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Section 4. Mathematics
Section 4. Mathematics
Drushinin Victor Vladimirovich, Lazarev Alexey Alexandrovich, National research nuclear University "MEPHI" Sarov physical-technical Institute Department of mathematics
E-mail: vvdr@newmail.ru
One power sums of natural integers at arbitrary intervals
Abctract: The formulas for calculating of sums of the arbitrary powers of natural numbers with different intervals and the initial number are received. Results for powers from «1» to «4» and intervals from «3» to «4» are given. Applications of the results are reviewed.
Keywords: partial sum of one power series, the sum of the natural numbers in the same powers.
The formulas for calculating of sums of the arbitrary powers of natural numbers with different intervals and the initial number are received. Results for powers from «1» to «4» and intervals from «3» to «4» are given. Applications of the results reviewed.
The question about sums of natural numbers is old problem, and it was discussed in the works of Euler, Bernoulli, and other mathematicians [1; 2; 3]. We introduce the notation of such sums in the form of
^ (n;t;l)=Z0 + (k -1)4. (1)
k=1
In (1) m — interval sum, i. e. the difference between two consecutive natural numbers, n — the number of summands, t — the power of natural number, l — the first summands. The sum of n summands of an arithmetic progression {a0 + ak} in our notation has form Sa (n ;1;a0) .The references [4; 5; 6] the formula for the sum with interval «1» up to t = 7. For example,
, x n(n +1) . x S, (n;1;l)(2n +1) S (n;1;l)= V ,; S, (n;2;l)= " ^-
S2 (n;1;l) = X(2k -1) = n2;
k=l
s n(4n2 - 2) S2 (n;2;l)= v 3 2
S2 (n;3;l) = 2n4 - n2.
As for sums at intervals m = 3;4;5;... , we did not find the relevant formulas, and in this paper we present the expressions and give general formulas without the use of the Bernoulli numbers.
First we get in our method of the general formula relating the sum of consecutive integers in different powers S1 (n;t ;l). In the literature there is this bundle, but it is presented in the form of disordered sum of summands and Bernoulli numbers. We write
S, (n ;2;1)= 22 =]T (1 + (k -1))2 = n + ifk + fk 2=1
+YCIYks = 1 + JP& (n -1;s;1).
(3)
2 1V 3
S (n;3;1) = ( (n ;1;1))2;
, S, (n;2;l)(3n2 + 3n -1) • S, (n;4;1) = -^-'-
Generalizing (1), we can write a general recurrence relation
(2)
S> ;1) =S (n + +1 -1 - ^c:+lSl (s;1)
(4)
In addition in [5] the general formula for arbitrary
n
powers t has form Sj (n;t;1) = ^k' with using of the
k=1
binomial coefficients Cq = q!/ s!(q - s) and Bernoulli numbers Bk. Thus, the question about sums at intervals of «1» was decided. Similarly, the question was resolved for sums with interval «2». For example, in the references there are only three such formulas:
which allows for the previous sums at intervals «1» to get the following sums. We found no such correlation in the literature.
Then turn to the intervals «2», «3», «4», and so on. First look at an example.
S3 (4;3;1) = 1 + 43 + 73 +103 = 1 + (l + 3 • 3 + 3 • 32 + 33) + + (1 + 3 • 6 + 3 • 62 + 63 ) + (l + 3 • 9 + 3 • 92 + 9^ = (5) = 1 + 3 + 32S (3;1;1) + 33S, (3;2;1) + 33S, (3;3;^.
!=0
s=0
One power sums of natural integers at arbitrary intervals
Equation (5) shows that the sum with interval «3» can be expressed by the sums with interval «1», which we can find by (4). Generalizing (5) we have
Sm (n;t;1) = 1 + i[mkCktSl (n -1;6;l). (6)
k=0
Using (4, 6), we received a new relatively si mple analytical formulas for calculating of the sums with m = 2;3;4;5, t = 1;2;3;4 and with arbitrary n. (See table № 1).
Always Sm (n;0;l) = n; Sm (1;t;l) = V. Next, the question is appear about calculating of such sums at an arbitrary initial summand l. In the reference books such question is not entered, but we decided it. We received the such relationship
+
(n ;t ;l ) = YfktSm (n; k ;l -1). (8)
k=0
For example, S3 (n;2;2) = S3 (n;l;l) + 2S3 (n;1;l) + S3 (n;2;1) = (6n3 + 3n2 - n)/2. Indeed, S3 (5;2;2) = = 22 + 52 + 82 +112 +142 = 410.
We do not present other analytical expressions sums, as they are bulky and the reader can get them himself by the formulas (4,6). Application amounts formulas is extensively: save computer time, the solution ofnonlinear Diophantine equations, analysis of the type in Fermat's last theorem and its generalizations, generating prime numbers.
Table № 1. Formulas sums Sm (n;t;l), where m — interval, n -number of summands, t — the powers of summands, the first number is «1»
s (6n3 - 3n2 - n) ,(n;2;l) = ^---1;
. (n;3;l) = n(3n - l)(2 -3n -4);
( ) n(l62n4 - 135n3 -90n2 + 60n +13)
An ;4;1) = io ;
, , , , , , (16«3 - 12n2 - n) i (n;1;l) = n(2n -1); S4 (n;2;l) = ^---'-;
S4 (n;3;l) = n(2n -l)(8n2 -4n - 3);
, n(768n4 - 960n3 - 160n2 + 360n + 7) 54 (n ;4;1) = -i---1;
■,(n= nö Ss(;2;1) = n-45n +
;(n;3;i) = n(-(2 -^-8);
( ) n(750n4 - 1125n3 + 50n2 + 360n - 29)
- (ftj4j1 ) ~ ,
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. Korn. G., Korn. T. Handbook of mathematics, Science, GHML, M., 1974. P. 31, 135.
5. Gradstein I. S. , Ryzhik I. M. Tables of integrals, sums, series, and products. GIFML, Moscow, 1962. P. 15-16.
6. A brief Physico-Technical reference. under.ed. by K. P. Yakovlev, GIFML, M., 1969. P. 9.
