Sums of Powers

You have reached the "Sums of Powers" Pages of Rajesh Ram (maths@nivedh.com).   Click here for a Graphics Version of this page

Other Links : PELL NUMBERS FORMULAE ,  SUMS OF POWERS,  FIBONACCI FORMULAE, RAJESH's MATHS PAGE, TRIANGLE NUMBERS THAT ARE PERFECT SQUARES

Also see Sums of Cubes , Sums of Powers - Ramanujan and his Number 1729

This page contains some of the identities that I have found. I  am only adding the ones which I think are very beautiful results. 

IF  T(m) = (a^2+2ab+b^2)^m + (a^2)^m + (b^2)^m - (a^2+ab+b^2)^m - (a^2+ab)^m - (ab+b^2)^m - (-ab)^m
THEN
1) T(3)T(6 ) / ( T(4)T(5) ) = 3 / 4  
2) T(3)T(5 ) / ( T(4)T(4) ) = 5 / 6 
SOP1
IF  X(m) = (a^2+ab+b^2)^m - (a^2+ab)^m - (ab+b^2)^m - (-ab)^m
Y(m) = (a^2+ab+b^2)^m + (a^2+ab)^m + (ab+b^2)^m + (-ab)^m
THEN
1) X(3)X(5)/X(4)^2 = 15/16  
2) X(11)X(6)/X(10)X(7) = 66/70   
3) X(5) / (X(3)Y(2)) = 5/6
4) X(11) / (X(7)Y(4)) = 11/14
5) X(8) / (X(5)Y(3)) = 4/5
6) X(10) / (X(6)Y(4)) = 5/6		 SOP2
IF A + B + C =0,  X = B - C,  Y = C - A, Z = A - B, A1 = -AB, B1 = -BC, C1 = -CA, X1 = -XY, Y1 = -YZ, Z1 = -ZX
THEN
1)  (A^m + B^m + C^m)*(X^m + Y^m + Z^m) / (A*X)^m + (B*Y)^m + (C*Z)^m)
      = m*3^( (2m-3*(1 - (-1)^m)) / 4)*2^( (2-m)*(1 + (-1)^m) / 4),  for m = 2,3,4,5,7
2)  (A1^m + B1^m + C1^m -(A1+B1+C1)^m )*(X1^m + Y1^m + Z1^m - (X1+Y1+Z1)^m) / {(-A1*X1-B1*Y1- C1*Z1)^m - (-A1*X1)^m - (-B1*Y1)^m - (-C1*Z1)^m) }  
=  m * 3^(m-3),  for m = 3,4,5		 SOP4
IF  a + b +c = 0 and T(n) = ( a^n + b^n + c^n ) / n  THEN
1)  T(13) / T(7)  -  T(11) / T(5) = T(3)^2
2)  T(19) / T(7)  -  T(17) / T(5) = T(5)T(7) / T(4)
3)  T(2)T(17) + T(5)T(7)T(11)/T(4)  = T(19)
4)  T(2)T(11) + T(3)^2  T(7) = T(13)
5)  T(3)T(7) = T(5)^2 = (T(2)T(3))^2		 SOP5
IF A = a^2 + 2ab,  B = -b^2 - 2ab,  C = b^2 - a^2
    X = x^2 + 2xy,  Y = -y^2 - 2xy,   Z = y^2 - x^2
    U1 = Bx^2 - 2Axy + Cy^2,  V1 = Cx^2 - 2Bxy + Ay^2, W1 = Ax^2 - 2Cxy + By^2
    U2 = Cx^2 - 2Axy + By^2,  V2 = Ax^2 - 2Bxy + Cy^2, W1 = Bx^2 - 2Cxy + Ay^2
THEN
U1^6 + V1^6 + W1^6 - U2^6 - V2^6 - W2^6
= 3ABC(A-B)(B-C)(C-A)XYZ(X-Y)(Y-Z)(Z-X)		 SOP6