EQUATIONS WHOSE ROOTS ARE T I E nth POWERS OF T I E ROOTS OF A GIVEN CUBIC EQUATION

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1 EQUATIONS WHOSE ROOTS ARE T I E nth POWERS OF T I E ROOTS OF A GIVEN CUBIC EQUATION N. A, DRAiMand MARJORIE BICKNELL Ventura, Calif, and Wilcox High School, Santa Clara, Calif. Given the cubic equation x - c ^ 2 + c 2 x - c = 0, with roots r l 5 r 2, r j the problem of this paper is to write the equation._ o, n n IK o, n n n n n IK n n n (1) x 6 - (r i + r 2 + r )x* + ( r ^ + r ^ g + r 2 r )x - r ^ ^ = x C ( l? n ) x 2 + C ( 2 s n ) X ~ C ( 5 n ) = whose roots are r^, r 2, r, and whose coefficients a r e expressed in t e r m s of the coefficients c l 5 c 2, c s of the given equation 0 This paper extends to the cubic equation a study initiated by the solution of a s i m i l a r problem for the quadratic by the s a m e authors [ 1 1. Just as a special quadratic equation leads to a relationship between the n Fibonacci number and a sum of binomial coefficients, so does a special cubic equation th r e l a t e the n m e m b e r of a Tribonacci sequence to a sum of products of b i - nomial coefficients. Some Lucas identities also follow. The summation for initial values of powers of roots by elementary theory yields the first five entries in the table below. r e v e a l s an iterative pattern; namely, that if Examination of this sequence c ( l j n ) = r i + r 2 + r? n > 0 s then C ( l, n ) = C i C ( l 9 n - l ) " C 2 C (l n-2) + C C (l n-) J which is easily proved, since each root, and hence sums of the r o o t s, satisfies the original equation,, 267

2 268 EQUATIONS WHOSE ROOTS ARE THE nth POWERS [Oct. Sums through eighth powers appear in the table below. The right-hand column gives the sum of the absolute values of the coefficients for each value of n. n a > c d n ) n n n = r i + r 2 + r Coefficient Sums for n c l C? cl *\ cf c? c 7 i \ - 2c 2-0 ^ 2-4 c i c ^ c 2 5 l c 2 + c ^ ^ - 5C2C - 6cfc 2 + 9c 2 lc2-2 c + 6c^c - 120^20-7c5c 2 + Ucfcjj - 70^1 + 7cjc - 21c5c 2 c + 7c \c z + lc i - 8cfc c^c - 16cfc + 2c + c c l c ~ ozc-jc2co + 24c j C ^ c ^ c -" 8c 2 c r c It is possible to perceive the generalized number pattern for sums of n powers of the roots by extending the table above and breaking down the sum in t e r m s of coefficients of powers of c. If ip is the coefficient of c / n! in the sum n n n r l + r 2 + r then 0 = c^ - nc^" 2 c 2 + n(n - ) c f " 4 c / 2! - n(n - 4)(n - 5 ) c ^ " 6 c /! +... ip 1 = n c j " - n(n - 4 ) ^ " c 2 + n(n - 5)(n - 6)c 1 ~ cf/21 - n(n - 6)(n - 7)(n - 8)cf" 9 cf/! + -- ip 2 = n(n - 5)cf* 6 - n(n - 6)(n - 7)c?' 8 c 2 "+ n(n - 7)(n - 8)(n - 9)c?~ i 0 cjj/2! - n(n - 8)(n - 9)(n - 10)(n - l l ) c f ~ 1 2 c ^ /! + -..

3 1967] OF THE ROOTS OF A GIVEN CUBIC EQUATION 269 leading to the three equivalent expressions below 0 For CjCgCg ^ 0, (2) r? + r f + r n [n/j = c % k / k!, k=o [(n-k)/2] \~^, ^ m n-2m-k m. _..,.... _ rt.., ^ k = 2 ^ t" 1 ) c i c 2 n(n - m - 2k - l )! / ( n - 2m - k)!m;, m=o [n/j [ (n-k)/2 () r? + rf + r n = J ] ^ (-1) n(n - m - 2k - 1)1 n-2m-k m k (n - 2m - k) Jm! k! C l 2 [n/][(n-k)/2] n V V* (-1) n / n - m - 2 k - l \ / n - m - k \ (4) T 1 + r 2 + r = 2Lt 2^ n - m - k \ k ) \ m ) Ky n-2m-k m k X c i c 2 c, where [n] is the greatest integer ^ n and I J is a binomial coefficient. Notice that the coefficients of c$ are the s a m e as the coefficients which a r o s e in studying the roots of the quadratic in [1J. The reiterative relationship of the t e r m s c n. suggests a proof by mathematical induction for the three formulae listed, and such a proof has been written by the authors. For the sake of brevity, the proof is omitted. A derivation of the above formulas could also be done using Waring's formula and Newton's identities (see [ 2 ] ). Thus far, we have found a way to e x p r e s s the coefficient for x 2 in our general problem. The coefficient for x, C (2,n) n n n n n n ' = r i I > 2 + r i I > + r 2 r has a similar computation. In the auxiliary cubic equation, n n w n n w n n. _ (x - r i r 2 )(x - r t T )(x - r 2 r ) = 0 notice that c.. is the coefficient of x 2. When n = 1, the above cubic (z, n; b e c o m e s, upon multiplication,

4 270 EQUATIONS WHOSE ROOTS ARE THE n t h POWERS [Oct. x - ( r ^ + r 1 r + r 2 r )x 2 + (rl r 2 r + r 1 r^r + r 1 r 2 r ) x - r^r^r = X - C 2 X 2 + CjCgX - C = 0. Comparing this equation with the equations of our original problem, we can apply the three formulae already derived for c M v to find c / 0 s if we r e - ^ J J (l,n) (2,n) place c 1 by c 2? c 2 by c^cg, and c by c. For example, from Equation (4), if c ^ C g fi 0, our formula for c, 2. becomes [n/] [(n-k)/2] <K v n v n + v n r n + r n r n - V V (-1) n(n - m - 2k - 1)! (5) r t r 2 + r i r + r 2 r - ^ Z, ^ n ~ ^ n ~ ^ k )! m! k! m n-2m-k 2k+m In p r a c t i c e, when raising the roots of a given equation, It is simpler to utilize the method of iterating functions than to substitute into the formulae, especially as n becomes l a r g e r. An example worked by each method follows. Given the equation x - 6x 2 + l l x - 6 = 0, write the cubic whose r o o t s a r e the fourth powers of the roots of the given equation, without solving for the roots. (A) By substitution; F r o m the table given e a r l i e r or from Equations () and (5), the desired cubic Is x - (cf - 4CjC 2 + 2c 2 + 4c 1 c )x 2 + (e ^ 0 ^ + 2G\C\ + 4c 2 c )x - c = 0. Substituting c i = 6, c 2 = 11, c = 6 yields x - 98x x = 0

5 1967] OF THE ROOTS OF A GIVEN CUBIC EQUATION 271 with roots l 4, 2 4, and 4. As a check, the roots of the given equation are 1, 2, and. (BJ By iteration: To get c.., we wish to write the sequence C ( l, 0 ) ' c ( l, l ) ' C ( l, 2 ) ' C ( l, ) ' C ( l, 4 ) ' Now C ( l, 0 ) = ' C ( l, l ) = C i = 6 > C ( l, 2 ) = 2 1 " 2C 2 = 6-22 = 14 By the iteration relationship, (1,) = C i C ( l 5 2 ) - C 2 C ( l, l ) + ^ ( l 5 0 ) C ( l 4 ) = 6 ( 6 ) ~ n ( 1 4 ) + 6 ( 6 ) Similarly, c r2 / o m =, c / 0 1N = 11, c / 0 Q^ = c 2-20^ = 49. '(2,0) " "' "(2,1) " " ' "(2,2) Since C (n,2) = C 2 C ( 2, n - l ) ~ C * C c (2 5 n-2) + C c (2,n-) substitution yields c ) = 251 and c ^ = 19, yielding the s a m e cubic as in (A). Next let us turn our attention to several special cubic equations. F i r s t, for the cubic x - x 2 - x - 1-0, c ( l j 0 ) -, c ( l j l ) - 1, c ( l ^ 2 )

6 272 EQUATIONS WHOSE ROOTS ARE THE n t h POWERS. [oct. and oux repeating multipliers in the iterative relationship a r e 1, 1, 1. Then, c ( l, ) = 1 ( ) + 1 ( 1 ) + 1 ( ) = 7 ' (1 4) = X = U ' ' " ' C ( l, n ) = C ( l, n - 1 ) + C ( l, n - 2 ) + C ( l, n - ) - For this particular equation we have a species of Tribonacci n u m b e r s, any t e r m after the third being the sum of the three preceding t e r m s, with the entry t e r m s, 1,. By Equation (4), the n t e r m T in this Tribonacci sequence is [n/] [(n-k)/2] ( - l ) m n / n - m - 2 k - l \ / n - m - k \ n i J JL-J n - m - k \ k / \ m / Notice that the sums of the coefficients in the table given for. c,.,. are these ( l, n ) s a m e numbers. It is interesting to recall that the special equation x 2 - x - 1 = 0 led to a formula relating the n member of the Fibonacci sequence to a sum of binomial coefficients in the e a r l i e r study of the quadratic equation [ 1 ], Considering the special equation x - x 2 + x - l = 0 with roots 1, ±V^T, we can write the following from Equation (4): [ n / ] [(n-k)/2] - I l if n is odd Zu Z. n - m - k ^ k H m I. f Q = x ' ' ' number Of more interest, however, are the following identities for the n L, defined by J n' Lucas

7 1967] OF THE ROOTS OF A GIVEN CUBIC EQUATION 27 Li = 1, L 2 =, Ln = ~L n -i + L n _ 2. We substitute in Equation (), using ; r i = a = (1 + N/5)/2, r 2 = p = (1 - N/5)/2, and letting r vary. If r = 1, Equation () cannot be used directly because c 2 = a/ + ( + a = 0, and 0 is not defined, But, by following the derivation for Equation (), it is seen that, if c 2 = 0, c^g 0 9 [ n / ] /ofx n n n \-^ n(n - 2k - 1)1 n-k k (!) r, + r 2 + r = ( n - k)!k! C * * k=o Since c 1 = a + fi + 1 = 2, c = a/ = - 1, and n J I L = a + B p n substitution gives K ]. k n-k (-1) 2 n(n - 2k - 1)] L n + X LJ (n - k)i k! k=o In general, if r = P, P ^ 1, P - 1, P ^ 0,

8 274 EQUATIONS WHOSE ROOTS ARE THE n POWERS Oct OF THE ROOTS OF A GIVEN CUBIC EQUATION Equation () gives [n/] [(n-k)/ 2 ] m + k. ^ m - s k, ^ m k T + n n - V V (-1) n(n - m - 2k - 1)1 (p + 1) (p-1) p n + p 2-J 2-J (n - 2m - k>! m* k! Similarly, Equation ( 1 ) gives the following two identities using r t = a, r 2 = p, r = - I/N/5", and the known relationship for Fibonacci n u m b e r s, F n = {a n -/ n )/M5. Below, n is taken to be 2s + 1 and 2s respectively. f ( 2 S + l ) / J, n, w ^k,2s-k + i s+i _ v-* (2s + 1) (2s - 2k)! (-1) 4 F 9 2S+1 ~ X/ ~ /_j / r t _ s-k+i Q (2s - k + l )! k! 5 ^ 1/58+1 = E [ 2 s / J k?s-k T + i /, s _ V 2s(2s - 2k - 1) I (-if 4 2 S k L 2 g + 1/5 - > : ^ (2s - k)j k! 5 S K REFERENCES 1. N. A. Draim and M, Bicknell, "Sums of the n Powers of the Roots of a Given Quadratic Equation, M The Fibonacci Quarterly, 4:2, April, 1966 y pp W. S. Burnside and A. W. Panton, An Introduction to the Theory of Binary Algebraic F o r m s, Dover, New York, 1960, Chapter 15. * *