CC BY
20
Section 4. Mathematics
DOI: http://dx.doi.org/10.20534/AJT-16-11.12-19-21
Druzhinin Victor Vladimirovich, Smagin Ivan Romanovich, National research nuclear University «MEPhI» Sarov Institute of physics and technology, Department of mathematics
E-mail: Sarov, vvdr@newmail.ru
A generalization of the sums of bernoulli for the case of fractional powers
Abstract: The formulas for approximate computation of the sums of the first n natural numbers raised to the same fractional power from «<0» to «<3» have been obtained. Different ratios between such sums have been found. The application of these results has been considered.
Keywords: sum of natural numbers, fractional power, Bernoulli numbers.
Let us consider sum _ 5 4 + 5 3 _ 1
x i x x.
2 3 6 For large values of t it is necessary to use (3) next.
The data to calculate the value of(l) with an arbitrary interval m and with arbitrary initial member of a (not «1») using the Bernoulli polynomials do not exist. There are also other simple approaches to their analysis. In previous articles [3; 4] we have proposed a faster way such calculations of generalized sums
B (n;t = 1 + 2t + 3t +... + n',
(1)
with integer non-negative t, which we call the sum of Bernoulli considered. These sums have been studied by many mathematicians. In 1617 Johann Faulhaber published the first book where he calculated these sums with powers up to «11» and later he published the second book with powers up to «17». Pierre Fermat suggested the quad- squares and cube-cubes methods to calculate such sums in the letter to Mersenne. Jacob Bernoulli obtained the polynomials named after him. Later Euler found the generating function for the Bernoulli polynomials, and Appel indicated derivative for them. The calculation of these amounts worked well as Jacobi and other mathematicians [1]. In the modern literature of the sum of (1) is calculated using the Bernoulli polynomials Bt (n) by the formula
b (n;t ) = t+Y [bm (n +1)-bm ]. (2)
These polynomials can be obtained by expansion in a power series generating function of the Euler
kekn » k'
= X ^ (n), (3)
S (n;m;t;a ) = X( + (k - l)m) = a' +(a + m )' +
(4)
ek -1
'!
where Bt — Bernoulli numbers, Bt = Bt (0) . For small values of t" 10 these polynomials can be taken from tables in [2]:
B0 (x) = 1 ;B1 (x) = x -—;B2 (x) = x2 -x + -;B3 (x) = x3 -
3 1 1
—x2 + — x; B. (x) = x4 - 2x3 + x2--;B5 (x) = x5 _
2 2 4W 30 5W
+ (a + 2m)).. + (a + (n-l)m) ,
at a random interval m, an arbitrary initial member a and with integer non-negative powers of t using recursive integral formula
S (n;m;t + l;a ) = a,+1 |m (t + l)Js ;t ;ljdz| +
i - a (t+i)js a it ;i)dz' (5)
This is the new formula and it is not in handbooks and monographs [1; 2; 5; 6]. It is obtained from the first principles exactly by the method of mathematical induction, and its use does not require knowledge of the Bernoulli polynomials. We show this relation for m = 1. As S (n;l;0;l) = B (n;0) = n, then
+at+1n
S (n;l;l;l) = B {n;\) = fk =
1 -i«]=y+2;
(6)
'=0
k=1
Section 4. Mathematics
S(n;1;2;l) = B(n;2) = ^k2 = 2J[ y + Z dZ
+n
1 - 2jly+f JdZ
k=1 \
n 3
n 2
n 6
S (n ;1; 3;1) = B (n;3) = — + — + —.
v ; y ; 4 2 4
(7)
(8)
This is the well-known equalities. Because th e function B(n;i) is continuous and integer t obeys the integral recurrent relations (7). On that base we found approximate expressions for the sums of Bernoulli with fractional powers. The formulas for calculating B (n ;t) with t = 0.5; 1.5; 2.5; 3.5 were given in the article of Indian mathematician Ramanujan [7] in 1915, were given the formula for calculating B (n ;t )with t = 0.5;1.5; 2.5; 3.5. In a recent paper in the Internet [8] in 2012, another Indian mathematician was given the formula for calculating such amounts for any 0 < t < 1. It is slightly different from our formula [9] provides less accuracy.
In this paper, we increased these results and generalize the equality (6-8) for the cas of fractional powers t, i. e. allowed to borrow any 0 < t < 3. The formulas are approximate due to irrationalities. The case 0 < t < 1 is
described in [9], but here we show a little more precise result
S1 (n; 1;0 < t < 1;1) = — + — -
(1 -1 )
(9)
t +1 2 2 (t +1)
Equation (9) corresponds to the three boundary conditions (see (6-8)): S1 (n;1;0;1) = n ;S1 (n;1;1;l) = = B (n;l); S1 (l;1;t ;l) = 1 .The calculations for (9) in the whole range of variation of and n from 1up to 107 in comparison with the exact value of S ives an absolute error of s < |0.07|. Table 1 shows the results.
For the next line segment 1 < t < 3, we have obtained the General calculation formula
n n
S2 (n;1;1< t < 3;1) =-+ —
n ' t +1 2
tn
(2 -1 )(3 -1 )
,(10)
12 12 (t +1) Formula (10) also satisfies five boundary conditions: S2 (n;1;1;l) = B(n;1); Ss ft;S;S;Sh = yft;S);S2 ft;g;3Sh = = yft;3); S2(l;1;i;l) = 1; S2(1;1;0;1) = n. Note that the first two summands in (9-10) coincide with the first two terms in B (n;t). Absolute error (10) in comparison with the exact value ofS (n;1;1< t < 3;1) iseven smaller than in the first interval s < |0.007| |. In table.2 shows the results for t = 1.5;2.5.
Table 1. - Values of the exact S and S1 calculated by (9) quantities of amounts, s = S - S1
t n 2 10 100 1000 104 106
0.50 S 2.4142 22.4683 671.4629 21097.4559 666716.4592 666667166.4588
S, 2.3570 22.4359 671.4767 21097.4881 666716.4975 666667166.4998
s -0.0118 -0.0280 -0.0371 -0.0399 -0.0408 -0.0412
0.10 s -0.0037 -0.0071 -0.0080 -0.0081 -0.0081 -0.0081
0.25 s -0.0081 -0.0167 -0.0198 -0.0203 -0.0204 -0.0205
0.75 s -0.0097 -0.0271 -0.0424 -0.0511 -0.0560 -0.0602
0.90 s -0.0049 -0.0153 -0.0276 -0.0373 -0.0450 -0.0570
Table 2 — Values of the exact S and S2 calculated by (10), s = S - S2
t n 2 10 100 1000 104
1.50 S 3.8284 142.6723 40501.2245 12664925.9563 4000500012.4745
S, 3.8287 142.6728 40501.2250 12664925.9568 4000500012.4750
s -0.0003 -0.0005 -0.0005 -0.0005 -0.0005
2.50 S 6.6569 1068.2176 2907351.1987 9050897005.4402 28576428779761.9135
S, 6.6561 1068.2158 2907351.1964 9050897005.4378 28576428779761.9107
s 0.0007 0.0017 0.0023 0.0024 0.0025
Let us discuss applications of the obtained results. The final numerical amounts with exact or approximate analytical solution, play a huge role in mathematics and physics. They allow us to reduce calculations to a new pattern, to achieve specific goals. This parameter, in our case, t is possible to differentiate and integrate, which yields a new formula considered the amount. For example, there is the following relationship
g[In(1 + ( -1))}(1 + ( -1)) j-^. (11)
The obtained formulas can be used in discrete spaces and to give the magnitude of error between the squares of the histogram covering the curvilinear trapezoid. They can be useful for specific tasks. Equations (9-10) also allow to calculate multiple sums. For example, raising S(n;1;1.5;l) to the square, you can calculate the sums
2 X kA = S(n;l;3;l)-S2(n;l;1.5;l). (12)
k1 (<k2 )=1
Combination of formulas (9-10) gives formulas for the calculation of the sums of the sums T = aS(n;1;t1;l) + PS(n;l;t2;l)with arbitrary a and P. For example, the sum
V2 + 2->/3 + 3-44 + ... + 9y/l0 =
= S (n; 1; 1.5; l) - S (n; 1; 0,5; l) « 120.236.
References:
1. Prasolov V. V. Polynomials. MZIMA, - 2003, - P. 131.
2. Graham Z., Knuth D., Patashnik O. Concrete mathematics, - Moscow, "Mir", - P. 313, - 1998.
3. Druzhinin V. V.//NTVP, - No 5, - P. 18-20, - 2016.
4. Druzhinin V. V., Strahov A. V.//AJTNS, - 2016. - No. 9-10. - P 15-17.
5. Gradshteyn I. S., Ryzhik I. M. Tables of integrals, sums, series and products. GIMPL, - Moscow, - 1962. - P. 15-16.
6. Korn G., Korn T. Handbook of mathematics, science, GIMPL, - M. - 1974, - P. 31, 135.
7. Ramanujan S.//J. Indian Math. SOC. - 1915. - No VII, - P. 173-175.
8. Snehal Shekatkar URL://https://arxiv.org/abs/1204.0877v2.
9. Druzhinin V. V., Smagin I. R.//NTVP,- 2016. - No 6, - P. 18-21.
Here is an example on a specific task. Imagine a pyramid, consisting of ten cubes are stacked on top of each other. Edges ofa cube are reducedbylaw: . The height ofthe pyramid is equal to S (10; 1; 0. 5;1) = 22.468 (see table 1). The volume of a pyramid is equal to S (10;1;1.5;1) = 142.673 (see table 2).
