,..,,.,,.,.,..,,.,,..,,,. 2

A.A. (1906) (1907). 2008.7.4 1.,.,.,,.,,,.,..,,,.,,.,, R.J.,.,.,,,..,.,. 1

,..,,.,,.,.,..,,.,,..,,,. 2

1, 2, 2., 1,,,.,, 2, n, n 2 (, n 2 0 ).,,.,, n ( 2, ), 2 n.,,,,.,,,,..,,. 3 x 1, x 2,..., x n,...,, a 1, a 2,..., a n,...,. x a = z 1, 2 15-4, 1906, 135-156 2 P.L. 3. 3

. 2 (z 1 + z 2 + + z n ) 2,,. z1 2 + z2 2 + + zm 2 + 2z 1 z 2 + 2z 1 z 3 + + 2z n 1 z n., z z k 0, (z 1 + z 2 + + z n ) 2, z 1, z 2,..., z n 2,,, z z k, E(z 1 + z 2 + + z n ) 2 < n k=1 Ez 2 k. 4, z, z,..., z (ω), z, p, p,..., p (ω)., z k, z = z, z,..., z (ω) a k, a k,..., a (ω) k., z z k, p z a k + p z a k + + p (ω) z (ω) a (ω) k., z, z k 0, p z + p z + + p (ω) p a k + p a k + + p (ω) z (ω), a (ω) k 4, E,, M.E.(=Mathematica Expectation). 4

., a k > a k > > a (ω) k, 5, p (i) z (i) a (i) k < p (i) z (i) (i) p a (i) k = 0. 6, zn 2, n. z k, k, x 1 + x 2 + + x k 1, E(z 1 + z 2 + + z n ) 2 < n k=1 Ez 2 k,., (z 1 +z 2 + +z n ) 2 = z 2 1 +z 2 2 + +z 2 n+2z 1 z 2 +2(z 1 +z 2 )z 3 +2(z 1 +z 2 +z 3 )z 4 +., x 1 + x 2 + + x n, n, 1, 2, 3. 1. 2a, 2b. 2.. 3. a + b, a b. 5, P.L. 2 716-719, Korkine Comptes Rendus,96 6,. 5

,, a,, 1 a + b,, a, a + b. 2.,,,., x 1, x 2,..., x n,...,. x 1 + x 2 + + x n, n, A., A, A p,, A p.,, p. 7,, p, p, p 1 p = q, 1 p = q, 1 p = q p = pp + qp. 8,,, p = p 1 p + p, p = 1 + p p p, p = p 1 p 1 p. x, x k, A, k, 0 1., a = a k = p 7, A p, n p = P (x n = 1). 8 p = P (x n = 1/x n 1 = 1) + P (x n = 1/x n 1 = 0). 6

., x x k a k x k a k x + a k a z z k, Ex x k p 2. x x k, k A,,, A p, A, k A Rk., Ez z k = p(rk p), R k., k, R k. k = 1, R 1 = p, R 2 = R 1 p + (1 R 1 )p = p p + q p, R 3 = R 2 p + (1 R 2 )p,...... R m+1 = R m p + (1 R m )p = p + (p p )R m. R m+1 = p + (p p )R m,, R m = p + C(p p ) m, C R 1 = p., Ez z k = pq(p p ) k 7

, E(z 1 + z 2 + + z k 1 )z k = pq ( p p + (p p ) 2 + + (p p ) k 1)., p < p, (z 1 + z 2 + + z k 1 )z k, E(z 1 + z 2 + + z k 1 )z k < npq. p > p, (z 1 + z 2 + + z k 1 )z k.,. pq(p p ) 1 p + p ( E(z 1 + z 2 + + z n ) 2 < npq 1 + 2(p p ) ) < npq(1 + p p ) 1 p + p 1 p + p,., A, p,, 1. 3. z z k,,., k, A m P m,k. P m,k = V m,k + U m,k., U m,k V m,k P m,k. (1) U m,k A, (2) V m,k A, 8

., ξ 9 Φ k = U 0,k + U 1,k ξ + U 2,k ξ 2 + + U k 1,k ξ k 1 Ψ k = V 1,k ξ + U 2,k ξ 2 + + V k,k ξ k Ω k = P 0,k + P 1,k ξ + P 2,k ξ 2 + + P k,k ξ k = Ψk + Φk. Ω k, t t k., U m,k = q V m,k 1 + q U m,k 1 V m,k = p V m 1,k 1 + p U m 1,k 1. Φ k Ψ k, Φ k = q Ψ k 1 + q Φ k 1 Ψ k = p ξψ k 1 + p ξφ k 1. Φ Ψ, Φ k+1 (p ξ + q )Φ k + (p p )ξφ k 1 = 0 Ψ k+1 (p ξ + q )Ψ k + (p p )ξψ k 1 = 0. Ω k = Φ k + Ψ k Ω k+1 (p ξ + q )Ω k + (p p )ξω k 1 = 0. Ω k A + Bt 1 (p ξ + q )t + (p p )ξt 2 t t k., A B, t. A B, k k = 1, 2 Ω k. Ω 1 = q + pξ, 9. Ω 2 = qq + (qp + pq )ξ + pp ξ 2 9

., Ω 0 = 1. t, 2, A + (B + p ξ + q )t = Ω 0 + Ω 1 t = 1 + (q + pξ)t. A = 1 B = (p p )ξ + q q, B = (p p )(qξ + p)., 1 + (p p )(qξ + p)t 1 (p ξ + q )t + (p p )ξt 2 = Ω 0 + Ω 1 t + Ω 2 t 2 +., 10.,,..,,. 1, x 1 + x 2 + + x n +,, n 1), a, b. 2),., x 1 + x 2 + + x k 1 10 1907. 10

, x k., (z 1 + z 2 + + z k 1 )z k. (z 1 + z 2 + + z n ) 2, n ( m n a ) 2P a,b m,n a + b m=0,., P a,b m,n n m, P a,b m,n = 1 2 n a(a + 1) (a + m 1)b(b + 1) (b + n m 1) 1 2 m 1 2 (n m)(a + b)(a + b + 1) (a + b + n 1).,. mp a,b m,n = m(m 1)P a,b m,n = n m=0 n m=0 na a + b P a+1,b m 1,n 1 n(n 1)a(a + 1) (a + b)(a + b + 1) P a+2,b m 2,n 2 mp a,b m,n = m(m 1)P a,b m,n =.,, m=0 n m=0 na a + b n(n 1)a(a + 1) (a + b)(a + b + 1) P a,b m,n = 1 n ( m n a ) Pm,n a,b a + b., : n m=0 ( 2na ) n m(m 1)Pm,n a,b a + b 1 m=0 11 mp a,b ( a ) 2P m,n + n 2 a,b a + b n m,n m=0 P a,b m,n.

, n ( m n a a + b m=0 ) P a,b m,n = nab(n + a + b) (a + b) 2 (a + b + 1).,., ε, ε m n a a + b +ε, n 1.. β., ξ = ( a ) 2 a + b ( m n a ) 2 a + b ( b ) 2 a + b.,,., ξβ > ab (a + b) 2 (a + b + 1) ξ2 ab ξ < (a + b) 2 (a + b + 1), β., m µ, (a 1)n b + 1 (a + b 2)µ (a 1)n + a 1 (a + b 2)µ., µ, n, n a a 1 a + 1 a + b 2., n, m n a 1 a + b 2 ( m ) 2., n g a g = a + b 12

,, n ab, 0 (a + b) 2 (a + b + 1). a = b = 1, m = 0, 1, 2, 3,..., n, 1,. n + 1 1907 1 16. 2,.,,,,. x 1, x 2,..., x k, x k+1,..., k, x k, x k+1 x 1, x 2,..., x k 1.,,, β, γ,...., p,, p,β, p,γ,..., p β,, p β,β, p β,γ,............, x k, x k+1, p x k, x k+1., x k = β 13

, x k+1 = γ p β,γ.,, k., x k x 1, x 2, x 3,..., β, γ,..., p (k), p (k) β, p(k) γ,.... p,, p,β, p,γ,..., p β,, p β,β, p β,γ,............,., p, + p,β + p,γ + = 1 p β, + p β,β + p β,γ + = 1........... p (k), p (k) β, p(k) γ,..., p (k) + p (k) β + p (k) γ + = 1, k., p, p β, p γ,... 14

, p (k), p (k) β, p(k) γ,..., p (k+1) p (k+1) β = p, p (k) = p,β p (k)......... + p β, p (k) β +, + p β,β p (k) β +,. x 1, x 2,... x k,..., a k x k. A (i), A (i) β, A(i) γ,..., x k =, x k = β, x k = γ,... x k+i.,. a k = p (k) a k+i = p (k) A (i) A (i) A (i) β + p (k) β + p(k) γ γ +, β = p, A (i 1) = p β,a (i 1) + p (k) β A(i) β + p,β A (i 1) β + p β,β A (i 1) β........., + p(k) γ A (i) γ +, a k+i, A (i), A (i) β, A(i) γ,... 15 + p,γ A (i 1) γ +, + p β,γ A (i 1) γ +,

, i,.,, a k+i A (i), A (i) β, A(i) γ,...., A (i 1)., A (i 1), A (i 1) γ,... β, A (i) A (i) β A (i) A (i) β = (p, p β, )A (i 1) + (p,β p β,β )A (i 1) β +., (p, p β, ) + (p,β p β,β ) + = 0 p, p β,, p,β p β,β,...,,., 1., p,, p,β, p,γ,..., p β,, p β,β, p β,γ,...., A (i 1), A (i 1), A (i 1) γ,..., β (i 1) A (i 1), A (i 1), A (i 1) γ,... β 16

, A (i) A (i) β < h (i 1)., h p, p β,, p,β p β,β,..., 1. A (i), A (i) β, A(i) γ,...,., A (i) A (i) β, A (i), A (i) β, A(i) γ,... (i), (i) < H (i 1)., H 0 1., (i) i 0., i, a k+i, A (i), A (i) β, A(i) γ,...,., (i), (i) < CH i., C H 0 < H < 1. 2 (x 1 a 1 + x 2 a 2 + + x n a n ) 2 17

, x k a k = z k, 2., Ez k (z 1 + z 2 + + z k 1 ) < D(H + H 2 + + H k 1 ) E(x 1 a 1 + x 2 a 2 + + x n a n ) 2 < Gn., D G., (x 1 a 1 + x 2 a 2 + + x n a n ) 2 (x 1 + x 2 + + x n na) 2, 2., (a 1 a + a 2 a + + a n na) 2 a = im i a k+i., n,., n, ( x1 + x 2 + + x n n ) 2 a 0.,., ε η, ε < x 1 + x 2 + + x n n a < ε, n 1 η.,,. 1907 3 25 18

( ) a 1 < a 2 < < a n, b 1 > b 2 > > b n, p j, j = 1, 2,..., n j, k (a j a k )(b j b k ) < 0, p j p k j, k p j p k (aj a k )(bj b k ) = 2 j j,k p k a k b k 2 j p j a j p k a k < 0 k, p k a k b k < k k p k a k p k b k k., a j = z (j), b j = a (j) k, j = 1, 2,..., ω. 19

A.A. 1907 (Buetin de Académie Impériae des Sciences de St.-Pétersbourg) 1907 14,,. 11.,. 12, 2.,, 13,, 14.,,,,, 15,,.,.,..,, E E,, p., E, E,, 11,, 2 15 4 135-156, 1906 12 P.L. 1 688-694 13 481-492 14, e x2 d m e x2 dx = 0 Buetin de Académie m des Sciences de St-Pétersbourg, 15 20

p 1.,, E,, E p 2.,,,,. E, F., 1 p, 1 p 1, 1 p 2,, q, q 1, q 2,.,. E F, p q. p, p 1, p 2,, p = pp 1 + qp 2 (1),., E,,. p, p 1, p 2, q, q 1, q 2, δ = p 1 p 2 (2) p, q, δ. (1) q = 1 p, q 1 = 1 p 1, q 2 = 1 p 2 (3) 21

{ p 1 = p + δq, p 2 = p δq, q 1 = q δq q 2 = q + δq (4).,, n E.,. P m,k, E k m.,, P m,k Pm,k o P m,k, E k,., P m,k = Pm,k o + P m,k (5). ϕ k = P 0 m,kξ m, ψ k = P m,kξ m, ω k = P m,k ξ m (6), ξ., (5). ω k = ϕ k + ψ k (7), k k + 1,,. { P o m,k+1 = q 1P m,k + q 2P o m,k P m,k+1 = p 1P m 1,k + p 2P o m 1,k, (8) {. ϕ k+1 = q 1 ψ k + q 2 ϕ k ψ k+1 = p 1 ξψ k + p 2 ξϕ k (9) 22

, (9) ϕ ψ,. { ϕ k+2 (p 1 ξ + q 2 )ϕ k+1 + (p 1 p 2 )ξϕ k = 0, ψ k+2 (p 1 ξ + q 2 )ψ k+1 + (p 1 p 2 )ξψ k = 0. ω k+2 (p 1 ξ + q 2 )ω k+1 + (p 1 p 2 )ξω k = 0 (10), t, ω 0 Ω(ξ, t) = ω 0 + ω 1 t + ω 2 t 2 + ω 3 t 3 +... (11) ω 2 (p 1 ξ + q 2 )ω 1 + (p 1 p 2 )ξω 0 = 0 (12), Ω(ξ, t) = L 0 + L 1 t 1 (p 1 ξ + q 2 )t + (p 1 p 2 )ξt 2 L 0 = ω 0, L 1 = ω 1 (p 1 ξ + q 2 )ω 0., ω 1 = pξ + q, ω 2 = pp 1 ξ 2 + (pq 1 + qp 2 )ξ + qq 2, (12) ω 0 = 1,. L 0 = 1, L 1 = (p p 1 )ξ + q q 2 Ω(ξ, t), (4) Ω(ξ, t) =., ω n. 1 δ(qξ + p)t 1 tpξ + q + δ(qξ + p)t + δξt 2 (13) 23

. Ω(ξ, t) (13) n E., n E m (13), k m k P m,n., (m pn) k P m,n., m pn., pn m., m(m 1) (m i + 1), d i ω n dξ i ξ = 1., d i Ω(ξ, t) dξ i ξ = 1, t. ξ = 1, { } d i Ω(ξ, t) dξ i = 1 2 i pti (1 t) 2 ξ=1 { } p 1 t + δq 1 δt (14)., i,. i = 1, 2, 3, 4 24

, E(m) = np, E ( m(m 1) ) = n(n 1)p 2 + 2pqδ ( n 1 + (n 2)δ + (n 3)δ 2 +... ), E ( m(m 1)(m 2) ) = n(n 1)(n 2)p 3 + 6p 2 qδ ( (n 1)(n 2) + (n 2)(n 3)δ +... ) + 6pq 2 δ 2( n 2 + 2(n 3)δ + 3(n 4)δ 2... ), E ( m(m 1)(m 2)(m 3) ) = n(n 1)(n 2)(n 3)p 4 + 12p 3 qδ ( (n 1)(n 2)(n 3) + (n 2)(n 3)(n 4)δ +... ) + 36p 2 q 2 δ 2( (n 2)(n 3) + 2(n 3)(n 4)δ +... ) + 24pq 3 δ 3( n 3 + 3(n 3)(n 4)δ + 6(n 5)δ 2... ). (14),, m(m 1)... (m i + 1).,,,.. t { } d i Ω(ξ, t) dξ i ξ=1 t n. (14),. (i 1)(i 2) (i j) 1 2 j t i (1 t) i j+1 (1 δt) j 1 2 i pi j (δq) j t i (1 t) i j+1 (1 δt) j t, t n (n j)(n j 1) (n i + 1) 1 2 (i j) (n j 1)(n j 2) (n i) + jδ 1 2 (i j)) 25

j((j + 1) 2 n j 2)(n j 3) (n i 1) + δ +, 1 2 1 2 (i j).,.,, n j. n j., (n j λ)(n j λ 1) (n i λ + 1)., n (0 n., C 0 n i j + C 1 n i j 1 + C 2 n i j 2 + C i j (15)., δ n j, n. δ 2 < 1,. δ 2,.,. C 0, C 1, C 2,...,,., 1 2 (i j)c 0 = (1 δ) j (16)., (1 δ) j, n j. n, (16) C 0., m(m 1) (m i + 1) n. 26

, δ n 1, n., δ n 1., m(m 1) (m i + 1), p q, p, q i, p n,., n p, n p, [(m, i)] 0, 16 [(m, i)] 0 = (np) i + i(i 1) δq 1 δ (np)i 1 + i(i 1)2 (i 2) ( δq ) 2(np) i 2 + 1 2 1 δ + i(i 1)2 (i 2) 2 (i j + 1) 2 (i j) ( δq ) j(np) i j + (17) 1 2 j 1 δ. δ, δ n 1,., m., m,. m i = m(m 1) (m i + 1) + A 1,i m(m 1) (m i + 2) + + A j,i m(m 1) (m i + j + 1) + (18) A 1,i, A 2,i,..., A i 1,1 m,. 17 16 p + q = 1., (17) q. 17 A j,i. d i f(e x ) dx i = e xi f (i) (e x ) + A 1,i e (i 1)x f (i 1) (e x ) + A 2,i e (i 2)x f (i 2) (e x ) + 27

A j,i A 1,i = i(i 1), A j,j = 0, A j,i+1 = A j,i + (i j + 1)A j 1,i (19) 2. A 1,2 = 1 A 1,3 = 3, A 2,3 = 1 A 1,4 = 6, A 2,4 = 7, A 3,4 = 1 A 1,5 = 10, A 2,5 = 25, A 3,5 = 15, A 4,5 = 1 A 1,6 = 15, A 2,6 = 65, A 3,6 = 90, A 4,6 = 31, A 5,6 = 1 A 1,7 = 21, A 2,7 = 140, A 3,7 = 350, A 4,7 = 301, A 5,7 = 63, A 6,7 = 1 A 1,8 = 28, A 2,8 = 266, A 3,8 = 1050, A 4,8 = 1701, A 5,8 = 966, A 6,8 = 127, A 7,8 = 1. (19), A i,j = i(i 1) (i j) ( i j 1 + i j 2 + βi j 3 + ) 2 4 2j (20).,, β,... i. (20), (18) A j,i., (20) A j,0, A j,2, A j,2,..., A j,j 1, A j,j. (18), m(m 1) (m i + 1), m i. m i, n., δ, n., δ, n 1. 28

, p q, i, p n., n p, n p, [m i ] 0,, (17) (18) [m i ] 0 = (np) i + A 1,i (np) i 1 + A 2,i (np) i 2 + + { i(i 1)(np) i 1 + A 1,i (i 1)(i 2)(np) i 2 + + i(i 1) 2 (i j + 1) 2 (i j) (np) i j 1 2 j + (i 1)(i 2) 2 (i j) 2 (i j 1) + A 1,i 1 2 j + ( δq ) j 1 δ + } δq 1 δ (np) i j 1 (21), (21), (17).., m pn. (m np) k = m k km k 1 pn +,., (m pn) k k(k 1) m k 2 (pn) 2 + (22) 1 2 R (k) k nk + R (k) k 1 nk 1 + + R (k) i n i + (23) 29

., R (k) k, R(k) k 1,..., R(k) i,..., p, q, δ, δ, n, δ n 1., R (k) i p q k. p i, (m pn) k (23),., (23),., δ n δ n 1,. E F, (23) (m pn) k. E F, m n m, (4), p q, q p., m pn n m qn. m pn ±.,.,,., k, (m pn) k, p, q. k,. 30

(23), p, q, R (k) i,., R (k) i, k, p i, q i,, p + q = 1,., R (k) i 2i., p q R i (k), R (k) i k,. (24), (23), p q 2i k (24) (m pn) k, k 2, k = 2 1,, n. R (2 1) 2 1 = R (2 1) 2 2, k = 2 = = R (2 1) = 0 R (2) 2., R (2) = R (2) 2 1 = = R(2) +1 = 0 ( m np ) 2 1 im E = 0, (25) n n ( m np ) 2 im E = im R (2) n n n. (26). a 0 p + a 1 p q + a 2 p q 2 + + a 1 p q 1 + a p q +b 0 p +1 + b 1 p +1 q + b 2 p +1 q 2 + + b 1 p +1 q 1 + 31

, p, q. a 0 p + a 1 p q + a 2 p q 2 + + a 1 p q 1 + a p q, p q., (1 p), (1 p) 1, 1 p, 1, a 0 p + a 1 p q + a 2 p q 2 + + a 1 p q 1 + a p q, (a 0 + a 1 + + a )p q S = a 0 p (1 p) + a 1 p (1 p) 1 q + + a p q (27). p, q. p q R 2. p +1, q +1, p + q = 1, p q 2., R (2) S = 0,. R (2) = (a 0 + a 1 + + a )p q (28) a 0 p + a 1 p q + a 2 p q 2 + + a p q 32

, (m pn) 2, n p, n.,, [(m pn) 2 ] 0 [(m pn) 2 ] 0 = [m 2 ] 0 2 1 pn[m2 1 ] 0 + 2(2 1) [m 2 2 ] 0 1 2. a 0 p + a 1 p q + a 2 p q 2 ; + a p q, [m 2 ] 0, [m 2 1 ] 0, [m 2 2 ] 0,., (21), n., a 0, a 1,..., a.,. a j : ( δ ) 1 δ = ( + j)( + j 1)2 ( + 1) 2 A j,2 j! 2 ( + j 1)( + j 2) 2 2 ( 2) 1 j! 2(2 1) + 1 2 ± 2(2 1) ( + 2) ( 1)! ( + j 2)( + j 3) 2 8 1) 2 ( 2) A j,2 2 j! j + 1)j 2 (j 1) 2 2 2 1 A j,+1 j! 33

, (20), 18 a j = 2 (x + j )(x + j 1) 2 (x + 1) 2 (x ) ( δ x=0 A j,x j! 1 δ = 2 x 2 ( δ ) j j!2 4 2( j) 1 δ ( 1) ( j + 1) ( 2δ ) j = 1 3 5 (2 1) j! 1 δ, ( a 0 + a 1 + a 2 + + a = 1 3 5 (2 1) 1 + 2δ ). (29) 1 δ,, (16). δ,. (29), n, ) j a 0 + a 1 + a 2 + + a. (28),.,,, ( 1 + δ ) im n R(2) = 1 3 5 (2 1) 1 δ pq ( m pn ) 2 ( 1 + δ ) im E = 1 3 5 (2 1) n n 1 δ pq (30) ( m pn ) 2 1 im E = 0 n n ( m pn ) 2 ( 1 + δ ) im E = 1 3 5 (2 1) n n 1 δ pq.,, np + t 1 2pq 1 + δ 1 δ n < m < np + t 2 2pq 1 + δ 1 δ n 18 k x=0f(x) = f(k) k 1 k(k 1) f(k 1) + f(k 2) ± f(0) 1 2 34

, n, 1 π t2 t 1 e t2 dt., n, m E, p, q, t 1, t 2, δ..,, A.M..,, E,,.,, E p, p,..., p (n)..., (1), p (n) = p 1 p (n 1) + p 2 (1 p (n 1) ) (31)., p (n) = p + (p p)δ n 1 (32)., p, n p (n).,,., n E ω n, ω k+2 (p 1 ξ + q 2 )ω k+1 + (p 1 p 2 )ξω k = 0.. 35

Ω(ξ, t),,., ω 1 pξ +q p ξ + q., Ω(ξ, t),. (ξ, t) = (p p)(ξ 1)t 1 [pξ + q + δ(qξ + p)]t + δξt 2 (33) Ω(ξ, t),.,, Ω(ξ, t) (ξ, t)., m(m 1) (m i + 1), t t n ξ = 1 d i (ξ, t) dξ i. (33),., [ d i (ξ, t) dξ i ]ξ=1 = i!(p p)t i [ p (1 t)(1 δt) 1 t + δq ] i 1 (34) 1 δt m(m 1) (m i + 1),, p p n., p n, p q i 1., (m pn) k, p p,, p, n, p, q k 1. 36

, E F, p q, q p, p q = 1 p,, E(m pn) k,,.,, (m pn) k, n k/2, n 0., A.M., n ( m pn n ) 2 1 ( m pn ) 2 n., E,,. 37

1917 10, 1906 (1), 1951,, (2). (2),., (1) Independence.. (1) Rasprostranenie zakona bo~xih qise na veiqiny, zaviswie drug ot druga (2) Issedavanie zameqateago suqa zavisimyh ispytani 1907, 61-80) 38