-P" -p. and V n = a n + ft\ (1.2)

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R. S. Melham School of Mathematical Sciences, University of Technology, Sydney, PO Box 123, Broadway, NSW 27 Australia {Submitted November 1997-Final Revision April 1998) 1. INTRODUCTION Define the sequences {UJ and {VJ for all integers n by K = PK-I-IK_ 2, V 0 = 2, v x = P, ( 1 1 ) where/? and q are real numbers with q(p 2-4q) ^ 0. These sequences were studied originally by Lucas [6], and have subsequently been the subject of much attention. The Binet forms for H and V are where u a" a- -P" -p and V n = a n + ft\ (1.2) p+ > 2 - -Aq a = "-" 2 - and p^ezjzer ( 1. 3 ) are the roots, assumed distinct, of x 2 - px + q = 0. We assume further that a I j3 is not an n^ root of unity for any n. A well-known relationship between U n and V n is K = U n+i -qu n _ u (1.4) which we use subsequently. Recently, Melham [7] considered functions of a 3-by-3 matrix and obtained infinite sums involving squares of terms from the sequences (1.1). Here, using a similarly defined 4-by-4 matrix, we obtain new infinite sums involving cubes, and other terms of degree three, from the sequences (1.1). For example, closed expressions for y^- and y ^ ^ - arise as special cases of results in Section 3 [see (3.4) and (3.5)]. Since the above mentioned paper of Melham contains a comprehensive list of references, we have chosen not to repeat them here. Unfortunately, one of the matrices which we need to record does not fit comfortably on a standard page. We overcome this difficulty by simply listing elements in a table. Following convention, the (i, j) element is the element in the /* row and j * column. 1999] 269

2. THE MATRIX A k x By lengthy but straightforward induction on n, it can be shown that the 4-by-4 matrix A = 'o 0 0 -q 3 ^ 0 0 q 2 3pq 2 0 -q -2pq -3p 2 q 1 p p 2 p 3, (2.1) is such that, for nonnegative integers n, A n is as follows: -fu^u* 3TT3 \ -1 -fulpn -q'ul WUl-P n qklutu^+u^uu) q\ul + 2U ^U U n+l ),2TT2 3qW n U +1-3qU n _ x Ul -q(u 3 +2U n _ l U n U n+l ) -q(2u 2 U +l+ U n _ l U 2 +1) -W n ut +l Ul u 2 u n+1 u n ul +l u n+l J To complete the proof by induction, we make repeated use of the recurrence for {U }. For example, performing the inductive step for the (2, 2) position, we have -q\ul + 2t/ _ 1 t/ / + i) + 3pq 2 U 2 U +1 = q 2 U n [U n (-qu n ) + 2U n+l (-qu _ l ) + 3pU n U n+1 ] = q 2 U n [U (U n+2 - pu n+l ) + 2U +l (U n+l ~pu n ) + 3pU n U n+l ] = q 2 U [2U2 +l + U n U +2 ] = q 2 [2U 2 +lu n + U n+2 U 2 ], which is the required expression. When p = 1 and q - - 1, the matrix A becomes fo o o O 0 0 1 3 0 1 2 3 1 1 1 1 ^ which is a 4-by-4 Fibonacci matrix. Other 4-by-4 Fibonacci matrices have been studied, for example, in [3] and [4]. The characteristic equation of A is X* - p (p2 _ 2 q)x 3 + q(p 2-2q)(p 2 -q)x 2 -pq 3 (p 2-2q)X + q 6 = 0. Since p = a + (3 and q-a^, it is readily verified that a 3, a 2 /?, a/3 2, and J3 3 are the eigenvalues A j (J = 1, 2, 3,4) of A. These eigenvalues are nonzero and distinct because of our assumptions in Section 1. Associated with,4, we define the matrix A ktx by A, x - X A, (2.2) 270 [AUG.

where x is an arbitrary real number and k is a nonnegative integer. From the definition of an eigenvalue, it follows immediately that xa 3k, xa 2k (3 k, xa k j3 2k, and xj3 3k are the eigenvalues of A kx. Again, they are nonzero and distinct. 3. THE MAIN RESULTS Let f{z)-t^=qa n z n be a power series whose domain of convergence includes the eigenvalues of A kt x. Then we have, from (2.2), f(aj = ZX4" * = % n * n A k \ (3.1) w=0 The final sum in (3.1) can be expressed as a 4-by-4 matrix whose entries we record in the following table. if, f) (ID 1 (1,2) 1 (1,3) (1,4) (2,1) (2,2) (2,3) (2,4) (3,1) (3,2) (3,3) (3,4) (4,1) (4,2) (4,3) (4,4) (/, j) element of f(a k x ) w=0 n=q 3q 2 ±a n x"u 2 kn. x U kn #1=0 q 2 ±a n x"(2ulu kn _, + U kn+ Vl_ x )\ I ^Zv n (^+2t/, _,C/ t C/, +1 ) 3q 2 ta n x"ulu kn+] -3q±a n x"u k^ul -qic, n x"(ul+2u kn _ x U kn U k^) M=0 M=0 w=0 1 1 Ya x"u kn Ui +] w=0 1 1 1999] 271

On the other hand, from the theory of functions of matrices ([2] and [5]), it is known that where / is the 4-by-4 identity matrix, and where c Q, c x, c 2, and c$ can be obtained by solving the system \c Q + c x xa 3k + c 2 x 2 a 6k + c 3 x 3 a 9k = f(xa\) = f(xa 3k ), c 0 + c l xa lk fi k + c 2 x 2 a 4k p 2k + c 3 x 3 a 6k p 3k = f{xx\) = f(xa 2k j3 k ), c Q + c x xa k p 2k + c 2 x 2 a 2k p 4k + c 3 x 3 a 3k j3 6k = f(xx k 3) = f{xa k p 2k \ C Q + cytjff 3 * + C 2 * 2^6* + C,* 3^* = /(x^4) = /(*/? 3 *). With the use of Cramer's rule, and making use of the Binet form for U n, we obtain, after much tedious algebra, -f{xa 3k )(?> k f(xa 2k p k )a k p 3k C U k U 2k U 3k {a-p) 3+ U 2 U 2k (a-p) 3 f{xa k p 2k )a 3k p f(xp 3k 3k\~6k )a ( + U 2 U 2lc (a-p) 3 ' U k U 2k U 3k (a-p) f(xa 3k )P 3k (a 2k +p 2k +a k p k ) f(xa 2k p k )(a 3k +p 3k +a 2k p k ) xa 2k U k U 2k U 3k (a-py xa 2k U 2 U 2k (a - /J) 3 + f(xak p 2k )(a 3k +P 3k +a k P 2k ) f(xp 3k )a 3k (a 2k +p 2k +a k p k ) xp 2k u 2 u 2k (a-py xp 2k u k u 2k u 3k (a-py -f(xa ik )p k {a 2k +p 2k +a k p k ) f(xa 2k p k )(a 3k +p 3k + a k p 2k ) x 2 a 3k U k U 2k U 3k {a-pf x 2 a 3k p 2k U 2 U 2k (a- pf f(xa k p 2k )(a 3k +p 3k +a 2k p k ) f{xp 3k )a k (a 2k +p 2k + a k p k ) x 2 a 2k p 3k + UlU 2k {a-pf x 2 p 3k U k U 2k U 3k (a-p) 3 : 3 ' <3 f(xa 3k ) f(xa 2k p k ) x 3 a 3k U k U 2k U 3k (a-p) 3 x 3 a 3k p 2k U 2 U 2k (a-p) 3 f(xa k p 2k ) f(xp 3k ) + x 3 a 2k p 3k U 2 ku 2k (a-pf x 3 p 3k U k U 2k U 3k (a-p) 3 ' The symmetry in these expressions emerges if we compare the coefficients of f(xa 3k ) and f{xp 3k ) and the coefficients of f(xa 2k p k ) and f(xa k p 2k ). Now, if we consider (3.1) and (3.2) and the expressions for the entries of A", and equate entries in the (4, 1) position, we obtain Y,a n x n U 3 kn = c x xu 3 +c 2 x 2 U 3 2k +c 3 x 3 a (3.3) H=0 Finally, with the values of c h c 2, and c 3 obtained above, we obtain, with much needed help from the software package "Mathematica": 272 [AUG.

f a xnu 3 _ f(xa 3k )-3f(xa 2k P k ) + 3f(xa k p 2k )-f(xp 3k ) In precisely the same manner, we equate appropriate entries in (3.1) and (3.2) to obtain _ qf(xa 3k ) - (2a+p)f(xa 2k p k ) + (a + 2p)f(xa k p 2k ) - Pf(xp 3k ) (a-fif 2_, a n X "'~'krfjkn+\ a 2 f(xa 3k )- (a 2 + 2ap)f(xa 2k p k ) + (J3 2 + 2ap)f(xa k p 2k ) - P 1 f{x0 ik ) (a-py 'kn+\ H<*n*?Ul P 'kn+\ a 3 f(xa 3k ) - 3a 2 p/(xa 2k p k ) + 3ap l f(xa k p 2k ) - p 3 f(xp 3k ) (a-p) 3 (3.5) (3.6) (3.7) 2_j a n X *~>kn-y~>kn Pf(xa 3k ) - (a + 2P)f(xa 2k p k ) + (2a+P)f(xa k p 2k )-af(xp 3k ) ap(a-pf f J c* x»(u 3 n + 2U kn _ l U kn U kn+] ) ^ 3aP(f(xa 3k ) - f(xp 3k )) - (a + 2/?)(2a + P)(f(xa 2k p k ) - f(xa k 0 2k )) ap(a-pf (3.8) (3.9) 2_, a n X (2U kn U kn+l + U/cn-lUkn+l) 3a 2 pf(xa 3k ) - a(a + 2P) 2 f(xa 2k p k )+p(2a+p) 2 f(xa k p 2k ) - 3ap 2 f(xp 3k ) ap(a-pf (3.10) " =0 (3.11) = P 2 f(xa 3k ) - p(2a + p)f(xa 2k p k ) + a(a + 2p)f(xa k p 2k ) - a 2 f(xp 3k ) a 2 p 2 (a-pf f,a n x"(2u 2 knu kn _ l + U kn+l Ul-d " =0 (3.12) 3ap 2 f(xa 3k ) - p(2a + pff(xa 2k p k ) + a(a + 2pf f (xa k p 2k ) - 3a 2 pf( X p 3k ) a 2 p 2 (a-pf 1999] 273

P 3 f{xa 3k ) - 3aJ3 2 f(xa 2k p k ) + 3a 2 Pf(xa k p 2k ) - a 3 f(xj3 3k ) a 3 p 3 {a-p) 3 From (3.4) and (3.9), we obtain (3.13) 2^ an xn u kn -iu kn u kn+l = aj3(f(xa 3k ) - f{xp 3k )) - {a 2 + ap+p 2 )(f {xa 2k p k ) -f(xa k j3 2k )) ap(a-/3) 3 (3.14) Similarly, (3.5) and (3.10) and then (3.8) and (3.12) yield, respectively, CO _ a 2 pf(xa 3k )- a(a 2 + 2p 2 )f{xa 2k p k )+P(2a 2 + 0 2 )f{xa k p 2k ) - ap 2 f(xp 3k ) ap(a-p) 3 (3.15) Z J a n X "^kn+v^kn-l (3.16) = ap 2 f(xa 3k ) - p(2a 2 + p 2 )f{xa 2k p k ) + a(a 2 + 2p 2 )f(xa k p 2k ) - a 2 Pf( X p 3k ) a 2 p 2 (a-pf Finally, from (1.2), we have V 3 = U 3 kn+l - 3qU 2 kn^u kn _ x + 3q 2 U kn+l U 2 kn^ - q 3 U 3 kn^. This, t gether with (3.7), (3.13), (3.15), and (3.16), yields f j a x"v 3, = f(xa 3k ) + 3f(xa 2k p k ) + 3f(xa k p 2k )+f(xp 3k ) (3.17) after some tedious manipulation involving the use of the equality ap-q. 4. APPLICATIONS We now specialize (3.4) and (3.17) to the Chebyshev polynomials to obtain some attractive sums involving third powers of the sine and cosine functions. Let {^(0}»=o an d {^(OK?=o denote the Chebyshev polynomials of the first and second kinds, respectively. Then S (t)=^" e] sin#, t = cos9, n>q. T n {i) = cosno Indeed, {S w (0}*=o and {2T n (t)} =0 are the sequences {U n } =0 and {V n } =0, respectively, generated by (1.1), where p - 2 coso and q - 1. Thus, a = cos$ + isin9 = e 10 and P = cos0-isin0 = e~ w, 274 [AUG.

which are obtained from (1.3). Further information about Chebyshev polynomials can be found, for example, in [1]. We use the following well-known power series, each of which has the complex plane as its domain of convergence: S m 2 = Z Vo^lM (2w + l)! > ' ( 4 1 ) *=?LtMr> (4.2) ti (2«)i ' 7 2 w +l s i n h z = 5<^r <43) «* x - ^ j. (4.4) Now, in (3.4), taking U = sin?i0/sin# and replacing/by the functions in (4.1)-(4.4), we obtain, after replacing all occurrences of k0 by ft V (~^T x2n+l $i n3 (2 n +1)# _ 3cos(xcos^)sinh(x sin ft) - cos(xcos3^) sinh(xsin 3ft) (.. + 0 (2n + l)! " 4. C -^ ^ (-l)"x 2w sin 3 2/?^ _ -3 sin(xcos^) sinh(x sin ft) + sin(xcos3^) sinh(x sin 3ft).. ( % <2*ji " 5 ' ' } yn x 2n+l sin 3 (2/2 +X)(j) _ 3 cosh(x cos^) sin(x sin ft) - cosh(x cos3^) sin(x sin 3ft) (. { 0 (2» + l)! " 4 ' } ^ x 2w sin 3 2/^ _ 3sinh(xcos^)sin(xsin^)-sinh(xcos3^)sin(xsin3^) (. n + 0 - ( 2 ^ r - = ( 4 8 ) -A Similarly, in (3.17), taking V n = 2cosn0 and replacing/by the functions in (4.1)-(4.4), we obtain, respectively, V (~l) Wy2w+1 cos 3 (2/? +1)^ _ 3 sin(xcos^) cosh(x sin ft) + sin(xcos3^)cosh(x sin 3ft),. ( f 0 (2^Tl)! - 4 ' j Y^ (-l)"x 2/7 cos 3 2?i^ _ 3 cos(x cos^) cosh(x sin ft) + cos(x cos3^) cosh(x sin 3ft) (. = + 0 (2n)\ 4 ' ( - > ^ x 2 " +1 cos 3 (2/i +1)^ 3 sinh(x cos^) cos(x sin ft) + sinh(x cos3^) cos(x sin 3 ft) / / j 1 A = ( 4 H ) S (2* + l)! 4" ' y, x 2 " cos 3 2/?^ _ 3 cosh(x cos^) cos(x sin ft) + cosh(x cos3^) cos(x sin 3ft) (. = ( 4 1 2 ) k (2»)i 4 Finally, we mention that much of the tedious algebra in this paper was accomplished with the help of,! Mathematica". 1999] 275

ACKNOWLEDGMENT The author gratefully acknowledges the input of an anonymous referee, whose suggestions have improved the presentation of this paper. REFERENCES 1. M. Abramowitz & I. A. Stegun. Handbook of Mathematical Functions. New York: Dover, 1972. 2. R. Bellman. Introduction to Matrix Analysis. New York: McGraw-Hill, 1970. 3. O. Brugia & P. Filipponi. "Functions of the Kronecker Square of the Matrix Q." In Applications of Fibonacci Numbers 2:69-76. Ed. A. N. Philippou et al. Dordrecht: Kluwer, 1988. 4. P. Filipponi. "A Family of 4-by-4 Fibonacci Matrices." The Fibonacci Quarterly 35.4 (1997):3-08. 5. F. R. Gantmacher. The Theory of Matrices. New York: Chelsea, 1960. 6. E. Lucas. "Theorie des Fonctions Numeriques Simplement Periodiques." Amer. J. Math. 1 (1878): 184-240, 289-321. 7. R. S. Melham. "Lucas Sequences and Functions of a 3-by-3 Matrix." The Fibonacci Quarterly 37.2 (1999): 111-16. AMS Classification Numbers: 11B39, 15A36, 30B10 276 [AUG.

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