aa r X i v : . [ m a t h . G M ] J a n AN UNUSUAL IDENTITY FOR ODD-POWERS
PETRO KOLOSOV
Abstract.
In this manuscript we provide a new polynomial pattern. This pattern allowsto find a polynomial expansion of the form x m +1 = x X k =1 m X r =0 A m,r k r ( x − k ) r , where x, m ∈ N and A m,r is real coefficient. Introduction and Main Results
We begin our mathematical journey from investigation of the pattern in terms of finitedifferences ∆ of cubes x . Consider the table of finite differences ∆ of the polynomial x x x ∆( x ) ∆ ( x ) ∆ ( x )0 0 1 6 61 1 7 12 62 8 19 18 63 27 37 24 64 64 61 30 65 125 91 366 216 1277 343 Table 1.
Table of finite differences ∆ of x It is easy to observe that finite differences ∆ of polynomial x may be expressed accordingto the pattern ∆(0 ) = 1 + 6 · ) = 1 + 6 · · ) = 1 + 6 · · · ) = 1 + 6 · · · · x ) = 1 + 6 · · · · · · · + 6 · x Date : January 5, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Polynomials, Polynomial identities.
1N UNUSUAL IDENTITY FOR ODD-POWERS 2
Furthermore, the polynomial x turns into x = (1 + 6 ·
0) + (1 + 6 · ·
1) + (1 + 6 · · ·
2) + · · · + (1 + 6 · · · · · · + 6 · ( x − x = x + ( x − · · x − · · x − · · · · · +( x − ( x − · · ( x − x as x = x − X k =0 k ( x − k ) + 1 = x X k =1 k ( x − k ) + 1since that term k ( x − k ) is symmetrical over x for k = 0 , , .., x . Now we can assume that P k k ( x − k ) + 1 has the implicit form as x = X k A , k ( x − k ) + A , k ( x − k ) , where A , = 6 , A , = 1. The main problem we meet is to generalize above pattern forsome power t >
3. Let be a conjecture
Conjecture 1.1.
For every m ∈ N there are exist A m, , A m, , ..., A m,m such that x m +1 = x X k =1 A m, k ( x − k ) + A m, ( x − k ) + A m, k ( x − k ) + · · · + A m,m k m ( x − k ) m . Consider the case m = 1 x = x X k =1 A , k ( x − k ) + A , k ( x − k ) We evaluate the coefficients A , , A , as follows x = x X k =1 A , kx − A , k + A , x = A , x x X k =1 k − A , x X k =1 k + x X k =1 A , Furthermore, by means of Faulhaber’s formula [1] we collapse the sums x = A , x x + x − A , x + 3 x + x A , xx = A , x + 3 x − A , x + 3 x + x A , xx = A , x − x A , x Multiply both part by 6 and moving 6 x to the left part gives A , x − A , x + 6 A , x − x = 0 x ( A , −
6) + x (6 A , − A , ) = 0 N UNUSUAL IDENTITY FOR ODD-POWERS 3
Since that x ≥ ( A , − A , − A , = 0Which gives A , = 6 and A , = 1. Therefore, x = x X k =1 k ( x − k ) + 1 . Consider the case m = 2. Let be x = x X k =1 A , k ( x − k ) + A , k ( x − k ) + A , As above, we replace the sums by means of Faulhaber’s formula [1] A , x − A , x + 30 A , x
30 + A , x − A , x − x = 0 A , x − A , x + 30 A , x + 5 A , x − A , x − x = 0Substituting x = 1 we get 30 A , −
30 = 0, hence A , = 1. Moving x out of the braces weget x ( A , −
30) + 5 A , x − x ( A , − A , + 5 A , ) = 0It produces the following system of equations ( A , −
30 = 0 A , − A , + 5 A , = 0Which leads to the conclusion A , = 30 , A , = 0 , A , = 1. Finally, we get anotherpolynomial identity x = x X k =1 k ( x − k ) + 1 . Theorem 1.2.
For every x, m ∈ N there are A m, , A m, , . . . , A m,m , such that x m +1 = x X k =1 m X r =0 A m,r k r ( x − k ) r , where A m,r is real coefficient. N UNUSUAL IDENTITY FOR ODD-POWERS 4
Therefore, conjecture 1.1 is true. For m > x = x X k =1 k ( x − k ) + 1 x = x X k =1 k ( x − k ) + 1 x = x X k =1 k ( x − k ) − k ( x − k ) + 1 x = x X k =1 k ( x − k ) − k ( x − k ) + 1 x = x X k =1 k ( x − k ) + 660 k ( x − k ) − k ( x − k ) + 1 x = x X k =1 k ( x − k ) − k ( x − k ) + 491400 k ( x − k ) − k ( x − k ) + 1Moreover, since that k ( x − k ) is symmetric over x , we can conclude that x m +1 = x X k =1 m X r =0 A m,r k r ( x − k ) r = x − X k =0 m X r =0 A m,r k r ( x − k ) r Coefficients A m,r may be calculated recursively [2] as follows A m,r := (2 r + 1) (cid:0) rr (cid:1) , if r = m ;(2 r + 1) (cid:0) rr (cid:1) P md =2 r +1 A m,d (cid:0) d r +1 (cid:1) ( − d − d − r B d − r , if 0 ≤ r < m ;0 , if r < r > m, (1.1)where B t are Bernoulli numbers [3]. It is assumed that B = . Reader may found moreinformation concerning coefficients A m,r in OEIS [4, 5]. To check formulas, use the Wolframmathematica Package[6]. References [1] Levent Kargın, Ayhan Dil, and M¨um¨un Can. Formulas for sums of powers of integers and their reciprocals. arXiv preprint arXiv:2006.01132 , 2020. https://arxiv.org/abs/2006.01132 .[2] Petro Kolosov. On the link between Binomial Theorem and Discrete Convolution of Polynomials. arXivpreprint arXiv:1603.02468 , 2016. https://arxiv.org/abs/1603.02468 .[3] Eric W Weisstein. ”Bernoulli Number.” From MathWorld – A Wolfram Web Resource. http://mathworld.wolfram.com/BernoulliNumber.html .[4] OEIS Foundation Inc. (2020). The On-Line Encyclopedia of Integer Sequences. https://oeis.org/A302971 .[5] OEIS Foundation Inc. (2020). The On-Line Encyclopedia of Integer Sequences. https://oeis.org/A304042 .[6] Petro Kolosov. Supplementary Mathematica Programs. 2020. https://github.com/kolosovpetro/Mathematica-scripts . Email address : [email protected] URL ::