与系数相关的表达式的极值问题是Newman多项式相关研究中的一个热点。令h<sub>i</sub>(x)是一类系数全为1的Newman多项式,借助不等式和组合的方法,讨论了与h<sub>i</sub><sup style="margin-left:-6px;">3</sup>(x)、h<sub>i</sub><sup style="margin-left:-6px;">4</sup>(x)系数相关表达式的取值,给出了该表达式的极值,从n的不同取值对结论进行了推广。<br>The extreme value problem of the expression related to coefficients is a hot spot in the research of Newman polynomials. Letting h<sub>i</sub>(x) be a kind of Newman polynomials with all coefficients of 1, the value of the coefficient correlation expression of the h<sub>i</sub><sup style="margin-left:-6px;">3</sup>(x) and h<sub>i</sub><sup style="margin-left:-6px;">4</sup>(x) is discussed by method of inequality and combination,and the extremal properties of the expression are given, and the conclusion is generalized from different values of n.
与系数相关的表达式的极值问题是Newman多项式相关研究中的一个热点。令 h i ( x ) 是一类系数全为1的Newman多项式,借助不等式和组合的方法,讨论了与 h i 3 ( x ) 、 h i 4 ( x ) 系数相关表达式的取值,给出了该表达式的极值,从n的不同取值对结论进行了推广。
Newman多项式,系数,极值性质
Changji Li
Tibetan-Chinese Bilingual School, Aba Teachers University, Wenchuan Sichuan
Received: Mar. 10th, 2022; accepted: Apr. 13th, 2022; published: Apr. 20th, 2022
The extreme value problem of the expression related to coefficients is a hot spot in the research of Newman polynomials. Letting h i ( x ) be a kind of Newman polynomials with all coefficients of 1, the value of the coefficient correlation expression of the h i 3 ( x ) and h i 4 ( x ) is discussed by method of inequality and combination,and the extremal properties of the expression are given, and the conclusion is generalized from different values of n.
Keywords:Newman Polynomials, Coefficients, Extremal Properties
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多项式是代数中的重要内容之一,系数受限的多项式及其相关性质是多项式研究中的热点问题之一。Newman多项式是指形如 f i ( x ) = ∑ i = 0 m a i x i , a i ∈ { 0 , 1 } 的多项式,这是一类系数受限的特殊多项式。有关Newman多项式的研究成果较多,如文献 [
定理1 当 h i ( x ) = ∑ i = 0 m a i x i , a i = 1 时,有 lim i → ∞ inf ( γ ( 3 ) ) = 3 4 。
定理2 当 h i ( x ) = ∑ i = 0 m a i x i , a i = 1 时,有 lim i → ∞ inf ( γ ( 4 ) ) = 2 3 。
令 h i ( x ) = ∑ i = 0 m a i x i = 1 + x + x 2 + x 3 + ⋯ + x i − 1 + x i ,
易知, ( # h i ) = i + 1 , ( deg h i ) = i ,又
h i 3 ( x ) = ( 1 + x + ⋯ + x i − 1 + x i ) 3 = 1 + 3 x + 6 x 2 + ⋯ + i ( i + 1 ) 2 x i − 1 + ( i + 1 ) ( i + 2 ) 2 x i + i 2 + 5 i 2 x i + 1 + i 2 + 6 i − 6 2 x i + 2 + ⋯ + i 2 + 3 i + 2 2 x 2 i + i ( i + 1 ) 2 x 2 i + 1 + ( i − 1 ) i 2 x 2 i + 2 + ⋯ + 3 x 3 i − 1 + x 3 i
当 i ≡ 0 ( mod 2 ) 时,多项式 h i 3 ( x ) 中 x 3 2 i 的系数最大,此时有
ζ ( h i 3 ) = ( i 2 + 1 ) + ( i 2 + 2 ) + ⋯ + ( i − 1 ) + i + ( i − 1 ) + ⋯ ( i 2 + 2 ) + ( i 2 + 1 ) = 3 ( i + 1 ) ( i + 2 ) 4
所以 lim i → ∞ inf ( γ ( 3 ) ) = lim i → ∞ i ⋅ 3 ( i + 1 ) ( i + 2 ) 4 ( i + 1 ) 3 = 3 4 。
当 i ≡ 1 ( mod 2 ) 时,多项式 h i 3 ( x ) 中 x 3 i − 1 2 , x 3 i + 1 2 的系数最大,此时有
ζ ( h i 3 ) = i + 3 2 + i + 5 2 + ⋯ + i + ( i + 1 ) + i + ⋯ + i + 5 2 + i + 3 2 + i + 1 2 = 3 ( i + 1 ) 2 4
所以 lim i → ∞ inf ( γ ( 3 ) ) = lim i → ∞ i ⋅ 3 ( i + 1 ) 2 4 ( i + 1 ) 3 = 3 4 。
综上,对任意正整数 i ,均有 lim i → ∞ inf ( γ ( 3 ) ) = 3 4 ,定理1得证。
令 h i ( x ) = ∑ i = 0 m a i x i = 1 + x + x 2 + x 3 + ⋯ + x i − 1 + x i ,
易知, ( # h i ) = i + 1 , ( deg h i ) = i ,又
h i 4 ( x ) = ( 1 + x + ⋯ + x i − 1 + x i ) 4 = ( 1 + 2 x + 3 x 2 + 4 x 3 + ⋯ + i x i − 1 + ( i + 1 ) x i + ( i − 1 ) x i + 1 + ⋯ + 3 x 2 i − 2 + 2 x 2 i − 1 + x 2 i ) 2 = ∑ r = 0 i ( ∑ j + k = r + 2 , j , k > 0 j k ) x r + ( i ⋅ 1 + ( i + 1 ) ⋅ 2 + i ⋅ 3 + ( i − 1 ) ⋅ 4 + ⋯ + 2 ⋅ ( i + 1 ) + 1 ⋅ i ) x i + 1 + ( ( i − 1 ) ⋅ 1 + i ⋅ 2 + ( i + 1 ) ⋅ 3 + i ⋅ 4 + ⋯ + 3 ⋅ ( i + 1 ) + 2 ⋅ i + 1 ⋅ ( i − 1 ) ) x i + 2
+ ( ( i − 2 ) ⋅ 1 + ( i − 1 ) ⋅ 2 + i ⋅ 3 + ( i + 1 ) ⋅ 4 + i ⋅ 5 + ⋯ + 2 ⋅ ( i − 1 ) + 1 ⋅ ( i − 2 ) ) x i + 3 + ⋯ + ( 1 ⋅ 1 + 2 ⋅ 2 + ⋯ + i ⋅ i + ( i + 1 ) ⋅ ( i + 1 ) + i ⋅ i + ⋯ + 2 ⋅ 2 + 1 ⋅ 1 ) x 2 i + ( 1 ⋅ 2 + 2 ⋅ 3 + ⋯ + ( i − 1 ) ⋅ i + i ⋅ ( i + 1 ) + ( i + 1 ) ⋅ i + i ( i − 1 ) + ⋯ + 3 ⋅ 2 + 2 ⋅ 1 ) x 2 i + 1 + ( 1 ⋅ 3 + 2 ⋅ 4 + ⋯ + ( i − 1 ) ⋅ ( i + 1 ) + i ⋅ i + ( i + 1 ) ⋅ ( i − 1 ) + i ⋅ ( i − 2 ) + ⋯ + 4 ⋅ 2 + 3 ⋅ 1 ) x 2 i + 2
+ ⋯ + ( 1 ⋅ ( i + 1 ) + 2 ⋅ i + 3 ⋅ ( i − 1 ) + ( i − 1 ) ⋅ 3 + i ⋅ 2 + ( i + 1 ) ⋅ 1 ) x 3 i + ( ∑ j + k = i + 2 , j , k > 0 j k ) x 3 i + ( ∑ j + k = i + 1 , j , k > 0 j k ) x 3 i + 1 + ( ∑ j + k = i , j , k > 0 j k ) x 3 i + 2 + ⋯ + ( ∑ j + k = 3 , j , k > 0 j k ) x 4 i − 1 + ( ∑ j + k = 2 , j , k > 0 j k ) x 4 i
结合排序不等式,易知多项式 h i 4 ( x ) 展开式中 x 2 i 的系数最大,此时有
ζ ( h i 4 ) = 1 2 + 2 2 + ⋯ + i 2 + ( i + 1 ) 2 + i 2 + ⋯ + 2 2 + 1 2 = 2 ⋅ i ( i + 1 ) ( 2 i + 1 ) 6 + ( i + 1 ) 2 = ( i + 1 ) ( 2 i 2 + 4 i + 3 ) 3
所以 lim i → ∞ inf ( γ ( 4 ) ) = lim i → ∞ i ⋅ ( i + 1 ) ( 2 i 2 + 4 i + 3 ) 3 ( i + 1 ) 4 = 2 3 。
综上,定理2得证。
本文主要探讨了一类Newman多项式 f i 中关于相关系数的表达式 γ ( n ) = deg ( f i ) ζ ( f i n ) ( # f i ) n ( ( # f i ) = o ( deg f i ) )的极值问题,将 n 的值从2的情形推广到了3和4的情形。当条件 ( # f i ) = o ( deg f i ) 取消时,本文猜测 n = 3 和4时 γ ( n ) = deg ( f i ) ζ ( f i n ) ( # f i ) n 的极值情况将会和 n = 2 时发生改变的情形相似,也会发生改变,在此情形下, ( # f i ) deg ( f i ) 的极值相应会有怎样的变化,这些将作为下一步研究的方向。
阿坝师范学院科研项目(20170101, ASB21-04, 202007013)。
李昌吉. 一类Newman多项式的性质Properties of a Class of Newman Polynomials[J]. 理论数学, 2022, 12(04): 561-564. https://doi.org/10.12677/PM.2022.124062