A FORMULA FOR F k (x)y " k T O r - B O N A C C I P O L Y N O M I A L S AND ITS GENERALIZATION M.N.S.SWAMY Sir George Williams Uiversity, Motreal, P.O., Caada. INTRODUCTION Some years ago, Carlitz [] had asked the readers to show that (D E F k 2 ~ k ~ = 2 -F +2 ad (2) E L k 2 ' k ~ =3(2 )-L +2f Fiboacci ad Lucas umbers, Recetly, Kig [2] geeralized these results to ob- where F ad L are the/7 tai the expressios: - (3) y T y -i<-l = (T y + T. )y - T y - T ^ ad o y 2 -y- i (4) T k 2 ~ k - = T 2 (2")-T +2 * where the geeralized Fiboacci umbers T are defied by T = T -i + T. 2t Tj = a, T 2 = b. The purpose of this article is to geeralize these results to sums of the form XF k (x)y, HL k (x)y ~, T,H k (x)y ~, where F k (xj, L k (x) ad H k (x) are, respectively, Fiboacci, Lucas ad geeralized Fiboacci Polyomials, ad the fially to exted these results to r-boacci polyomials. 2. FIBONACCI AND LUCAS POLYNOMIALS AS COEFFICIENTS The Fiboacci polyomials F (x) are defied by [3] (5) F (x) =xf - (x) + F - 2 M with FQ(X) =, Ff(x) =. Now cosider the sum S= E Fk(x)y ' k = y - +xy - 2 +Y. lxf k -i<x) + F k 2 (x)ly - k 3 Hece, - y'- +xy - 2 + xy- F k y"- k +y' " f F k y ~ k 2 y"- +xy- {S-F M} + y' 2 {S- F _ 7 My - F (x)} (y 2 -xy- )S = y + -yf + (x)- F (x). 73
74 A FORMULA FOR ' F k (x)y ~ k AND [FEB. / Lettig (6) G (x,y) = y + -yf + (x)-f (x), we may write S as (7) S - f ^ " " * = f ^ - j, <?,fr,w *. The Lucas polyomials /.^ W are defied by [3] (8) L (x) = xl-jm + L^M with LQ(X) = 2, L-j(x) = x. It may be show by iductio or otherwise that Hece, L (x) = F + (x) + F. (x). Y. L k (x)y -«= F k+ (x)y - k +Y. Fk-lW" = ". F k (x)y + ' k ^ F k (x)y 2 --k - W K " ^ - Z F k (x)y ~ ' k - F (x)y - ^±^^"^ _,», usig (7) Therefore = ^ * ^ ~ ^ ( ^ 2 ^ * FM) - {F+lM + F-M} Gi(x,y) (9) ^ f r ^ ^ - ^ - ^ ^ ^ ^ ^. By lettig x =, y = 2 \ results i (7) ad (9), we obtai () F k 2 ' k = 2 + - F +3 = 2".F 3 - F +3 ad () L k 2 ~ k = 2 +2 -L +3 = 2"-L 3 -L +3 which are the results of Carlitz []. Further, by lettig A- = y = 2 i (7) we get (2) P k 2 ' k = P +2-2 + = P +2-2.P 2, where P is the th Pell umber. 3. GENERALIZED FIBONACCI POLYNOMIALS AS COEFFICIENTS Let us defie the geeralized Fiboacci polyomials H (x) as (3) H (x) = xh. (x) + H. 2 M with Hg(x) ad Hj(x) arbitrary. It is obvious that the polyomials F (x) are obtaied by lettighq(x) =, Hi(x)= 7, while the Lucas polyomials L (x) are obtaied by lettig HQ(X) = 2 ad Hf(x)=. I fact, it ca be established that H (x) is related to F (x) by the relatio Hece, H (x) = H (x)f (x) + H (x)f - (x).
977] ITS GENERALIZATION TO r-bonacci POLYNOMIALS 75 Z H k (x)y ' k = H 7 (x) Z F k (x)y ~ k + H (x) Z i=k-l(x)y ~ k -»M G ^ +H M Z F kv "-l~ k, usig (7) = H (x,g < X >V) + H MG - y (x,y) Gi(x,y) The right-had side may be simplified to show that (4) Z H k (x)y ~ k = H l (x)y " +]^ y 2 -xy - + H (x)y ~yh + (x)-h (x) Some special cases of iterest obtaiable from (4) are, Z H k (x).x ' k = H +2 (x)-x H 2 (x), Z H km = - lh + (x) + H (x)-h (x)-h Q (x)l, Z (-D k+ H k (x) = t f(-v + {H + (x)-h (x)} + {H 7 (x)-h (x)}]. It should be oted that by lettig x = 7, HQ(X) = a ad H j(x) = b i (3), we geerate the geeralized Fiboacci umbers H defied earlier by Horadam [4]. From (4) it is see that for these geeralized Fiboacci umbers (5) E Hky^ - ^HlfylzJ^y^ y 2 - y - i ad <? (6) Z H k 2 ~ k = (2b+a)2 - H +3 = 2 -H 3 -H +3 which are the results obtaied by Kig [2]. 4. r-bonacci POLYNOMIALS AS COEFFICIENTS The /--boacci polyomials Fp(x) have bee defied by Hoggatt ad Bickell [5] as ad Fi r! r -2)M = = Fl%) = F ( r, M =, F< r> (x) =, F< 2 r >(x) = x r ~, (7) F ( r l r (x) = x r ~ F ( r i r^(x) +x r - 2 F ( r l r _ 2 (x) +.» + F ( r) (x). Let us ow cosider Deotig for the sake of coveiece / - F^Mv ~ k (8) F ( k r, (x) = R k we have, / = R iy ~ +x r - R y - 2 + tx l " R 2 + x r ' 2 Fli)y ' 3 +-+(x'" R r -i+x'" 2 R r _ 2 + - + xr )y - r + (x r ~ 7 R k^ +x r ~ 2 R k - 2 + -+R k r )y - k = R Y "- +x r - f [R iy ~ +R 2 V ~ 2 + -+ R r -iy ~ r+ + x r - 2 y- 2 [R y - + R 2y - 2 + - + R r - 2 y -" 2 l
76 A FORMULA FOR Z F k (x)y ~ k AND [FEB. / + xy- ( '- >[R iy - ] +x r -\- Y, R k y"- k +x r - 2 y- 2 E *&"* r r- Hece, +... + xy- (r - ) "' ' R k y - k +y~ r j: R k y ' k. 2 ly r = Riy +r ~ + (xyr R k y"- k + (xyr 2 JT R k y ~ k -r+ -r + -+xv E R k v ~ k + i: R k y ' k = R iy +r - + Hxyf- + (xy) " 2 + - + (xy) + till -( xy r i R -( Xy r 2 R k y ' k -- Thus, - (xy) E R ky - k - E R k y ' k -r+2 -r+ l\y r -E (xya - R y +r - -y r - (x r - R +x r - 2 R + -+R. r+ ) - y r ~ 2 (x r - 2 R +x r ~ 3 R -i + -+R - r+2 ) -y r - 3 (x r ~ 3 R +x r ~ 4 R ^ + -+R -r+3) - -y(xr + R -is-r Deotig ow G< r) (x,y) = y +r - - F^My'- - y r ~ 2. (x^f^m ^ ^ F ^ M + +F ( r ] r+2 (x)j (9) - y r - 3 [x r ~ 3 F( r >(x) + x r - 4 F<fl,(x) + - + F< r l r+3 (x)] we have ylxf^m + F < r l (x)j - F l r) (x) (2) / = F< r> (x)-y - k ILLJM} G\ r) (x,y) ' The above result for r-boacci polyomials may be cosidered as a geeralizatio of the result (7) for Fiboacci polyomials. Let us ow see if we ca obtai for the r-boacci umbers [5], a result correspodig to () for Fiboacci umbers; it may be oted that the r-boacci umbers F' r ' we obtaied by lettigx= 7 i (7). We have from (2) that (2D E 4 r) -2"- k - ^ - ^ - / G ( j r, (,2) Now we have from (9), 2+r- _ G(r) (w = 2^-2^ + 2r-2 [F<r) +... + ^ j +2^3 [F(r) +... + fjrj^j +... + + 2IFM + F^+FM =
977 ITS GENERALIZATION TO /--BONACCI POLYNOMIALS 77 - f - 2 F ^ +2 r - 2 ff<%] + f- 3 [F^ + + F< r l r+3 ] + - + 2[F< r > + F&J + F^ - f~ 3 lf^+2 + F<?>,] + 2^3[F< % + Fjfj, + F<r> + + F^+3 ] + 2^[F< '> +... + FJll^l + - + 2[F^ + F^] + F^ - 2-3 [F< % + F< % + f W, y +2^F<«+ - + FlfJ^J + 2[F< '> + F^J + F?. Cotiuig the process, the above may be reduced as Hece, (22) Also 2 +r ' _ rj r Ul?) =? r - r ff (r) +... + F (r) = F (r) z u a,zj z Lr +r -i- -- t + j r +r+ G ( r> ri+r- (,2) = -F G\ r) (l2) = 2 r - (23) = 2' 2'-- -2 Therefore from (2), (22) ad (23) we get r(r) 9-k _ 9+r- p(r) _ 9 p(r) r(r) Z r k ' z z " ~ t +r+ ~ z mt r+ r +r+ The above result may be cosidered as a geeralizatio, for the r-boacci umbers, of the result of Carlitz [] for the Fiboacci umbers. REFERENCES. L Carlitz, Problem B-I35, The Fiboacci Quarterly, Vol. 6, No., I968, p. 9. 2. B.W. Kig, "A Polyomial with Geeralized Fiboacci Coefficiets," The Fiboacci Quarterly, Vol., No. 5, 973, pp. 527-532. 3. V. E. Hoggatt, Jr., ad M. Bickell, "Roots of Fiboacci Polyomials," The Fiboacci Quarterly, Vol., No. 3, 973, pp. 27-274. 4. A. F. Horadam, "A Geeralized Fiboacci Sequece," Amer. Math. Mothly, Vol. 68, 96, pp. 455-459. 5. M. Bickell ad V. E. Hoggatt, Jr., "Geeralized Fiboacci Polyomials," The Fiboacci Quarterly, Vol., No. 5, 973, pp. 457-465. * * * * * * * r- -E +r+ 2 k