2 A n = a n B n 2 B n A p B p A p B p = ( ) p a 0 a a p, p = 0,, 2,... (.3) ( a 0 = ) A p, B p {A p /B p } p=0.3. (.) p B p 0 A p lim (.4) p B p (.3) a p 0, (p = 0,, 2,... ) A p B p 0 (.4) B p 0 a p, b p 3

11 .4. (.) D C D a p, b p D (.) {A p /B p } D C.2.5. ([Eu48]) B p p =, 2, 3,... B p 0 (.5) + a 2 + a 3 + (.5) b 2 b 3 ρ p := a p+b p B p+, p =, 2, 3,... (.6) ρ ρ 2 (.7) + ρ + ρ 2 n ρ p (.7) n + ρ ρ 2 ρ p (.8) p= n n. n n ( Ap+ + A ) ( p A2 = + A ) ( A3 + A ) ( 2 An + + A ) n B p+ B p B 2 B B 3 B 2 B n B n p= = A B + A n B n. (.5) t 0 (ω) = ω, t p (ω) = a p b p + ω, (p =, 2, 3,... ), a = b = A /B = t 0 t (0) = n ( Ap+ + A ) p = A n. B p+ B p B n p= 4

12 (.3) a 2 B B 2 + a 2a 3 B 2 B 3 a 2a 3 a 4 B 3 B 4 + (.6) (.8) s = s p (ω) = + ρ p ω s = s p (ω) = ρ p ρ p +, p =, 2, 3,... ω n ω = 0 s s 2 s n (0) = + ρ ( + ρ 2 ( + ρ 3 + = + ρ ρ 2 + ρ ρ 2 ρ ρ ρ n n = + ρ ρ 2 ρ p (.8) n p= s s 2 s n+ (ω) = ρ ρ + ρ ρ n + ρ 2 ρ n ρ n+ ρ n+ +. ω (.7) ρ ρ + ρ n ρ n +. s s 2 s n+ (0) (.7) n =, 2, 3,... (.7) (.5) n (.8) n (.7) n B n = (.7) a n := ρ n, b n := + ρ n, (n =, 2, 3,... ) t 0 (ω) = + ω, t n(ω) = a n b n + ω n A n, B n A 0 /B 0 = t 0 (0) = / B 0 = A /B = t 0 t (0) = + a b = b / B = (.2) B 2 = b 2 B + a 2 B 0 = b 2 + a 2 =,..., B n = b n + a n = (.8) n (.5) (.7) B n =, (n = 0,, 2,... ) A n (.8) n ρ p ρ p (.6) (.2) :.6. ([Wa48]) p b p =, (p = 2, 3, 4,... ) ρ 0 = 0 ρ p = a p+( + ρ p ), p =, 2, 3,... (.9) + a p+ ( + ρ p ) 5

13 a = 0, ρ = ρ 0 = 0 a p+ ρ p =, p =, 2, 3,... (.0) + a p + a p+ + a p ρ p (.) {c p } p= p N c p 0 b 0 + c a c b + c c 2 a 2 c 2 b 2 + c 2c 3 a 3 c 3 b 3 + (.) (.) {c p } A p, B p (.) p (.) p c c 2 c p A p, c c 2 c p B p (.2) 0 2 p A p, B p A p, B p ( A p, B p ) p =, 2, 3,... C p 0 A p = C p A p, B p = C p B p (.2) A =, B = 0, A 0 = b 0, B 0 =, A p = b p A p + a p A p 2, B p = b p B p + a p B p 2 ( ) ( ) ( ) A p B p A p 2 B p 2 b p a p = A p B p det ( A p B p A p 2 B p 2 ) = A p B p 2 A p 2 B p 0 6

14 ( A p B p ( b p a p A p 2 B p 2 ) ( = ) A p A p 2 B p B p 2 ) ( A p B p ). A =, B = 0, A 0 = b 0, B 0 =, A p = b pa p + a pa p 2, B p = b pb p + a pb p 2 (.2) A p = A p C p = b pa p + a pa p 2 = b p C p C p C p C = C 0 = c p := C p 2 A p + a C p p A p 2. C p C p C p C p b p = c p b p, a p = c p c p a p. 2 p b p 0, c p := /b p, (p =, 2, 3,... ) (.) a p 0, (p =, 2, 3,... ), c o = c p c p c p a p = (.).4 (.) + a 2 + a 3 + (.3).8. (.3) : + a 2 a 2 + a 2 + a 3 a 2 a 3 + a 3 + a 4 a 3 a 4 + a 4 + a 5 a 4 a 5 + a 5 + a 6, (.4) a 5 a 6 + a 6 + a 7. (.5) (.3) t = t (ω) = ω, t = t p (ω) = 7, p = 2, 3, 4,... + a p ω

15 t t 2 t n () = A n /B n s p (ω) = t 2p t 2p (ω), (p =, 2, 3,... ) s (ω) = a 2p, s p (ω) =, p = 2, 3, 4, a p + a 2p + a 2p ω s s 2 s p () = t t 2 t 2p () = A 2p /B 2p.5 a p, b p a p 0, (p =, 2, 3,... ) b 0 + b + b 2 + b 3 + (.6) b p 0, (p =, 2, 3,... ) b 0 + a + a 2 + a 3 + (.7) p N a p = 0 :.9. ([Wa48]) (.) m a m = 0 (m > ) a n 0, (n =, 2, 3,..., m ) (.) k ( n) n B n 0 (.) A m /B m..3 k k (.2) B m 0 B m = B m+ = = 0 A p B m A m B p = (b p A p + a p A p 2 )B m A m (b p B p + a p B p 2 ) = b p (A p B m A m B p ) + a p (A p 2 B m A m B p 2 ) p m, m +, m + 2,... a m = 0 A p B m A m B p = 0, p m. p m p k A p /B p = A m /B m A m /B m 8

16 0 p > k B p 0.0. ([Wa48]) (.6) b 2p = 0, (p =, 2, 3,... ). (.2) B = b = 0, B 3 = b 3 B 2 + a 3 B = 0, B 5 = b 5 B 4 + a 5 B 4 = 0,... (.6) b 0 b 0.. ([Ko95]) b p (.8) b b 2 b 3 {A 2p }, {A 2p+ }, {B 2p }, {B 2p+ } F 0, F, G 0, G ( ) F G 0 F 0 G = (.9). (.2) a n =, (n =, 2, 3,... ) A 2p = b 2p A 2p + A 2p 2 = b 2p A 2p + b 2p 2 A 2p 3 + A 2p 4 = = b 2p A 2p + b 2p 2 A 2p b 2 A p = b 2r A 2r. r= b 2r {A 2p } p C > 0 A 2p C M := max{ A, A 0 } A = b A 0 + A b A 0 + A M( + b ), A 2 = b 2 A + A 0 b 2 A + A 0 M( + b ) b 2 + M M( + b )( + b 2 ). A n M( + b )( + b 2 ) ( + b n ) n =, 2, 3,... C := M ( + b p ) p= 9

17 b 2r {A 2p+ }, {B 2p }, {B 2p+ } (.3) A 2p+ B 2p A 2p B 2p+ = p (.9) F 0 /G 0, F /G.2. i + i + i + (.20) B 2 = B 5 = B 8 = = B 3p = = 0 b p (.8). a p, b p [Wa48, pp.29].6 a p / 0 c p = /b p (b p 0) a p /.3. + a 2 + a 3 + (.2) r n 0, (n =, 2,... ) r p + a p + a p+ r p r p 2 a p + a p+, p =, 2, 3,... (.22) a = r 0 = r = 0 a p :.4. ([ScWa40]) (.2) (.22) (.2) p B p 0, (p = 0,, 2,... ) ρ p = a p+b p B p+ ρ p r p, p =, 2, 3,..., (.23) 0

18 . (.22) p =, 2, 3,... r + a 2 a 2, r 2 + a 2 + a 3 a 3. B 2 = + a 2 0, B 3 = + a 2 + a 3 0 a 2 ρ = + a 2 r a 3, ρ 2 = + a 2 + a 3 r 2. p =, 2,..., k (k 2) B p+ 0, ρ p r p a k+2 = 0 a k+2 0 Case : a k+2 = 0 (.2) B k+2 = B k+ + a k+2 B k = B k+ 0 a k+2 B k ρ k+ = 0 r k+. B k+2 Case 2: a k+2 0 (.22) p = k+ r k+ +a k+ +a k+2 r k+ r k a k+ + a k+2 r k+ 0 r k+ = 0 0 a k+2 a k+2 0 r k+2 > 0 (.2) B k+2 = ( + a k + a k+2 )B k a k a k+ B k 2. (.22) B k+2 a k+2 B k = + a k+ + a k+2 a k+ akb k 2 a k+2 a k+2 B k + a k+ + a k+2 a k+ r k > 0. r k+ B k+2 0, ρ k+ r k+ a k+2 a k+2 (.2) (.22) + r r 2 r p A p A p B p = + ρ ρ 2 ρ p p= B p p =, 2, 3,... A p A p B p = ρ ρ 2 ρ p r r 2 r p (.24) B p.4.9 :.5. ([Wa48]) (.2) (.22) p N a p = 0 (.2) p=

19 .7 Worpitzky Worpitzky [Wo65].6. ([Wa48]) a C + a + a + a + (.25) a = /4 c (c R >0 ) v : if a = 2 v = { 4, + + 4a max 2, } + 4a if a 2 4. [Wa48, pp.35].7 (Worpitzky ). D C z D a p z a p+ = a p+ (z), (p = 0,, 2,... ) a p+ (z), p =, 2, 3,... (.26) 4 : (a) (.2) D (b) z (.27) (c) (a) /4 (.26) (.27). (a) : (.26) 3 + a 2 ( ) = a 2, a 2 + a 3 2 ( 4 4 ) = 4 4 a 3, p p a p + a p+ p 2(p + 2) = p(p 2) (p + 2)p p p + 2 p 2 p a p + a p+2, p = 3, 4, 5,... 2

20 p =, 2, 3,... r p := p/(p + 2) (.22) ( + r r 2 r p = ) 6 p p + 2 p= p= 2 = + (p + )(p + 2) p= ( = + 2 p + ) = 2. p + 2 p= (.2) p =, 2, 3,... a p+ /4 + r r 2 r p = 2 2 p N a p = 0 (b) : z = + 4 w (.28) w = x + a 3 + a 4 + x w 2 (.28) z w 4 2 z z w 2 z = z(2) = 2/3, + w 4 z( 2) = 2, z(2i) = 4/3 2i/3 z (c) : /4 (.26) p N a p+ = c (c > /4),.6 (.27) z = + a 2 /4 /4 = + 2a 2 a 2 /4 (.27) (b) w /4 z = /( + 2w) (.27) (b) (c) Paydon Wall [PaWa42] a p+ (z) t( t), ( < t /2) z t 2 t t 2 3

21 .8. D C (.2) a p D, (p = 2, 3, 4,... ) (.2).7 (.2) /4.8 ( g p )g p x p /.7 ( g p )g p x p / g n = /2, x p :.9. ([Wa48]) {g p } p= 0 g p <, p =, 2, 3,..., (.29) 0 < g p, p =, 2, 3,... (.30) : (a) g + ( g )g 2 x 2 + ( g 2)g 3 x 3 + (.3) x p, (p = 2, 3, 4,... ) (b) z S (.32) S = + p= g g 2 g p ( g )( g 2 ) ( g p ) (.33) x p =, (p = 2, 3, 4,... ) /S (c) z 2 g g (.34) 2 g. g ( g )g 2 ( g 2)g 3 (.35) 4

22 p =, 2, 3,... θ p := ( g p )g p+ P n, Q n (.35) n (.2) n = 2, 3, 4,... : P n = P n θ p P n 2, Q n = Q n θ p Q n 2, P 0 = 0, P = g, Q 0 =, Q =. (.36) Q n = ( g )( g 2 )( g 3 ) ( g n ) + g ( g 2 )( g 3 ) ( g n ) + g g 2 ( g 3 ) ( g n ) + + g g 2 g n ( g n ) + g g 2 g n, P n = Q n ( g )( g 2 ) ( g n ). (.37) (.29), (.30) n =, 2, 3,... Q n > 0 (.36) (.2) Q n+2 = ( θ n+ θ n+2 )Q n θ n θ n+ Q n 2, n = 2, 3, 4,... r p := θ p+q p Q p+, p =, 2, 3,... (.38) θ p+ = ( θ p θ p+ )r p r p r p 2 θ p. p =, 2, 3,... a p+ θ p+ r p + a p + a p+ r p ( θ p θ p+ ) = r p r p 2 θ p + θ p+ r 0 = r = 0, θ = a = 0. r p r p 2 a p + a p+, (.3) x p+, (p =, 2, 3,... ) (a) g + r r 2 r p p=.5 g (.35).5 P g + r r 2 r p n = lim = n Q n S p= 5

23 S (.33) (b) (c) : z = g, w, + ( g )w (.34) Scott Wall [ScWa40] Paydon Wall [PaWa42].9 (c).20. ([Vl0a]) p =, 2, 3,..., 0 g p < S := + p= g g 2 g p ( g )( g 2 ) ( g p ) + g x + ( g )g 2 x 2 + ( g 2)g 3 x 3 + (.39) x p, (p =, 2, 3,... ) S x p =, (p =, 2, 3,... ) S. (.39) (.3) x S (.3) /S(< ) (.39) x p, (p =, 2, 3,... ) (.39) S = x p =, (p =, 2, 3,... ) S = S :.2. ([PaWa42]) p =, 2, 3,..., 0 < g p < S (.39) x p, (p =, 2, 3,... ) x p = p N x p. : k N x k (.39) (.3) (.32) z z C (.39) x z = x z x = z = (.34) z = z = x z = x = z = g + ( g )x 2 z 2 6

24 z 2 = g 2 + ( g 2)g 3 x 3 + ( g 3)g 4 x 4 + z = x 2 z 2 = z 2 =, x 2 = (.3) x r <, x p, (p = 2, 3, 4,... ) (.3) :.22. ([Wa48]) p =, 2, 3,... 0 g p < 0 < g p (.39) x r < r, x p, (p = 2, 3, 4,... ).9 k, k 2 0 (.2) : r + a 2 ( + k ) a 2, r + a 2 + a 3 ( + k 2 ) a 3, (.40) r p + a p + a p+ r p r p 2 a p + a p+, p = 3, 4, 5, ([ScWa40]) k, k 2 > 0 + a 2 + a 3 + (.4) (.40). p N a p = 0.5 p = 2, 3, 4,... a p 0 (.40) r p > 0, + a p + a p+ > 0, r p + a p + a p+ a p+ > 0, (p =, 2, 3,... ) a = 0 k > 0 g p := r 2p+ + a 2p+ + a 2p+2 a 2p+2, p =, 2, 3,... r 2p+ + a 2p+ + a 2p+2 0 < g p < (.40) g = r 3 + a 3 + a 4 a 4 r 3 + a 3 + a 4 r a 3 + a 3 + a 4 ( + k ) a 2 a 3 ( + a 2 )( + a 3 + a 4 ) ; 7

25 ( g p )g p a 2p a 2p+ ( + a 2p + a 2p )( + a 2p+ + a 2p+2 ), p = 2, 3, 4,... a 2p a 2p+ ( + a 2p + a 2p )( + a 2p+ + a 2p+2 ) = ( g p )g p x p, p =, 2, 3,... g 0 = 0, x /( + k ), x p, (p = 2, 3, 4,... ) (.4) + a 2 + g x + ( g )g 2 x k 2 > 0 g p := r 2p+2 + a 2p+2 + a 2p+3 a 2p+3, p =, 2, 3,... r 2p+2 + a 2p+2 + a 2p+3 + r r 2 r p.24. ([ScWa40]) (.4) k, k 2 > 0 (.40) p = 2, 3, 4,... a p 0 b, b 2,... b =, a p+ = b p b p+, p =, 2, 3,... (.42) (.4) (.43) b b 2 b 3 B p (.4) p Q p (.43) p : Q p = b b 2 b p B p (.44) r r 3 r 2p Q 2p k ( + r b 2 + r 2 r 3 b r 2 r 2 3 r 2 2p 3r 2p b 2p ), r 2 r 4 r 2p Q 2p+ k ( + r 2 b 3 + r 2 2r 4 b r 2 2r 2 4 r 2 2p 2r 2p b 2p+ ), (.45) p =, 2, 3,..., k > 0 8

26 . (.40) (.42) r + b b 2 + k, r 2 b b 2 b 3 + b + b 3 + k 2, r p b p b p b p+ + b p + b p+ r p r p 2 b p+ + b p+, p = 3, 4, 5,... (.46) Q 0 =, Q = b =, Q 2 = + b b 2, Q 3 = b b 2 b 3 + b + b 3 2 Q 2 r Q 0 k b 2, Q 3 r 2 Q k b 3 (.47) k := min { k } k 2 r b 2, r 2 b 3, r Q 2 k( + r b 2 ), r 2 Q 3 k( + r 2 b 3 ). (.48) (.46) Q p+3 b p+ b p+2 b p+3 + b p+ + b p+3 r p+2r p b p+3 + b p+ Q p+ r p+2 b p+ Q p+ b p+ b p+3 b p+ b p+3 b p+ Q p Q p Q p+3 Q p+ r p r p+2 b p+3 b p+ ( Q p+ ) Q p. r p p =, 3, 5,... p = 2, 4, 6,... (.47) r 2p Q 2p Q 2p 2 + r r 3 r 2p k b 2p, r 2p Q 2p+ Q 2p + r 2 r 4 r 2p k b 2p+ (.49) (.48) (.45) A p+ A p B p+ B p = ( ) p+2 a a 2 a p+ B p B p+ = ( )p+2 b b 2 b 2 b 3 b p b p+ B p B p+ = Q p Q p lim sup Q p Q p+ = (.50) 9

27 .0 (.40) p =, 2, 3,... r p = k, k 2 > 0 + a 2 > a 2, + a 2 + a 3 > a 3, + a p + a p+ a p + a p+, p = 3, 4,.... (.5) p N a p = 0 (c.f..5). m, m 2,... > 0 m <, m p + m p+, p =, 2, 3,... (.52) a p+ R(a p+ ) m p, p =, 2, 3,... (.53) : + a 2 + R(a 2 ) + a 2 m > a 2, + a p + a p+ + R(a p ) + R(a p+ ) m p m p + a p + a p+ a p + a p+, p = 2, 3, 4, ([Wa48]) a p+ R(a p+ ) m p, (p =, 2, 3,... ) m <, m p + m p+, (p =, 2, 3,... ) + a 2 + a 3 + (.54) : (a) p N a p = 0, (b) p = 2, 3,..., a p 0 b =, a p+ = b p b p+, p =, 2, 3,... b p m p = /2, (p =, 2, 3,... ) Worpitzky [ScWa40].26 ( ). S C (.54) S { z R(z) /2} =: A a, a 2, A (.52) : 20

28 (a) p N a p = 0, (b) p = 2, 3,..., a p 0 b =, a p+ = b p b p+, p =, 2, 3,..., b p.27. ([Wa48]) z = x + iy, z = x iy x, y R + z + z + z + z + (.55) z R(z) 2, i.e., y2 x + 4. : n N a n A a n 0 b =, b 2 = z, b 3 = z z =,... b p.25 a 2p := z, a 2p+ := z, (p =, 2, 3,... ) : + z + z + z + z + z + z + (.56) B p (.56) p B p z + z + z + p B p = B p 2 (.56) z 2 2x z 2 2x z 2 (.57) u := z 2 2x z 2 2x z 2 u = z 2 ( + 2x) u ( + 2x) 2 4(x 2 + y 2 ) 0 y 2 x + /4 2

29 m p = /2, (p =, 2, 3,... ).25 S C b p a 2, a 3, S. S A z = a A c a S S ā S a 2p := a, a 2p+ := ā, (p =, 2, 3,... ).27.7 (.54) a 2, a 3, /4 a, a 2,... z 4/3 2/3 z /4 z R(z) /2 :.28. ([Wa48]) (.54) z, z 0 (.58) a 2, a 3,... A.26 (a) (b). (.9) + ρ p = + a p+ + a p + + a 3 + a 2 (.23) r p =, (p =, 2, 3,... ) ρ p + a p+ + a p + + a 3 + a 2. a 2, a 3,... z R(z) /2 a n (.58) Lane [La45] [ScWa4] 22

30 2 + ρ ρ 2 ρ p 0 2. a2 a2 2 (2.) b + z b 2 + z 2 b 3 + z 3 a p, b p C :, z p C, (p =, 2, 3,... ) a p a 2 p β p := I(b p ), y p := I(z p ), α p := I(a p ) z := (z, z 2, z 3,... ) z p z Jacobi Jacobi : 2.. ([Wa48]) (2.) p B p (z), (p =, 2, 3,... ) z : b + z a a b 2 + z 2 a B p (z) = 0 a 2 b 3 + z 3 a (2.2) a p 2 b p + z p a p a p b p + z p B p (z) x p C (b + z )x a x 2 = 0, a x + (b 2 + z 2 )x 2 a 2 x 3 = 0, a p x p + (b p + z p )x p = 0 (2.3) 23

31 x, x 2,..., x p p p (b r + z r ) x r 2 a r (x r x r+ + x r x r+ ) = 0. (2.4) r= y r > 0, (r =, 2, 3,..., p), p r= x r 2 > 0 r= p p (β r + y r ) x r 2 α r (x r x r+ + x r x r+ ) > 0 (2.5) r= r= (2.3) (2.4) y r > 0 x = x 2 = x 3 = = x p = 0 y r > 0, (r =, 2, 3,..., p) B p (z) ([Wa48]) ξ, ξ 2, ξ 3,... p p β r ξr 2 2 α r ξ r ξ r+ 0 (2.6) r= r= (2.5). : y r ξ r R : x r := u r + iv r, (u r, v r R) (2.5) ( p ) ( p p ) p β r u 2 r 2 α r u r u r+ + β r vr 2 2 α r v r v r+ + r= r= y r 0 r= r= p y r x r 2 r= p y r x r (6.) p =, 2, 3,... ξ, ξ 2, ξ 3,... R r= r= ( ) r= (2.7) p p β r ξr 2 2 α r ξ r ξ r+ 0 (2.8) 2.2 : 2.4. ([WaWe44a]) (2.) y r > 0, (r =, 2, 3,... ) B p (z) 0 24

32 Wetzel Wall [WaWe44b] a p, b p 2.5. ([WaWe44b]) (2.) : (i) β p 0, p =, 2, 3,..., (ii) {g p } p=0 0 g p α 2 p+ = β p+β p+2 ( g p )g p+.. : ξ = ξ 2 = = ξ n = ξ n+ = 0, ξ n 0 (2.6) β n 0 ξ 3 = ξ 4 = ξ 5 = = 0 (2.6) β ξ 2 2α ξ ξ 2 +β 2 ξ2 2 0 α 2 β β 2 α 2 = β β 2 ( g 0 )g, g 0 = 0, 0 g. β = 0 g = 0 ξ 4 = ξ 5 = = 0 (2.6) { (β ( g 0 )) /2 ξ (β 2 g ) /2 ξ 2 } 2 + β2 ( g )ξ 2 2 2α 2 ξ 2 ξ 3 + β 3 ξ β = 0 g = 0 { (β ( g 0 )) /2 ξ (β 2 g ) /2 ξ 2 } 2 = 0 β > 0 ξ := (β 2g ) /2 ξ 2 (β ( g 0 )) /2 { (β ( g 0 )) /2 ξ (β 2 g ) /2 ξ 2 } 2 = 0 ξ2, ξ 3 α 2 2 β 2 β 3 ( g ) α 2 2 = β 2 β 3 ( g )g 2 0 g 2 g = β 2 = 0 g 2 = 0 ξ 5 = ξ 6 = = 0 {(β ( g 0 )) /2 ξ (β 2 g ) /2 ξ 2 } 2 + {(β 2 ( g )) /2 ξ 2 (β 3 g 2 ) /2 ξ 3 } 2 + β 3 ( g 2 )ξ 2 3 2α 3 ξ 3 ξ 4 + β 4 ξ α 2 3 β 3 β 4 ( g 2 ) α 2 3 = β 3 β 4 ( g 2 )g 3 0 g 3 g 2 = β 3 = 0 g 3 = 0 : p β r ξr 2 2 r= p α r ξ r ξ r+ = g 0 β ξ 2 + ( g p )β p ξp 2 r= (2.6) + p r= { (β r ( g r )) /2 (β r+ g r ) /2 ξ r+ }

33 : 2.6. ([Wa48]) 2.5 g 0 = 0 (2.6) p p β r ξr 2 2 α rξ r ξ r+ 0 r= r= r =, 2, 3,..., p ξ r R α r α r. 2 p p β r ξ r 2 2 α r ξ r ξ r+ = g 0 β ξ r 2 + ( g p )β p ξ p 2 r= r= p { } 2 + (β r ( g r )) /2 ξ r (β r+ g r ) /2 ξ r+ r= 2.7. ([Wa48]) 2.5 (ii) : {g p } 0 g n, α 2 n β n β n+ ( g n )g n, n =, 2, 3,.... (2.9) 2.7 a 2 n R(a 2 n) = 2α 2 n 0 g n, a 2 n R(a 2 n) 2β n β n+ ( g n )g n, n =, 2, 3,... (2.0) p =, 2, 3,... a p =: c 2 p + a 2 + a 3 + = i i c2 i c2 2 i β p =, (p =, 2, 3,... ) (2.0) 0 g n, c 2 n R(c 2 n) 2( g n )g n, n =, 2, 3,... (2.) g n = /2, (p = 0,, 2,... ) z R(z) /2 ( g 0 )g < /2, ( g n )g n + ( g n )g n+ < /2, (n =, 2, 3,... ) (.53) {g p } 26

34 2.2 I(z ) > 0, I(z r ) 0, (r = 2, 3, 4,... ) : 2.8. ([DeWa45]) t = t p (w) = b p + z p a2 p, p =, 2, 3,... (2.2) w β p = I(b p ) 0, α 2 p = I(a p ) 2 β p β p+ ( g p )g p, 0 g p, p =, 2, 3,... (2.3) I(w) β p+ g p I(t) β p g p +y p y p = I(z p ), (p =, 2, 3,... ). I(w) β p+ g p y p > 0 I(w) β p+ g p β p ( g p ) β p ( g p ) + y p α 2 p β p ( g p ) + y p. a 2 p R(a 2 p) = 2α 2 p α2 p = { a 2 p I(ia 2 p) } /2 w + ia 2 p 2{β p ( g p ) + y p } a 2 p 2{β p ( g p ) + y p }. w i + 2{β p ( g p ) + y p } 2{β p ( g p ) + y p }. a 2 p ( ) a 2 p β p + y p I β p g p I(t) β p g p w y p y p 0 β p I(a 2 p/w) β p g p ( ) a 2 p β p + y p I β p g p + y p I(t) β p g p + y p. w t = t 0 (w) = /w I(w) β g 0 + y, y > 0 K 0 (z): t + i 2(β g 0 + y ) 2(β g 0 + y ) (2.4) 27

35 β g 0 +y > 0 I(w) β 2 g t = t 0 (t (w)) = /{b + z (a 2 /w)} I(w) g β 2 I(t (w)) g 0 β + y (K (z) ) K 0 (z) K (z) w = I(w) β 2 g t 0 (t ( )) = A (z)/b (z) K (z) a = 0 K (z) = /(b + z ) = A (z)/b (z) p K p (z) I(w) g p β p+ t = t 0 t t p (w) = a2 a 2 p b + z b 2 + z 2 b p + z p a 2 p/w, I(z r ) 0, r = 2, 3, 4,... (2.5) t 0 t t p ( ) = A p (z)/b p (z) K p (z) a p = 0 p m K p (z) = t = t 0 t t m ( ) (2.4) K 0 (z) K (z) K 2 (z) (2.6) t K p (z) : I(t) 0, t. (2.7) g 0 β + y g 0 β g 0 β + y > 0, y r 0, (r = 2, 3, 4,... ) (2.) ( ) Ap (z) I 0, B p (z) A p (z) B p (z), (2.8) g 0 β + y y g 0 β + y > 0, g r 0, (r = 2, 3, 4,... ) p A p (z)/b p (z) K 0 (z) 2.9. ([DeWa45]) (2.3) g p β p > 0, (p =, 2, 3,... ) I(z r ) 0, (r =, 2, 3,... ) B p (z) 0. p =, 2, 3,... a p 0 B p 0 p N a p = 0 (2.2) B p (z) = (b p + z p )B p (z) a 2 pb p 2 (z) (2.8) B p 0.9, 2.4, 2.9 : 2.0. ([Wa48]) (2.) #{p N: a p = 0} I(z r ) > 0, (r =, 2, 3,... ) (2.) p =, 2, 3,... g p β p > 0 I(z r ) 0, (r =, 2, 3,... ) (2.) 28

36 p =, 2, 3,... a p 0 I(z ) > 0, I(z r ) 0, (r = 2, 3, 4... ) z r C 2 Case ( ): K p (z) r p (z) lim p r p (z) = 0 f(z) lim p A p (z)/b p (z) = f(z) Case 2( ): K p (z) r p (z) c > 0 lim p r p (z) = c 2 r p (z) X p+ := A p a a 2 a p, Y p+ := X = 0, Y =, X 0 =, Y 0 = 0 B p a a 2 a p, p =, 2, 3,... (2.9) a p X p + (b p + z p )X p a p X p+ = 0, a p Y p + (b p + z p )Y p a p Y p+ = 0, p =, 2, 3,... (2.20) (2.) a 0 = (2.5) X p+ Y p X p Y p+ = a p, p =, 2, 3,... (2.2) 0 g p, α 2 p = β p β p+ ( g p )g p, p =, 2, 3,... (2.22) t = t 0 t t p (w) = A p(z)w a 2 pa p (z) B p (z)w a 2 pb p (z) (2.9) t = X p+w a p X p Y p+ w a p Y p (2.23) w = a p Y p /Y p+ t = K p (z) C p C p w = a p Y p /Y p+ I(w) = g p β p+ a p Y p Y p + 2ig p β p+ (2.24) C p = ( ) ap Y p X p+ + 2ig p β p+ a p X p Y p ( ). (2.25) ap Y p Y p+ + 2ig p β p+ a p Y p Y p 29

37 t 0 t t p ( ) = A p (z)/b p (z) K p (z) (2.25) (2.2) r p (z) = C p X p+. Y p+ 2r p (z) = gp β p+ Y p+ I ( ). (2.26) a p Y p Y p+ (2.20) Y p a r Y r Y r + (b r + z r ) Y r 2 a r Y r+ Y r, r =, 2, 3,.... Y 0 = 0 2r p (z) = p (b r + z r ) Y r 2 r= I(a p Y p+ Y p ) = p ( ) a r Yr+ Y r + Y r+ Y r = ap Y p+ Y p. r= p ( ) p α r Yr+ Y r + Y r+ Y r (β r + y r ) Y r 2. (2.27) r= p+ r= β r Y r 2 p r= α ( ) r Yr+ Y r + Y r+ Y r + p r= y r Y r 2 ( g p )β p+ Y p+ 2 (2.28) r= 2r p (z) = g 0 β + ( p r= y r Y r 2 + ), (β r ( g r )) /2 Y r (β r+ g r ) /2 2 Y r+ p =, 2, 3,.... (2.29)

38 K p (z) C p (z) C p (z) p A p (z), B p (z) (2.6) C p = a p 2 A p B p a 2 pa p B p + 2ig p β p+ A p B p a p 2 B p B p a 2 pb p B p + 2ig p β p+ B p B p r p (z) = C p A p = B p a a 2 a p 2 B p + ia 2 pb p 2 a 2 pb p 2 + (2g p β p+ ) B p 2. (2.30) (2.) z p := 0, b p := i, (p =, 2, 3,... ) (2.) i a2 i a2 2 i (2.3) B p + ia 2 pb p = ib p+ g p := /2, (p =, 2, 3,... ) β p = (2.3) r p = r p (0) = a a 2 a p 2 B p+ 2 a 2 pb p 2. c =, a 2 p = c p c p+, Q p = c c 2 c p B p, p =, 2, 3,... r p = ( i) c p+ Q p+ Q p Q p+ + Q p. (2.32) c i c 2 i c 3 i (2.33) i c + c 2 + c 3 + p =, 2, 3,... a p 0 # {p N : a p = 0} 2.0 c p lim p r p = 0 c p a 2 p R(a 2 p), p =, 2, 3,

39 + a a2, + a 2 p + a 2 p+ a 2 p + a 2 p+, p =, 2, 3,... + c c 2 2 c c 2 +, c p c p+ c p+2 + c p+2 + c p c p+2 + c p, p =, 2, 3,... (.47) Q 2 Q 0 2 c 2, Q 3 Q c 3 2 c 3 Q 2 2 ( + c 2 ), Q 3 2 ( + c 3 ) (2.34) (.49) Q 2p Q 2p c 2p, Q 2p+ Q 2p + 2 c 2p+. (2.35) (2.32) (2.34) (2.35) r 2p + p r= c 2r, r 2p + p r= c 2r+ (2.36) {K p (z)} r > r 2 > r 3 > lim p r p (z) = 0 p K p (z) a 2 := (a 2, a2 2,... ) D := { z R(z) /2} R p = p r= c r, a a 2 a p 0, 0, k ( p ) a k = 0 0 a 2 G D : 2.. ([PaWa42]) M > 0 + a 2 + a 3 + z R(z) 2, z < M 32

40 2.4 ( g p )g p+ (0 g p, p = 0,, 2,... ) 2.7 β p =, (p =, 2, 3,... ) (ii) α 2 p+ ( g p)g p+ ( g p )g p+ x p+ / (x p+ C) (2.) g p 2.2. {a p } p= p = 0,, 2,... 0 g p {g p } p=0 a p+ = ( g p )g p+ {g p } {a p } p= 2.3. /4, /4, /4, g p = /2, (p = 0,, 2,... ) 2.4. ([Wa48]) a, a, a, 0 a 4. : n =, 2, 3,... 0 g n ( g n )g n > 4 (2.37) ( g n ) + g n ( g n )g n > 2 2. g n > g n, (n =, 2, 3,... ) g 0 < g < g 2 < < {g n } g lim n g n = g ( g) + g ( g) g 2 2. ( g)g = /4 g = /2 : n = 0,, 2,... g n := + 4a 2 33

41 2.3 g p := ( ), p = 0,, 2,... 2 p ([WaWe44b, DeWa45]) g p m p g p M p, (p = 0,, 2,... ). {a p } {g p } p =, 2, 3,... a p = ( g p )g p p =, 2, 3,... m p : 0 if m p =, m 0 = 0, m p+ = a p+ (2.38) if m p < m p m 0 = 0 m 0 g 0 k ( ) m k g k m k = m k < a k+ = 0 2 m k = m k+ = 0 g k+ m k < a k+ > 0 m p (2.38) m k+ = a k+ m k = ( g k)g k+ m k ( g k)g k+ g k = g k+. p = 0,, 2,... m p g p m p m p 0 m p m p < m p (2.38) ( m p ) a p+ = m p+ ( m p ) m p = = m p g p g p = a p+ = ( g p )g p+ = 0 ( m p )m p+ = a p+ M p : M p = a p+ a p+2 a p+3, p = 0,, 2,... (2.39) k {p+, p+2,... } a k = 0 0 a p+ = 0 M p = g p a p+2 = 0 a p+ > 0 M p = a p+ = ( g p )g p+ 0 a p+, a p+2,..., a p+k > 0, a p+k+ = 0, (k > 0) M p = ( g p ) gp+ ( g p+)g p+2 ( g p+k 2)g p+k ( g p+k )g p+k. g p+k = M p = g p g p+k <.5 ( ) M p = ( g p ) T p+k p+ (g) (2.40) 34

42 p+k T p+k p+ (g) := + r=p+ g p+ g p+2 g r ( g p+ )( g p+2 ) ( g r ) (2.4) M p > m p a p+r > 0, (r =, 2, 3,... ) (2.40) T p+(g) = + r=p+ g p+ g p+2 g r (2.42) ( g p+ )( g p+2 ) ( g r ) M p m p T p+ (g) = M p 0 M p, (p = 0,, 2,... ) a p+ = 0 M p =, ( M p )M p+ = 0 = a p+ a p+ > 0 ( g p )g p+ > 0 0 < g p+ M p (2.39) M p = a p+ M p+, a p+ = ( M p )M p+. a p > 0, (p =, 2, 3,... ) M p = g p + r=p+ g p+ g p+2 g r =. (2.43) ( g p+ )( g p+2 ) ( g r ) m p = M p, (p = 0,, 2,... ) : 2.6. ([Wa48]) {a p } p= m p T0 (m) = m p m 0 = 0 : 2.7. ([Wa48]) M 0 = : 2.8. ([Wa48]) {a p } p= m p 0 m p <, (p = 0,, 2,... ) 0 < M p, (p = 0,, 2,... ) a p+ a p+2 p = 0,, 2,... 35

43 2.5 p =, 2, 3,... a p = a [, /4] {m p } m 0 = 0, m = a, m 2 = a/( a), m 3 = a a a,... a a a p m p = 4a 2 ( p + 4a r=0 + 4a M p = a a = + ) r, p = 0,, 2,... (2.44) 4a, p = 0,, 2,... (2.45) ([Wa48]) {c p } p= {c p } p= n c p <, n =, 2, 3,... (2.46) p=. p =, 2, 3,... c p 0 (2.46) 0 p= c p = c < g := c g, g 2,..., g k g c k+ < c c 2 c k = g ( g )g 2 ( g 2 )g 3 ( g k )g k = ( g )( g 2 ) ( g 2 )g 3 ( g k )g k ( g 2 ) ( g 2 )g 3 ( g k )g k = ( g 2 )( g 3 ) ( g 3 )g 4 ( g k )g k = ( g k 2 )( g k ) ( g k )g k ( g k ) ( g k )g k = ( g k )( g k ) ( g k ). 0 g k+ < g k+ c k+ = ( g k )g k+ 36

44 2.9 0 r /2 r, r 2, r 3,... 37

45 3 Jacobi Stieltjes Jacobi Jacobi Jacobi Stieltjes 3. Stieltjes Jacobi Stieltjes Jacobi Jacobi Jacobi Stieltjes 3.. Stieltjes : k z + + k 2 k 3 z + +. (3.) k 4 z C p =, 2, 3,... k p R >0 (3.) /k c + z c c 2 c 2 + c 3 + z c 3 c 4 c 3 + c 4 + z (3.2) c p := (3.) k p k p+, p =, 2, 3,... (3.3) T p (w) = /k c + z c c 2 c 2 + c 3 + z c 2p c 2p w (3.4) T p (z + c 2p + c 2p+ ) = A 2p+2 B 2p+2, T p (z + c 2p ) = A 2p+ B 2p+. (3.5) (3.) Jacobi Jacobi 38

46 a2 a2 2 b + z b 2 + z 2 b 3 + z 3 z := (z, z, z,... ) Jacobi : b + z a2 b 2 + z a2 2 b 3 + z (3.6) 3.2. Jacobi (3.6) M > 0 p =, 2, 3,... a p M 3, b p M 3 (3.7) M Jacobi (3.7) : 3.3. ([Wa48]) Jacobi (3.6) N > 0 n = 2, 3, 4,... n n b p u p v p a p (u p v p+ + u p+ v p ) N n n u p 2 v p 2 (3.8) p= p= u p, v p C. : : n n b p u p v p a p (u p v p+ + u p+ v p ) p= p= n n b p u p v p p= + a p (u p v p+ + u p+ v p ) p= M 3 n n u p 2 v p 2 + 2M 3 n u p 2 N := M p= n = M u p 2 p= p= n v p 2. p= p= p= p= n v p 2 : r p u p = v p =, u r = v r = 0 b p N r p, s p + u p = v p+ =, u r = v s = 0 a p N M := 3N p= 39

47 (3.8) Jacobi b p u p v p a p (u p v p+ + u p+ v p ) (3.8) N Jacobi Jacobi 3.4. z /4 z /4 z a p = /2, b p = 0, (p =, 2, 3,... ) a p = /2 = M/3 M = 3/2 n n n b p u p v p a p (u p v p+ + u p+ v p ) p= p= = a p (u p v p+ + u p+ v p ) p= 2M 3 n n u p 2 v p 2 N = p= n = u p 2 p= p= n v p ([Wa48]) M > 0 Jacobi z M. M > 0 Jacobi b + z a2 b 2 + z a2 2 b 3 + z n p= a 2 n (b n + z)(b n + z) a 2 n (b n + z)(b n + z) a n 2 b n + z b n + z a 2 n ( b n z ) ( b n z ) M 2 /9 (M/3 M) (M/3 M) = 4..7(a) z M 40

48 Stieltjes Jacobi 3.6. b + z a2 b 2 + z a2 2 b 3 + z, z C Jacobi p =, 2, 3,... a p, b p R Jacobi α r = I(a r ) = 0, β r = I(b r ) = 0, (r =, 2, 3,... ) 2 p r= β rξr 2 2 p r= α rξ r ξ r+ 0 Jacobi Jacobi : 3.7. ([WaWe44b]) Jacobi z a2 z a2 2 z (3.9) N {a 2 p} [Wa48, p.5] 3.8. ([Wa48]) N Jacobi I := [ N, N] d(k, I) = inf{d(x, y) : x K, y I} > 0 K z, z 2 C d(z, z 2 ) := z z : (3.8) x p R, b p = 0, (p =, 2, 3,... ) n 2 a p x p x p+ p= n x 2 p. p= n p= n x 2 p 2 a p x p x p+ p= n p= n x 2 p, n = 2, 3, 4,.... (3.0) p= n x 2 p 2 a p x p x p+ 0 (3.) p= 4

49 {a 2 p} : {a 2 p} (3.) x p ( ) p x p n n ( ) 2p x 2 p 2 ( ) 2p+ a p x p x p+ 0. p= n 2 p= a p x p x p+ p= n x 2 p. p= (3.9) /z a2 /z2 z z := /z 2 a2 2 /z2 + a2 z + a2 2 z + I z = [, ] I z = [, ] 3.9. ([Wa48]) p = 0,, 2,... g p [0, ] + ( g 0)g z + ( g )g 2 z + (3.2) L := [, ] d(k, L) = inf{d(x, y) : x K, y L} > 0 K C 3.2 Jacobi Jacobi Jacobi r z + s + r 2 z + s 2 + r 3 z + s 3 + (3.3) p =, 2, 3,... r p, s p C r p 0 A p (z), B p (z) p deg A p (z) = p, deg B p (z) = p z r p, s p f 0 := f 0 (z) = a 00 z n + a 0 z n + + a 0n, f := f (z) = a z n + a 2 z n a n (3.4) 42

50 p, q = 0,, 2,..., n a pq C f 0 f = r z + s + r 2 z + s r p ( 0), s p f 0, f r n z + s n (3.5) f p = (r p z + s p )f p + f p+, p =, 2, 3,..., n, f n+ = 0, f n = c 0 (3.6) (n p) f p (p = 2, 3,..., n ) (3.5) p =, 2, 3,..., n r p, s p C Jacobi (3.6) f 0 = f 0, f = f, r p, s p C f p = (r pz + s p)f p + f p+, p =, 2, 3,... n, f n+ = 0, f n = c 0 : deg f p = n p p = 0 = f 0 f 0 = { (r r )z + (s s ) } f + (f 2 f 2). f 2 f 2 n 2 deg f = n r = r, s = s, f 2 = f 2 p = 2 r 2 = r 2, s 2 = s 2, f 3 = f 3 p 3 z r p, s p (3.6) a 00 z n + a 0 z n + + a 0n a z n + a 2 z n a n = r z + b z n + b 2 z n b n a z n + a 2 z n a n, r = a 00 a, b = a 0 r a 2, b 2 = a 02 r a 3,..., b n = a 0n, b z n + b 2 z n b n a z n + a 2 z n a n = s + a 22z n 2 + a 23 z n a 2n a z n + a 2 z n a n, s = b a, a 22 = b 2 s a 2, a 23 = b 3 s a 3,..., a 2n = b 2n s a n. r, s, f 2 r p, s p, f p+ a 00, a,..., a pp,... 0 (3.5) a 00, a,..., a pp,

51 : a 00, a 0, a 02,..., a, a 2, a 3,..., r = a 00 a, b = a 0 r a 2, b 2 = a 02 r a 3, b 3 = a 03 r a 4,..., s = b a, a 22 = b 2 s a 2, a 23 = b 3 s a 3, a 24 = b 4 r a 4,..., r 2 = a a 22, b 22 = a 2 r 2 a 23, b 23 = a 3 r 2 a 24, b 24 = a 4 r 2 a 25,..., s 2 = b 22 a 22, a 33 = b 23 s 2 a 23, a 34 = b 24 s 2 a 24, a 35 = b 25 r 2 a 25,..., r 3 = a 22, a 33 b 33 = a 23 r 3 a 34, b 34 = a 24 r 3 a 35, b 35 = a 25 r 3 a 36,...,, (a pq = b pq = 0 for q > n). (3.7) r p, s p 3.0. f 0 = z 3 + (2 + i)z 2 + (3 + i)z + (2i + 2), f = 2z 2 + iz + 2 (3.8) f f 0 = 2 z + + i z 4i z + 27i 32 (3.9) Jacobi 3.3 (3.5) f 0, f 3.. ([Fr46]) f 0 = a 00 z n + a 0 z n + + a 0n, f = a z n + a 2 z n a n 44

52 f /f 0 (3.5) p =, 2, 3,... D p 0 : a, a 2, a 3, a 4, a 5, a 6, a 7, a 8,..., a 00, a 0, a 02, a 03, a 04, a 05, a 06, a 07,..., 0, a, a 2, a 3, a 4, a 5, a 6, a 7,..., 0, a 00, a 0, a 02, a 03, a 04, a 05, a 06,..., 0, 0, a, a 2, a 3, a 4, a 5, a 6,..., (3.20) 0, 0, a 00, a 0, a 02, a 03, a 04, a 05,..., 0, 0, 0, a, a 2, a 3, a 4, a 5,..., 0, 0, 0, a 00, a 0, a 02, a 03, a 04,...,, (a 0p = a p = 0 for p > n); a a 2 a 3 a 4 a 5 a a 2 a 3 a 00 a 0 a 02 a 03 a 04 D 0 = a 00, D = a, D 2 = a 00 a 0 a 02, D 3 = 0 a a 2 a 3 a 4, a a 2 0 a 00 a 0 a 02 a a a 2 a 3. (3.5) a 00, a, a 22,..., a nn 0 D 0 = a 00 0, D = a 0 D p f 0, f D p (f 0, f ) := D p (3.20) D p (f 0, f ) = ( ) p a 2 D p (f, f 2 ), p = 2, 3,..., n (3.2) D p (f 0, f ) = ( ) p a 2 D p (f, f 2 ), D p (f, f 2 ) = ( ) p 2 a 2 22D p 2 (f 2, f 3 ), D p 2 (f 2, f 3 ) = ( ) p 3 a 2 33D p 3 (f 3, f 4 ), D 2 (f p 2, f p ) = ( ) a 2 p,p D (f p, f p ) D (f p, f p ) = a pp D p (f 0, f ) = ( ) p k= k a 2 a 2 22 a 2 p,p a pp = ( ) p(p ) 2 a 2 a 2 22 a 2 p,p a pp 0. (3.22) (3.5) n N 0 D n 0 D 0 0, D 0 a 00, a 0 (3.22) p = 2 D 2 = ( ) 2 2 a 2 a a 22 0 a nn 0 45

53 3.4 Jacobi f /f 0 (3.5) : a 0 = r, a p = r p r p+, p =, 2, 3,..., n, (3.23) b p = s p r p, p =, 2, 3,..., n (3.24) f = a 0 f 0 b + z a b 2 + z a n b n + z (3.7) a 0 = a a 00, b p = (3.25) a p = a p+,p+ a p,p, p =, 2, 3,..., n, (3.26) b pp a p,p, p =, 2, 3,..., n (3.27) a 00 = (3.22) (3.26) D p+ = a 0 a a p D p, p = 0,, 2,..., n. (3.28) a 0 = D, a p = D p D p+, p =, 2, 3,..., n (3.29) D 0 D 2 p (3.25) Jacobi p f /f 0 z f c p := f 0 z p+ (3.30) (.3) A p+ (z) B p+ (z) A p(z) B p (z) = p=0 a 0a a p, B p+ (z)b p (z) p = 0,, 2,..., n, (3.3) A p, B p (3.25) p deg A p = p, deg B p = p, (p = 0,, 2,..., n ) deg B p+ (z)b p (z) = 2p + A p+ /B p+ A p /B p /z 2p+ A p+ = c 0 B p+ z + c z c 2p z 2p + c (p) 2p z 2p+ + c z (p) 2p+ 2p+2 +, p =, 2, 3,..., n (3.32) p(< n) f /f 0 2p f /f 0 46

54 3.2. ([Gr4]) p =, 2, 3,..., n L p > 0, x p R, x k x l (k l) f f 0 = n p= L p z x p (3.33) f /f 0 p =, 2, 3,..., n b p R a p > 0 (3.25) F (ξ, ξ 2,..., ξ n ) := n p,q= a pq ξ p ξ q, (a pq = a qp ) (3.34) ξ, ξ 2,... R n p= ξ2 p > 0 F (ξ, ξ 2,..., ξ n ) > 0 F 2 F n N : a a 2 a n a 2 a 22 a 2n D n =. (3.35) a n a n2 a nn 3.2. (3.33) l n c p c p = L l z x l = p=0 L l x p l z p+ n x p k L k, p = 0,, 2,... k= c p 2 x s x t (t s, s, t =, 2, 3,..., n) u m (< n) 0 X 0 +X u+x 2 u 2 + +X m u m u = x k, (k =, 2, 3,..., n) 0 2 m p,q=0 c p+q X p X q = x k u=x (X + X u + + X m u m ) 2 L(u), L(x k ) = L k 3.3 c 0 c c p c c 2 c p+ p = > 0, c p c p+ c 2p p = 0,, 2,..., n (3.36) 47

55 t c 0 z 2n + c z 2n 2n + + c 2n z + t z 2n+ = p=0 c p z p+ + t z 2n+ (3.37) 3. D p D 0 = 0, D = c 0 =, D 2 =,..., D n = n, D n+ = D n+ (t) = t n + D n+ (0) (3.36) D p > 0, (p = 0,, 2,..., n) t D n+ > 0 (3.37) Jacobi (n + ) a 0 b + z a b 2 + z a n b n + z a n b n+ + z, b p R, (3.29) a p > 0, (p = 0,, 2,... ) (3.32) n 2n A n (z) B n (z) = p=0 c p z p+ + c (n) 2n z 2n+ + ( ) f (z)b n (z) A n (z)f 0 (z) = P z P (/z) /z f A n z (n ) f 0 B n z n f (z)b n (z) A n (z)f 0 (z) 0 f /f 0 (3.25) A n (z) 0. (3.38) B n (z) 3.5 Stieltjes (3.5) z z f ( z) f 0 ( z) = + r z s r 2 z s r n z s n. f (z)/f 0 (z) p =, 2, 3,..., n s p = 0 p =, 2, 3,..., n s p = 0 f (z)/f 0 (z) zf (z 2 )/f 0 (z 2 ) Jacobi zf (z 2 )/f 0 (z 2 ) Jacobi s p 0 r p k p zf (z 2 ) f 0 (z 2 ) = k z + k 2 z + k 3 z + + k m z (3.39) 48

56 deg f 0 = n f 0 (0) = 0 m = 2n f 0 (0) 0 m = 2n f (z)/f 0 (z) z 2 z f (z) f 0 (z) = k z + + k 2 k 3 z + + T, T = k 2n z if f 0 (0) = 0, k 2n if f 0 (0) 0. (3.40) (3.40) (3.39) (3.40) (3.39) f (z)/f 0 (z) k p (3.7) : a 00, a 0, a 02,, a, a 2, a 3,, k = a 00 a, a 22 = a 0 k a 2, a 23 = a 02 k a 3, a 24 = a 03 k a 4,, k 2 = a a 22, a 33 = a 2 k 2 a 23, a 34 = a 3 k 2 a 24, a 35 = a 4 k 2 a 25,,. (3.4) 3.4 (3.40) p =, 2, 3,..., m zf (z 2 )/f 0 (z 2 ) D p 0 D p a, 0, a 2, 0, a 3, 0, a 4, 0,..., a 00, 0, a 0, 0, a 02, 0, a 03, 0,..., 0, a, 0, a 2, 0, a 3, 0, a 4,..., 0, a 00, 0, a 0, 0, a 02, 0, a 03,..., 0, 0, a, 0, a 2, 0, a 3, 0,..., (3.42) a 0 a 2 0 a 3 a 0 a 2 a D = a, D 2 = a 00 0 a 0, D a 0 0 a 02 = 0 a 0 a 2 0,... 0 a 0 0 a 00 0 a a 0 a 2 (3.20) : a a 2 a 3 a 4 a a 2 W = a 00 a 0, W a 00 a 0 a 02 a 03 2 =,... 0 a a 2 a 3 0 a 00 a 0 a 02 49

57 D 0 = a 00 = D 0 W 0, W 0 =, D = a = D W 0, D 2 = D W, D 3 = D 2 W, D 2p = ( ) p D p W p, D 2p+ = ( ) p D p+ W p. (3.43) : 3.4. ([Wa48]) (3.4) f 0, f f /f 0 p =, 2, 3,..., m (3.40) D p 0, (p = 0,, 2,..., n) n if f 0 (0) = 0, W k 0, (k = 0,, 2,..., m) m = n if f 0 (0) 0. (3.40) a 0 = k, a p = k p k p+, p =, 2, 3,..., m (3.44) (3.29) (3.43) a 0 z a a 2 z a 3 (3.45) a 0 = D D 0, a 2p = D p W p D p W p, a 2p = W p D p+ D p W p (3.46) 50

58 4 Jacobi ( c p /z p+) Jacobi {c p } Jacobi Jacobi 4. n = 0,, 2,... c n C c := {c n } u : (k 0 + k u + + k n u n ) dϕ c (u) = k 0 c 0 + k c + + k n c n. ϕ c (u) u u c k [0, ] {B p (u)} k p=0 c B p (u)b q (u)dϕ c (u) = 0 for p q : p = 0,, 2,..., m, (m > ) = 0 for p q, p, q m, B p (u)b q (u)dϕ c (u) 0 for p = q < m (4.) B p (u) = u p + β p u p + + β pp c 0 c c p c c 2 c p+ p = 0, c p c p+ c 2p p = 0,, 2,..., m (4.2) 5

59 0 B0(u)dϕ 2 c (u) = dϕ c (u) = c 0 0 = c < q < m B 0 (u)b q (u)dϕ c (u) = B q (u)dϕ c (u) = 0, B (u)b q (u)dϕ c (u) = ub q (u)dϕ c (u) = 0, B p (u)b q (u)dϕ c (u) = Bq 2 (u)dϕ c (u) = u p B q (u)dϕ c (u) = 0, u q B q (u)dϕ c (u) =: h p 0 q = h q q, q = 0,, 2,..., m, ( = ) (4.2) (4.2) B0(u)dϕ 2 c (u) = dϕ c (u) = c 0 = 0 0. a 0 := c 0 B (u) := u + b B 0 (u)b (u)dϕ c (u) = c + c 0 b = 0 B 0 (u)dϕ c (u) = a 0, ub 0 (u)dϕ c (u) = a 0 b a 0 0 b B n < m B 0 (u), B (u), B 2 (u),..., B n (u) p, q m (4.) a p, b p u p B p (u)dϕ c (u) = a 0 a a p 0, u p+ B p (u)dϕ c (u) = a 0 a a p (b +b 2 + +b p+ ), p = 0,, 2,..., n, B (u) = 0 B p (u) (4.3) B p (u) = (b p + u)b p (u) a p B p 2 (u), p =, 2, 3,..., n, B (u) = 0 (4.4) 52

60 B n+ (u) deg B n+ (u) = n +, B n+ (u) = (b n+ + u)b n (u) a n B n (u) + k 0 B 0 (u) + k B (u) + + k n 2 B n 2 (u) b n+, a n, k 0, k,..., k n 2 u p (u)b n+ (u)dϕ c (u) = 0, p = 0,, 2,..., n 2 k 0 a 0 = 0, k a 0 a = 0,...,k n 2 a 0 a a n 2 = 0 k 0 = k = = k n 2 = 0 u n B n+ (u)dϕ c (u) = 0, u n B n+ (u)dϕ c (u) = 0 p = n (4.3) u p B n (u)dϕ c (u) = 0, p = 0,, 2,..., n, u n B n (u)dϕ c (u) = a 0 a a n n = a 0 a a n n n 0 a n 0 b n+ p, q n + (p q) B p (u)b q (u)dϕ c (u) = 0, n + < m Bn+(u)dϕ 2 c (u) = u n+ B n+ (u)dϕ c (u) 0. n+ = 0 : 4.2. ([Wa48]) c = {c p } m > (4.2) (4.) B p (u) = u p + β n + + β nn, (p = 0,, 2,..., m) B 0 (u) =, B (u) = b + u, B p (u) = (b p + u)b p (u) a p B p 2 (u), p = 2, 3, 4,..., m; u p B p (u)dϕ c (u) = a 0 a a p, u p+ B p (u)dϕ c (u) = a 0 a a p (b + b b p+ ), p = 0,, 2,..., m (4.5) (4.6) (4.2) 4.3. {a p }, {b p } a p 0, (p =, 2, 3,... ) (4.5) (4.6) c 0, c,..., c 2m 53

61 4.2 Jacobi (4.5) B p (z) Jacobi a 0 b + z a b 2 + z a m b m + z (4.7) ( ) P := z p=0 c p z p+ (4.8) P (/z)b p (z) /z, /z 2,..., /z p u r B p (u)dϕ c (u) = 0, p = 0,, 2,..., p /z p+, /z p+2 u p B p (u)dϕ c (u) = a 0 a a p, u p+ B p (u)dϕ c (u) = a 0 a a p (b + b b p+ ) p A p (z) ( ) P B p (z) A p (z) = a 0a a p z z p+ a 0a a p (b + b b p+ ) z p+2 +, (4.9) B p (z) p = 0,, 2,..., m A p (z) B p (z) = c 0 z + c z c 2p z 2p + c 2p +. (4.0) z2p+ (3.32) Jacobi n 2n A n (z) B n (z) = p=0 c p z p+ + (p) c (n) 2n z 2n+ + (4.0) Jacobi p m =, 2, 3,... (4.2) {B m (z)} 2p P (/z) Jacobi p Jacobi {c p } Jacobi : 4.4. ([Wa48]) p = 0,, 2,... p 0 P (/z) Jacobi a p Jacobi p = a 0 a a p p, p = 0,, 2,..., ( = ) (4.) 54

62 P (/z) z p N a p = 0 p = 0 : 4.5. ([Wa48]) Jacobi p N a p 0 z (4.6) : (c 2 c ) (c 3 c 2 ) B p (z) = z p + β p z p + β p2 z p β pp, A p (z) = α p z p + α p z p β p,p ; β 00 =, c 0 β 00 = a 0, c β 00 = a 0 b = h 0 ; b = h 0 a 0, (β 0 β ) = ( b ), ( ) ( β 0 β β 0 β ) = a 0 a, = a 0 a (b + b 2 ) = h ; (c 4 c 3 c 2 ) b 2 = h 0 h, a 0 a0a ( (β 20 β 2 β 22 ) = (β 0 β ) (c 5 c 4 c 3 ) β 20 β 2 β 22 β 20 β 2 β 22 = a 0 a a 2, b b 2 ) = a 0 a a 2 (b + b 2 + b 3 ) = h 2 ; a (0 0 β 00 ), (4.2) b 3 = h h 2, a 0 a a 0 a a 2 (β 30 β 3 β 32 β 33 ) = (β 20 β 2 β 22 ) b b b 3 a 2 (0 0 β 0 β ), (α p0 α p α p2 ) = (β p0 β p β p2 ). (4.3) 55

63 c 0 c c p 0 c 0 c p c 0 2 (4.5) 2 (4.6) (4.3) (4.9) z 0, z, z 2,..., z 2p. Jacobi 4.6. c p =, p = 0,, 2,... p + ( ) z P = log z z = z + /2 z 2 + /3 z 3 + Jacobi β 00 =, c 0 β 00 = a 0 =, c β 00 = a 0 b = 2 ; (c 2 c ) (/4 /3) ( b = 2, (β 0 β ) = ( /2), ( ) ( β 0 β /2 ) = (/3 /2) /2 ) = a 0 a (b + b 2 ) = 2 = h ; = a 0 a = 2, a = 2 (/5 /4 /3) b 2 = 2, (β 20 β 2 β 22 ) = ( /2) (/6 /5 /4) /6 /6 ( = ( /6) /2 0 0 /2 = a 0 a a 2 = 80, a 2 = 5, = a 0 a a 2 (b + b 2 + b 3 ) = 20 = h 2; 56 ) (/2)(0 0 ),

64 b 3 = 2, (β 30 β 3 β 32 β 33 ) = ( 2/3 3/5 /20), (α 30 α 3 α 32 ) = ( /60). /2 + z /2 /2 + z /5 /2 + z = z 2 z + /60 z 3 3z 2 /2 + 3z/5 /20. z = log 2 = Stieltjes Stieltjes k z + + k 2 k 3 z + + k 4 Jacobi p =, 2, 3,... b p = 0 Stieltjes a 0 z a z a 2 z (4.4) a 0 z z 2 a a 2 z 2 a 3 (4.5) P (/z) (4.4) ( ) z P = c 0 z z 2 + c z 4 + c 2 z 6 + (4.6) /z z 2 z a 0 z a a 2 z a 3 (4.7) Stieltjes c 0 z + c z 2 + c 2 z 3 + (4.8) Jacobi c 0, 0, c, 0, c 2, 0, c 3,... c 0, c 0 0 c 0 0 c 0 c, 0 c 0 0, (4.9) c 0 c 2 57

65 0 c c 2 c p+ c 2 c 3 c p+2 Ω p =, c p+ c p+2 c 2p+ p = 0,, 2,... (4.20) (4.9) 0, 0 Ω 0, Ω 0, Ω, (4.8) Stieltjes (4.7) p 0, Ω p 0, p = 0,, 2,... (4.2) c 0, 0, c, 0, c 2, 0, c 3,... Jacobi (4.9) 0 Jacobi p = 0,, 2,... c 2p c p, c 2p+ 0 (4.2) b = 0, β = 0, b 2 = 0, β 2 = 0, b 3 = 0, β 3 = 0, β 33 = 0, b 4 = 0, β 4 = 0, β 43 = 0, b 5 = 0, β 5 = 0, β 53 = 0, β 55 = 0, (4.4) (4.8) Stieltjes (4.7) : 4.7. ([Wa48]) p = 0,, 2,... p 0, Ω p 0 (4.8) a p 0 Stieltjes 4.8. Stieltjes (4.7) p = 0,, 2,... a p > 0 p > 0, Ω p > 0 Jacobi (c.f. 4.5): 4.9. ([Wa48]) Stieltjes p N a p 0 z 58

66 4.4 Stieltjes Jacobi Stieltjes [St89] 4.0 (Stieltjes ). Jacobi b + z a b 2 + z a 2 b 3 + z (4.22) ( ) P = z p=0 ( ) p c p z p+ (4.23) c p+q = k 0p k 0q + a k p k q + a a 2 k 2p k 2q +, k 00 =, k rs = 0 if r > s k rs (r s) k k 0 k 0 0 k 02 k 2 k 22 0 = b a b a 2 b 3 0 k 0 k k 02 k 2 k k 03 k 3 k 23 k 33 0 (4.24) (4.25) : c p+q x p x q = (x 0 + k 0 x + k 02 x 2 + ) 2 p,q=0 + a (x + k 2 x 2 + k 3 x 3 + ) 2 + a a 2 (x 2 + k 23 x 3 + k 24 x 4 + ) 2 +. (4.26) p =, 2, 3,... a p 0 (4.26) P (/z) Jacobi (4.22) p =, 2, 3,... b = k 0, b p+ = k p,p+ k p,p (4.27) (4.23) Jacobi (4.22) Q(z) = p=0 c p z p p!, Q r(z) = 59 p=r k rp z p p!, (c 0 = ) (4.28)

67 Q(x + y) = Q(x)Q(y) + a Q (x)q (y) + a a 2 Q 2 (x)q 2 (y) + (4.29). p,q=0 c p+q x p p! yq q! = Q(x)Q(y) + a Q (x)q (y) + a a 2 Q 2 (x)q 2 (y) + y x (4.26) (4.25) C p+q := k 0p k 0q + a k p k q + a a 2 k 2p k 2q + C p,q+ = k 0p (b k 0q + a k q ) + a k p (k 0q + b 2 k q + a 2 k 2q ) + a a 2 k 2p (k q + b 3 k 2q + a 3 k 3q ) +, C p+,q = k 0q (b k 0p + a k p ) + a k q (k 0p + b 2 k p + a 2 k 2p ) + a a 2 k 2q (k p + b 3 k 2p + a 3 k 3p ) +. C p,q+ = C p+,q = C p+2,q = = C p+q+,0 = C p,q+2 = C p 2,q+3 = = C 0,p+q+. p + q = r + s C p,q = C r,s C p,q = C p+q C = p,q=0 C p+q x p x q C = (k 00 x 0 + k 0 x + k 02 x 2 + )(k 00 y 0 + k 0 y + k 02 y 2 + ) + a (k x + k 2 x 2 + k 3 x 3 + )(k y + k 2 y 2 + k 3 y 3 + ) + a a 2 (k 22 x 2 + k 23 x 3 + k 24 x 4 + )(k 22 y 2 + k 23 y 3 + k 24 y 4 + ) +. p, q > n x p, y q = 0 U 0 = k 00 x 0 + k 0 x + k 02 x k 0n x n, U = k x + k 0 x + + k n x n, U n = k nn x n, V 0 = k 00 y 0 + k 0 y + k 02 y k 0n y n, V = k y + k 0 y + + k n y n, V n = k nn y n 60

68 n p,q=0 C p+q x p y q = n a 0 a a p U p V p, (a 0 = ) p=0 k 00 k k 0n 0 k k n = 0 0 k nn C 0 C C p C C 2 C p+ p = C p C p+ C 2p = a p+ 0 a p ap 2 a 2 p a p. (4.23) Jacobi p =, 2, 3,... a p 0 p = a 0 a a p p, p = 0,, 2,..., ( = ). (4.30) x n = y n = 0, p, q > n + x p, y q = 0 p = k p,p+ a p+ 0 a p a2 p a p = (a 0 a a p )(b + b b p+ ) p (4.3) C C 2 C p+ C 2 C 3 C p+2 p = C p+ C p+2 C 2p+ (4.30) (4.3) (4.9) C p c p /z, /z 2,..., /z p p 0 c p a p, b p C p C p = c p, (p = 0,, 2... ) (4.24) (4.26) C p = c p (4.30) (4.3) Jacobi (4.22) sech k u e zu du = z + k z 2(k + ) + + z 3(k + 2) + (4.32) z 6

69 sech k u = kz2 2! + 4! k(3k + 2)z4 6! {k(5k2 + 30k + 6)}z 6 + (4.32) 0 sech k u e zu du ( = kz ! 4! k(3k + 2)z4 ) 6! {k(5k2 + 30k + 6)}z 6 + e zu du = e zu kz 2 du 0 0 2! e zu du + = z k k(3k + 2) + z3 z 5 (4.33) c = (c 0, c, c 2, c 3, c 4,... ) = (, 0, k, 0, k(3k + 2),... ), z k k(3k + 2) + z3 z 5 Jacobi sech k (x + y) = (cosh x cosh y + sinh x sinh y) k = sech k x sech k y ksech k+ x sinh x sech k+ y sinh y k(k + ) 2 sechk+2 x sinh 2 x sech k+2 y sinh 2 y = Q(x)Q(y) + a Q (x)q (y) + a a 2 Q 2 (x)q 2 (y) +, Q p (z) = sechk+p z sinh p z p! = zp z p+ p! + k p,p+ (p + )! +, a p = p(k + p ), k p,p+ = 0 Jacobi (4.25) Jacobi z + 2 t z + t t z + 2t t z + 3t

70 (4.25) t + t (4.24) = t t t 2t c 0 =, c 3 = t 2 + 4t +, t + t t 2 + 4t + t 2 + 0t + 7 3t c =, c 4 = t 3 + t 2 + t +, c 2 = t +, c 5 = t t t t +, c 6 = t t t t t Jacobi Jacobi Stieltjes z /z /z a 0 b z + a z 2 b 2 z + a 2z 2 b 3 z +, (4.34) a 0 a z a 2z (4.35) P (z) = c 0 + c z + c 2 z 2 + (4.36) 4.2. ([Vl0b]) Jacobi (4.34) M > 0 z M (4.36) M Stieltjes. {f p (z)} p=0 Jacobi ( Stieltjes ) z M k N p k f p (z) z M u (z) := f k (z), u p (z) := f k+p (z) f k+p 2 (z), p = 2, 3, 4,... 63

71 p= u p (z) = lim p f p (z) = u(z) z M u(z) z < M p= u p(z) p= u (n) p (z) = u n (z) f p (z) Jacobi (4.34) Jacobi (4.34) P (z) p n = 0,, 2,... u (n) p (0) = n!c n. u(z) = p= u (p) (0) z p = p! p=0 c p z p = P (z). p=0 6 Gauss Stieltjes 4.3. ([Vl04]) lim a p = 0, (a p 0) (4.37) p Stieltjes (4.35) f(z) f(z) K C lim a p = a 0 (4.38) p Stieltjes (4.35) Ω C\[/4a, ] f(z) K C\[/4a, ] [Wa48, pp.38] [Vl0b, pp.254] : 4.4. ([Vl0b]) {h p } p=0 h h = 0 L /4h K d(k, L) = inf{d(x, y) : x K, y L} > 0 h 0 K N(K) N + h nz + h n+z n > N(K) 64 + h n+2z + (4.39)

72 4.3. K (4.37) (4.38) /4a 4.4 N(K) N n > N(K) a n z a n+z a n+2z (4.40) K F n (z) 0 K 4.2 (4.40) F n (z) 4.9 F n (z) A m B m := a nz a n+z a n+2z Stieltjes (4.35) (n + m) A n (z) A m B m A n (z). B n (z) A m B m B n (z) A n (z) F n (z)a n (z) B n (z) F n (z)b n (z) K B n (z) F n (z)b n (z) 0 ( ) 65

73 5 Stieltjes Jacobi a p, b p Stieltjes Jacobi 0 dϕ(u) z u [St94a] ϕ(u) Van Vleck [Vl03] Jacobi 5. Stieltjes Stieltjes k p R, (p =, 2, 3,... ) Stieltjes (3.) Stieltjes 0 f(u)du z + u, f(u) > 0 p= L p z + x p, L p > 0, 0 x < x 2 < x 3 < u [a, b], (a, b R) f(u), ϕ(u) [a, b] n + u, u 2,..., u n u 0 := a u n+ := b : u 0 < u < u 2 < < u n+ k =, 2, 3,..., n + u k v k u k v k n+ S (u, v) := f(v k ) [ϕ(u k ) ϕ(u k )] k= := max{u k u k : k =, 2, 3,..., n + } 66

74 ϕ(u) f(u) Stieltjes lim S (u, v) = L L b a f(u)dϕ(u) Stieltjes [Wa48, pp.239] 5.. Stieltjes Stieltjes b a b b a a f(u)dϕ(u) ϕ(u)df(u) f(u)dϕ(u) = f(b)ϕ(b) f(a)ϕ(a) b a ϕ(u)df(u). (5.) Stieltjes f(u) ϕ(u) Stieltjes : b a b a {f (u) + f 2 (u)} dϕ(u) = f(u)d {ϕ (u) + ϕ 2 (u)} = b a b a b a f (u)dϕ(u) + f(u)dϕ (u) + b a b b k f(u)d {k 2 ϕ(u)} = k k 2 f(u)dϕ(u), a a f 2 (u)dϕ(u); f(u)dϕ 2 (u); k, k 2 : ( ) 2 [Wa48, p.244] 5.2. f(u) [a, b] ϕ(u) t > 0 f(u 0 ) > 0 ϕ(u 0 + t) > ϕ(u 0 t) u 0 (a, b) b a f(u)dϕ(u) > P (u) r 0 ϕ(u) r u 0 (a, b) t > 0 f(u 0 ) > 0 ϕ(u 0 + t) > ϕ(u 0 t) b a u k P (u)dϕ(u) = 0 for k = 0,, 2,..., n P (u) (a, b) n 67

75 Stieltjes. 5.2 Jacobi u p c p c p = u p dϕ c (u), p = 0,, 2,... {c p } Hausdorff {µ p } µ p = 0 u p dµ(u), p = 0,, 2,... µ(u) ( ) Stieltjes 5.4. (0, ) {µ p } 0 for u 0, µ(u 0) + µ(u + 0) µ(u) = for 0 < u <, 2 µ() for u > (5.2) µ p = 0 u p dµ(u), p = 0,, 2,... (5.3) µ(u) µ(u) (5.2) (5.3) Hausdorff [Wa48, p.259] 5.5. (, +) {d p } θ( ) for u <, θ(u 0) + θ(u + 0) θ(u) = for < u <, 2 θ() for u >, (5.4) θ(u) = θ( u) for < u < + (5.5) 68

76 d p = + u p dθ(u), p = 0,, 2,... (5.6) θ(u) 2 : 5.6. θ(u) u p := d 2p /2, (p = 0,, 2,... ) Hausdorff µ(u) = θ( u) Hausdorff µ(u) d 2p := 2µ p, (p = 0,, 2,... ) θ(u) = µ(u 2 ), (u 0) θ(u) = µ(u 2 ), (u < 0). µ p = 0 up dµ(u) = 0 u2p dµ(u 2 ) = 0 up dθ(u) = (/2) + u2p dθ(u) = d 2p /2 d 2p = + u2p dθ(u) = + t4p dµ(t 2 ) = + t2p dµ(t) = 2µ 2p d 2p+ = 0 Stieltjes 5.3 (5.6) d p = 0, (p = 0,, 2,... ) d p > 0 d 0 > 0 d 0 = 5.7. {d p } {d p } Stieltjes z ( g 0)g z ( g )g 2 (5.7) z 0 g p, (p = 0,, 2,... ) θ(u) + dθ(u) z u Stieltjes (5.7) (5.8). : {d p } p=0 Stieltjes (5.7) (5.7) I := [, +] d(k, I) = inf{d(x, y) : x K, y I} > 0 K [Wa48, p.255]: z ( g 0)g z ( g )g 2 + dθ(u) = z z u, (z ). 69

77 4.2 d p = zp+ z ( g 0)g z p=0 ( g )g 2 + = z /(z u) = p=0 up /z p+ dθ(u), (z ) z u d p = + u p dθ(u), p = 0,, 2,... : θ(u) {d p } p=0 d 0 = θ(u) {u p } m p=0 t > 0 p = 0,, 2,..., m θ(u p + t) > θ(u p t) (X 0 + X u + + X n u n ) 2 dθ(u) > 0, n = 0,, 2,..., m d 0 d d n d d 2 d n+ n = > 0, n = 0,, 2,..., m. d n d n+ d 2n : a 0 b + z a b 2 + z a 2 b 3 + z a m b m + z a 0 = d 0 = a, a 2,..., a m > 0 Jacobi 2m d 0, d, d 2,... (5.5) (5.8) z Jacobi z a z a 2 z a m z (5.9) p B p (z) c > 0 ++c c u r B p (u)dθ(u) = 0, r = 0,, 2,..., p, p m (5.9) 5.3 p < m B p (u) [ c, + + c] p u B p (u) > 0 u > B p (u) > 0 a p = B ( p( + c) + c B ) p+( + c), p =, 2, 3,..., m. (5.0) B p ( + c) B p ( + c) 70

78 a p > 0 0 < B p+( + c) B p ( + c) < + c, p = 0,, 2,..., m, (c > 0). B p+ ( + c) lim c 0 B p ( + c) = g p 0 g p, (p = 0,, 2,..., m ) (5.0) a p = ( g p )g p, (p =, 2, 3,..., m ) (5.9) (5.7) θ(u) n > 0, (n =, 2, 3,... ) (5.9) z a z a 2 z (5.7) /z ( g 0)g /z 2 ( g )g 2 /z 2 /z /z 2 =: z [Wa40]: 5.8. (0, ) Hausdorff µ p = 0 u p dµ(u), p = 0,, 2,... (5.) µ 0 µ z + µ 2 z 2 (5.2) µ 0 + ( g 0)g + ( g )g 2 + (5.3) µ g p, (p = 0,, 2,... ) (, + ) Hamburger [Wa48, pp.325] [Ha20] 7

79 5.9. {c p } p=0 c 0 = (, + ) c p = + u p dϕ(u), p = 0,, 2,... (5.4) ϕ(u) c 0 c c p c c 2 c p+ p = > 0, p = 0,, 2,... (5.5) c p c p+ c 2p : 5.0. (5.4) ϕ( ) = 0, ϕ(u) = [ϕ(u + 0) + ϕ(u 0)]/2, ( < u < + ) ϕ(u) Hamburger (, + ) Stieltjes [St94a]: 5.. {c p } p=0 [0, + ) c p = + 0 u p dϕ(u), p = 0,, 2,... (5.6) ϕ(u) c c 2 c p+ c 2 c 3 c p+2 Ω p = > 0, p = 0,, 2,... (5.7) c p+ c p+2 c 2p+ [Wa48, pp.327] Stieltjes Stieltjes [Wa48, p.329] [St94a] 5.2. {c p } p=0 (5.5) (5.7) (c p /z p+ ) Stieltjes k z k 2 k 3 z (5.8) k 4 Stieltjes (5.6) k p 72

80 6 Gauss Gauss Gauss 2 Stieltjes Gauss [Fr56] ([MeSc6, Kü02] ) ( g p )g p z 6. Gauss a, b C c {0,, 2, 3,... } F (a, b, c; z) = n=0 (a) n (b) n (c) n n! zn = + ab a(a + )b(b + ) z + z 2 a(a + )(a + 2)b(b + )(b + 2) + z 3 + c c(c + ) 2! c(c + )(c + 2) 3! (6.) (a) n Pochhammer (a) 0 := (a) n+ := (a) n (a+ n), (n = 0,, 2,... ) a b {0,, 2,... } F (a, b, c; z) 6.. F (,, 2; z) = z 2 + z2 3 z3 4 + = log( + z), z F ( k,, ; z) = + kz + 2 (k )z2 + 6 (k 2)(k )kz3 + = ( + z) k, ( z F 2, 2, 3 ) 2 ; z2 = z + 6 z z z7 + = arcsin(z), ( z F 2,, 3 ) 2 ; z2 = z z3 3 + z5 5 z7 7 + = arctan(z). (6.2) 73

81 (6.) z z/a a z 2 Φ(b, c; z) = + b b(b + ) b(b + )(b + 2) z + + c c(c + ) 2! c(c + )(c + 2) 3! + (6.3) (6.3) z z/b b z 2 Ψ(c; z) = + c z + c(c + ) 2! + c(c + )(c + 2) 3! + (6.4) (6.) z cz c Ω(a, b; z) = + abz + a(a + )b(b + ) z2 2! z 3 z 3 + a(a + )(a + 2)b(b + )(b + 2)z3 3! + (6.5) a b {0,, 2,... } 0 (6.) Gauss : F (a, b, c; z) = F (a, b +, c + ; z) a(c b) c(c + ) zf (a +, b +, c + 2; z). (6.6) F (a, b +, c + ; z) = F (a, b, c; z). a(c b) F (a +, b +, c + 2; z) z c(c + ) F (a, b +, c + ; z) (6.7) (6.7) a b b b + c c + F (a +, b +, c + 2; z) = F (a, b +, c + ; z). (b + )(c a + ) F (a +, b + 2, c + 3; z) z (c + )(c + 2) F (a +, b +, c + 2; z) (6.8) (6.7) (6.8) Gauss [Ga76]: a(c b) F (a, b +, c + ; z) = F (a, b, c; z) c(c + ) z (b + )(c a + ) z (c + )(c + 2) (6.9) (a + )(c b + ) (b + 2)(c b + 2) z z (c + 2)(c + 3) (c + 3)(c + 4). 74

82 {a p } z a 2p+ = (a + p)(c b + p) (c + 2p)(c + 2p + ), a 2p+2 = n = 0,, 2,... (b + p + )(c a + p + ), p = 0,, 2,... (c + 2p + )(c + 2p + 2) (6.0) P 2n (z) = P 2n+ (z) = F (a + n, b + n +, c + 2n + ; z), F (a + n, b + n, c + 2n; z) F (a + n +, b + n +, c + 2n + 2; z) F (a + n, b + n +, c + 2n + ; z) (6.) P n (z) = Gauss (6.9), a n zp n (z) n =, 2, 3,... (6.2) F (a, b +, c + ; z) F (a, b, c; z) = a z a 2z a n z a n zp n (z). (6.3) k N a, a 2,..., a k 0 a k = 0 F (a, b +, c + ; z)/f (a, b, c; z) z Gauss (6.3) p =, 2, 3,... a p 0 A p (z), B p (z) Gauss (6.3) p F (a, b +, c + ; z) F (a, b, c; z) F (a, b +, c + ; z) F (a, b, c; z) A n (z) B n (z) = = A n(z) a n zp n (z)a n (z) B n (z) a n zp n (z)b n (z). a a 2 a n z n B n (z) {B n (z) a n zb n (z)p n (z)} n A n (z)/b n (z) Gauss F (a, b +, c + ; z)/f (a, b, c; z) n Gauss Gauss Gauss : 6.2. ([Th67, Ri63, Vl0b]) Gauss (6.) C\[, ] F (a, b +, c + ; z)/f (a, b, c; z) F (a, b +, c + ; z)/f (a, b, c; z) C\[, ] Gauss K C\[, ] Gauss (6.) 75

83 . lim a (a + p)(c b + p) p = lim p p (c + 2p + )(c + 2p + ) = lim (b + p + )(c a + p + ) p (c + 2p + )(c + 2p + 2) = C 4.2 Gauss F (a, b+, c+; z)/f (a, b, c; z) Gauss F (a, b+, c+; z)/f (a, b, c; z) g 2p = c a + p c + 2p, g 2p+ = c b + p, p = 0,, 2,... (6.4) c + 2p + F (a, b +, c + ; z) = F (a, b, c; z) ( g 0)g z ( g )g 2 z ( g 2)g 3 z (6.5) b = 0 c c Gauss a F (a,, c; z) = c z c a c(c + ) z c(a + ) (c + )(c + 2) z 2(c a + ) (c + 2)(c + 3) z (6.6) {b p } z b 2p+ = (a + p)(c + p ) (c + 2p )(c + 2p), b 2p+2 = Gauss (6.6) h 2p = a + p c + 2p 2, h 2p = F (a,, c; z) = h z (p + )(c a + p), (c + 2p)(c + 2p + ) p = 0,, 2,.... (6.7) ( h )h 2 z p, c + 2p p = 0,, 2,... (6.8) ( h 2)h 3 z. (6.9) (6.4) 0 < g p <, (p = 0,, 2,... ).9 g F (a, b +, c + ; z) 0 F (a, b, c; z) 2 g 0 g 0 2 g 0 c a F (a, b +, c + ; z) c c F (a, b, c; z) c + a c + a, z (6.20) 76