1 22.3 3?f ?# ?é ? 3?f ?￥?l ? d=[ T? ? ? 3?f ?￥?é ? 3?f ?D? ￥?1" 2 !?  {a n }/? T a) ? G(x) = a 0 + a 1 x + a 2 x 2 + … + a n x n + … ? G(x)1 {a n }￥ 3?f ? . L è {C(m,n)}￥ 3?f ?1 G(x)= 1 + C(m,1)x + C(m,2)x 2 + … = (1+x) m ó??? ? k, {k n }￥ 3?f ?1 G(x) = 1+ kx + k 2 x 2 + k 3 x 3 + … = kx?1 1  3?f ?￥?l 3 d=[ T" ?  ? ? ? ? ? ? ? > +?? = < = ? ? ? ? ? ? ? ? 0 ! )1)...(1( 01 00 n n nrrr n n n r ? r1 L ? n1? ? d=[ T? ? ! α1 L ?5B M x,y |x/y|<1μ ! )1)...(1( ,)( 0 n n n yx n yx n nn +?? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? =+ ∑ ∞ = ? ααα αα αα ?  d=[ T? ? 4  ? α = m HM?=[ T? ?    ,)( 0 ∑ ∞ = ? ? ? ? ? ? ? ? ? =+ n nmnm yx n m yx ,)1( 0 ∑ ∞ = ? ? ? ? ? ? ? ? =+ n nm z n m z ! )1)...(1( ,)( 0 n n n yx n yx n nn +?? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? =+ ∑ ∞ = ? ααα α α αα ? d=[ T? ? ? 5 1|| 1 )1( 1 )1( 1|| 1 )1( )1( 1 )1( 0 0 < ? ? ? ? ? ? ? ? ?+ = ? =? < ? ? ? ? ? ? ? ? ?+ ?= + =+ ∑ ∑ ∞ = ? ∞ = ? zz n nm z z zz n nm z z n n m m n n n m m ∑ ∞ = += ? = +++= ? = 0 2 2 )1( )1( 1 ,2 ...1 1 1 ,1 n n xn x m xx x m ? α = -m H  ? ? ? ? ? ? ? ? ?+ ?= ?++? = +????? = ? ? ? ? ? ? ? ?? = ? ? ? ? ? ? ? ?α n nm n nmmm n nmmm n m n n n 1 )1( ! )1)...(1()1( ! )1...(1)((  =[ T? ? ? 6  L??é 1. b n =αa n , 5 B(x)= αA(x) 2. c n =a n +b n , 5 C(x)=A(x)+B(x) e?é 3 )()()(, 0 xBxAxCbac n i inin ?== ∑ = ? 5 3?f ?￥?é —L?De 7  )()(, 0 xAxxB lna ln b l ln n = ? ? ? ≥ < = ? 5   ,......,,,0,...,0,0 ...,,...,, 1, 0 10 nlll l n bbb aaa ++ 43421 ?  5 b n =a n+l ,5 l l n n n x xaxA xB ∑ ? = ? = 1 0 )( )(  ...,, ...,,,...,, 10 110 bb aaaa ll + 3?f ?￥?é —Mê 8 6 x xA xBab n i in ? == ∑ = 1 )( )(, 0 5 ... 1 1 ... 1 1 1 1 )( ... ... ... 10 10 101 00 + ? ++ ? + ? = +++= += = x xa x xa x axB xaxaxaxb xaxaxb ab n n n n nnn n 7 x xxAA xBaAab n i ni in ? ? === ∑∑ ∞ = ∞ = 1 )()1( )()1(, 0 5 l ? O 3?f ?￥?é — p? 9 Dí?é 8b n =α n a n , 5 B(x)=A(αx) p?Ds?é 9b n =na n , 5 B(x)=xA’(x) 10b n = 1+n a n , 5 ∫ = x dxxA x xB 0 )( 1 )(  3?f ??é —DíD±s 10 1 ó??  {a n }1? a n ￥?wZ? , p 3?f ? G(x) ?¨) ?￥?é?/ ?×1) ? ∑∑ ∑ ∑∑ ∑ ∑ ∞ = ? ? ∞ = ? ? ∞ = ? ∞ = ∞ = ∞ = ∞ = ? ? ? ? ? ? ? ? ? ? ? += ?? ?? += ???? += +?? += ? ? ? ? ? ? ? ? =+ ?= + = ? 1 12 1 1 1 1 1 1 1 2 1 2 1 2 1 0 2 1 2 1 0 0 1 22 2 )1( 1 )!1(2!2 )!22()1( 1 !2 )32...(531)1( 1 ! )1)...(1( 1)1( )1( 1 1 1 1 k k k k k k kk k k k k k k k k k n nn n n x k k k x kk k x k k x k k x k x x x x x 3?f ?D? ￥? 11 è 1 p?  {a n }￥ 3?f ?  1 a n = 7? 3 n  2 a n = n(n+1) 3  1 x xxxG n n n nn 31 7 )3(737)( 00 ? === ∑∑ ∞ = ∞ =    32 2 2 2 0 2 0 1 0 1 12 1 1 0 )1( 2 )' )1( ()( )1( )( )1( 1 )(, 1 )( )(),()( x x x x xG x x dxxG x xH x x xdxxH nxxHxHxnxdxxG x n n x n n n n x ? = ? = ? = ? = ? == === ∫ ∑ ∫ ∑∑ ∫ ∞ = + ∞ = ? ∞ = + L è 12 22.4 3?f ?￥?¨ ? p3?wZ? ?9 ?×"￥ rF? ? ???Z?￥3 ?? ?·s 13 è 1 a n – 5a n-1 + 6a n-2 = 0 G(x) = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + … -5x G(x) = -5a 0 x -5a 1 x 2 -5a 2 x 3 - … 6x 2 G(x) = +6a 0 x 2 +6a 1 x 3 + … (1-5x+6x 2 )G(x) = a 0 + (a 1 -5a 0 )x nn n n nn n nn a xx xx xx x xG 3425 3425 31 4 21 5 651 71 )( 00 2 ???= ?= ? ? ? = +? ? = ∑∑ ∞ = ∞ = p3?wZ? 14 p3?wZ? ? ∑ ∞ = = 1 )( n n n xhxH 3 ! {h n }￥ 3?f ?1 ? ? ? ? ? = ≥= ∑ ? = ? 1 2, 1 1 1 h nhhh n k knkn è 2 xxHxhxH xhhhx xhxhxH n n n n n k knk n l l l k k k ?=?= == ?= ∑∑∑ ∑∑ ∞ = ∞ = ? = ? ∞ = ∞ = )()( )( 1 22 1 1 11 2 15 ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? =? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? +?= ??= ?? = =+ ∑∑ ∑ ∞ = ∞ = ∞ = ? ? 1 22 1 1 22 1 2)1( 1 22 2 )1( ])4( 1 22 2 )1( 1[ 2 1 2 1 41( 2 1 2 1 2 )41(1 )( 0,)(-)( 11 2 2 1 12 1 2 1 2 1 n n n h x n n n x n n n x n n n x x xH xxHxH n n n n nnn n n n n n n  p3?wZ? ? 16 S={n 1 ?a 1 , n 2 ?a 2 ,…, n k ?a k }￥ rF? ?ü ^??Z? x 1 + x 2 + …+ x k = r x i ≤ n i ￥dμ? ?3￥? ? 3?f ? )...1)...(...1)(...1()( 21 k nnn yyyyyyyG +++++++++= ￥ y r ￥" ? ×"￥ r-F? ? 17 è 3 S ={ 3?a, 4?b, 5?c }￥ 10 F? ? 3 3?f ? G(y) = (1+y+y 2 +y 3 )(1+y+y 2 +y 3 +y 4 )(1+y+y 2 +y 3 +y 4 +y 5 ) = (1+2y+3y 2 +4y 3 +4y 4 +3y 5 +2y 6 +y 7 )(1+y+y 2 +y 3 +y 4 +y 5 ) = (1 + … +3y 10 +2y 10 +y 10 + …) N = 6 {a,a,a,b,b,b,b,c,c,c}, {a,a,a,b,b,b,c,c,c,c}, {a,a,a,b,b,c,c,c,c,c}, {a,a,b,b,b,b,c,c,c,c}, {a,a,b,b,b,c,c,c,c,c}, {a,b,b,b,b,c,c,c,c,c} ×"￥ r-F? ? ? 18 ' ? x 1 + x 2 + …+ x k = r x i 11 ? ? ∑∑ ∑ ∞ = ∞ = ∞ = ? ? ? ? ? ? ? ? ?+ =? ?++? = ? +????? = ? =++= 00 0 1 )1( ! )1)...(1)(()1( )( ! )1)...(1)(( )1( 1 ...)1()( r rrr r r r r k k y r rk y r rkkk y r rkkk y yyG N = ? ? ? ? ? ? ? ? ?+ r rk 1 ??Z?3￥? ? 19 {K?Hq x 1 + x 2 + … + x k = r l i ≤ x i ≤ n i 3?f ? )...(... )...)(...()( 1 11 222111 knn nll nllnll yyy yyyyyyyG +++ ++++++= + ++ {" ? p 1 x 1 + p 2 x 2 + … + p k x k = r x i ∈ N 3?f ? ...)1(... ...)1...)(1()( 2 22 2211 +++ ++++++= kk pp pppp yy yyyyyG ??Z?3￥? ? ? 20 è 4 1 X, ? 2? 2 X, ? 1? 4 X, ? 2?ù ? ? 't× Z?μ $ 3 1 x 1 + 2 x 2 + 4 x 3 = r 0 ≤ x 1 ≤ 2, 0 ≤ x 2 ≤ 1, 0 ≤ x 3 ≤ 2 G(y) = (1+y+y 2 )(1+y 2 )(1+y 4 +y 8 ) = 1+y+2y 2 +y 3 +2y 4 +y 5 +2y 6 +y 7 +2y 8 +y 9 +2y 10 +y 11 +y 12 1121212121211 Z? 1211109876543210 ×  ??Z?3￥? ? ? 21 ?T×tb·H ?, ?b? ü H5 G(y)=(y -2 +y -1 +1+y+y 2 )(y -2 +1+y 2 )(y -8 +y -4 +1+y 4 +y 8 ) =5+3y+4y 2 +3y 3 +5y 4 +3y 5 +4y 6 +3y 7 +4y 8 +2y 9 +2y 10 +y 11 +y 12 ? 0 X 1 | 1 1,1 | 2 1,1,2 | 4 4 | 4 1,1,4 | 2,4 ? 1 X 1 | 1 2 | 1,1 4 | 1,2,1 ? 2 X 1,1 | 2 2 | 2 4 | 1,1,2 4 | 2,2 ??Z?3￥? ? ? 22 ?? ?￥·s ?·s￥?lDs ? ?í?·s ?μ?·s 23 4 = 4 4 = 1+3 4 = 2+2 4 = 2+1+1 4 = 1+1+1+1 4 = 4 4 = 1+3 4 = 3+1 4 = 2+2 4 = 2+1+1 4 = 1+2+1 4 = 1+1+2 4 = 1+1+1+1 ×ˉ 4 = 4 4 = 1+3 4 = 4 4 = 1+3 4 = 3+1 ?×ˉ í?μ? ·s￥?lDs ? 24 ' ?  | Ní?·s??? ? a 1 , a 2 , … , a n a 1 x 1 + a 2 x 2 + … + a n x n = N ? ?×ˉ )1)...(1)(1()( 21 n aaa yyyyG +++=  ?×ˉ )1)...(1)(1( 1 ...)1(... ...)1...)(1()( 21 2211 2 22 n nn aaa aa aaaa yyy yy yyyyyG ??? = +++ ++++++= í?·s 25 è 5 ￡ ü ???? ?? V[·BV U? 2é? ? . ??| ???? ? N·s? 2￥ a 2 0 , 2 1 , 2 2 , 2 3 , …, O? ?×ˉ . ∑ ∞ = = ? = ? ? ? ? ? ? = ++++= 0 4 8 2 42 842 1 1 ... 1 1 1 1 1 1 )...1)(1)(1)(1()( n n y y y y y y y y yyyyyG ? ?μ￥ n, a n = 1?ü￡ ü·B￥VE . L è 26 μK?Hq￥í?·s | Ní?·s??? ? a 1 , a 2 , … , a n a 1 x 1 + a 2 x 2 + … + a n x n = N K?Hq l i ≤ x i ≤ t i , M1Z?dμ? ?3￥ù5 27 è 6 ó? r, p N ?×ˉí?·s? k? ? k≤r￥ZE ? 3 N ?×ˉí?·s? k? ? k≤r￥Z? ? N ?×ˉí?·s??? ? k k≤r￥Z? S/ ? Ferrersm |m[ y =x?LJ 180 ¤? a￥ Ferrersm 16 = 6+5+3+2  k ≤ 4 ? ?? ???V 4￥·s . 16 = 4+4+3+2+2+1 ??·s ?￥ 3?f ?1 )1)...(1)(1( 1 )( 2 r yyy yG ??? = μK?Hq￥í?·s 28 (1) | N ?×ˉ1μ?·s? r??s￥Z? ?1 C(N-1,r-1) ! N= a 1 +a 2 +…+a r ^ ?@Hq￥·s5 7 NSSS riaS r i k ii =<<<< == ∑ = ...0 ,...2,1, 21 1 r-1? S i |′1 1,2,…,N-1ZE ?1 C(N-1,r-1). w ? N S ?i×ˉ￥μ?·sZ? ?1 ∑ = ? = ? ? ? ? ? ? ? ? ? ? N r N r N 1 1 2 1 1 (2) ? ?×ˉμ?·s? ?×ˉí?·s + ? ?   μ?·s 29 T< ?ˉ1? 3?f ?D? ￥?1" ×"￥ ? F?￥9 ?ZE Z?dμ? ?3￥ ?# 2T ? ?·s￥9 ? ?# 2T ? ? ?T< 5= E= , 17,18, 19, 20, 22, 26, 28