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Section 2. Mathematics
Drushinin Victor Vladimirovich, National research nuclear University "MEPHI" Sarov physics and technical Institute, Department of mathematics Sarov, E-mail: vvdr@newmail.ru
Single-degree of the sums of integers with stiping sings
Abstact: The formulas for calculating sums of the same integer degrees of an alternating real numbers with arbitrary difference and a initial number are obtained. Formulas of an arbitrary degree and difference of bases are presented. Integral and differential relations between such sums are found.
Keywords: the sum of the summands of an arithmetic progression, the sum of a single-degree alternating real numbers.
(1)
The question about the final sum of the positive single-degree numbers is an old problem and it is partially solved in modern mathematics [1; 2; 3; 4; 5]. In this article we look at a sum of the common form, which is an alternating single-degree function
2n k 1 ' T (2n;m;t; x )=X(-1) + (x + (k - l)m) = x'-(x + m )' +
k=1
(x + 2m) -(x + 3m) + ... + (x + (2n-2)m) -
-(x + (2n-l)m) .
Note that in mathematical handbooks, we have not found any analytical formulas for the calculation of such amounts. In (1) m — the difference between the bases of adjacent numbers, x — the initial number, m and x — any real numbers, 2n — amount of summand, t — non-negative integer degree of the number. Immediately, we note that T (2n ;m;0; x ) = 0;T (2;m;t; x ) = = x' -(x + m) . Earlier we considered [4; 5] positive sum of the form
S (n;m;t; x) =X(X + (k - l)m) = x' + (x + m ) + ...
k=1
... + (x + (n - l)rnj, (2)
for which the new common a3l and differential equations were obtained. The sums (1) can be expressed by S (n;m;t; x) as follows:
T (2n;m;t; x ) = S (n;2m ;t; x ) - S (n ;2m;t; x + m). (3)
Using the results of [4; 5], we can obtain a common formula for calculating sums in the form of an alternating generalized binomial of Newton
T(2n;m;t;x) = fC^ -(2m) •ak (n) -IV-k -(x + m)'-k]. (4)
k=0 ^
In (4) there are Ckt = t!/ k !(t - k)! — the binomial coefficients, and the sum of the form (2) a0 (n) = n, and when k > l ak (n ) = S (n - l;l;k;l) .Formulas for calculating ak (n) with a small k have the form
n2 n / \ n3 n2 n / \
a 3 (n ) =
n n n / N n n n n
4
3 30
6 5 r 4 2
z \ n n 5n n
a5 (n )=---+---;
5W 6 2 12 12
/ x n7 n6 n5 n5 n
a, (n) =----1-----1--;
6W 7 2 2 6 42
8 7 r-7 6 r-7 4 2
/ \ n n 7n 7n n
I7 (n )=---+---+ —
7W 8 2 12 24 12.
(5)
ak (n ) is the sum of successive natural numbers from « 1 » to « n -1 » in the degree of k. Always when k > l ak (l) = 0, ak (2 ) = 1. We have found the following recurrence relations between these functions with k > 1
ak+i(n)=(k+i)iI fa(n)dn I-nIk(n)dnIf, (6)
k +1
dak+1 (n )
dn
= ak (n ).
(7)
Equations (6) and (7) allow you quickly find the functions ak (n) and calculate an alternating arbitrarily large sums of the form (1).
Consider an example. Direct calculation T(6;1.5;2;0.3) = 0.32 -1.82 + 3.32 - 4.82 + 6.32 - 7.82 = = -36,45. Equation (4) gives
T (6;1.5;2;0.3 ) = 1-1-3-(-3.15) + 2 ■ 3 ■ 3-(-1.5) =-36.45.
Between the different values of T (2n;m;t ; x) there is a recurrence relation
^ dT (2n ; m;t ; x )
dx
= T (2n;m;t -1; x) (8)
/ \ u u
a1 (n ) =---;
2 2
/ \ il ft II
a2 (n ) =---+ -;
2W 3 2 6
If you want to calculate T(2n + 1;m;t;x), then this sums is determined by the formula
T (2n + 1;m;t; x ) = T (2n;m;t; x ) + (x + 2nm)'. (9)
Here is some concrete expression to calculate an alternating sums
T (2n;m;l; x) = -nm;T (2n ;m;2; x) = -nm [2 (x + nm)- m ]; T(2n;m;3;x) = -nm(3x2 -3xm + 6nxm-3nm2 + 4n2m2).
1
Single-degree of the sums of integers with stiping sings
References:
1. Sizyi S. V. Lectures on the theory of numbers, FIZMATGIZ, M., 2007.
2. Poznyakov S. N. and Rybin S. V. Discrete mathematics. M., Academy, 2008.
3. Dicson L. E. History of the Theory of Numbers, V. II, CPC, New York, 1971.
4. Druzhinin V. V., Lazarev A. A. Austrian Journal of Technical and Natural Sci-ence, No. 1-2, 20-21, 2016.
5. Druzhinin V. V., Lazarev A. A. NTVP, N. 2, 9-11, 2016.
