与位数码列有关的函数的一个性质

The Behavior of Functions Related to the Digit-Sequence

对于正整数ｎ＝ａｋｎ＾（ｎ）１０＾ｋｎ＋…＋ａ０＾（ｎ），…，ａ０（ｎ）是ｎ位数码，记ｓ（ｎ）＝ａｋｎ＾（ｎ）＋…＋ａ０（ｎ），ｓｍ（ｎ）＝（ａｋｎ（ｎ））ｍ＋…＋（ａ０（ｎ））ｍ，称１＾ｍ２＾ｍ３＾ｍ４＾ｍ５＾ｍ６＾ｍ７＾ｍ８＾ｍ９＾ｍ１＾ｍ０＾ｍ１＾ｍ１＾ｍ１＾ｍ２＾ｍ３＾ｍ１＾ｍ４＾ｍ…是ｍ阶位数码列，定义函数ｆｍ（χ）＝１＾ｍ＋２＾ｍχ＾１＋…

For positive integer n = a 10kn + + a0, we say that a, a0 are the digits of n, and denote s(n) = a + + a0 ), sm(n) = (a) m + + (a))m. We call 1m2m3m4m5m6m7m8m9m 1m0m1m1m1m2m1m3m1m4m the m-order sequence of digits, and define the function fm (x) = 1m + 2mx1 + + 9mx8 + 1mx9 + 0x10 + 1mx11 + 1mx12+ . Two theorems on fm(x) are given.