T O r - B O N A C C I P O L Y N O M I A L S. M.N.S.SWAMY Sir George Williams University, Montreal, P.O., Canada. n-1

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1 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 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

2 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 = F +3 = 2".F 3 - F +3 ad () L k 2 ~ k = 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 = 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).

3 977] 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 tx l " R 2 + x r ' 2 Fli)y ' 3 +-+(x'" R r -i+x'" 2 R r _ xr )y - r + (x r ~ 7 R k^ +x r ~ 2 R k R k r )y - k = R Y "- +x r - f [R iy ~ +R 2 V ~ R r -iy ~ r+ + x r - 2 y- 2 [R y - + R 2y R r - 2 y -" 2 l

4 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) " (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 IFM + F^+FM =

5 977 ITS GENERALIZATION TO /--BONACCI POLYNOMIALS 77 - f - 2 F ^ +2 r - 2 ff<%] + f- 3 [F^ + + F< r l r+3 ] [F< r > + F&J + F^ - f~ 3 lf^+2 + F<?>,] + 2^3[F< % + Fjfj, + F<r> + + F^+3 ] + 2^[F< '> FJll^l [F^ + F^] + F^ [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 B.W. Kig, "A Polyomial with Geeralized Fiboacci Coefficiets," The Fiboacci Quarterly, Vol., No. 5, 973, pp V. E. Hoggatt, Jr., ad M. Bickell, "Roots of Fiboacci Polyomials," The Fiboacci Quarterly, Vol., No. 3, 973, pp A. F. Horadam, "A Geeralized Fiboacci Sequece," Amer. Math. Mothly, Vol. 68, 96, pp M. Bickell ad V. E. Hoggatt, Jr., "Geeralized Fiboacci Polyomials," The Fiboacci Quarterly, Vol., No. 5, 973, pp * * * * * * * r- -E +r+ 2 k