GENERALIZED FIBONACCI AND LUCAS POLYNOMIALS AND

GENERALIZED FIBONACCI AND LUCAS POLYNOMIALS AND THEIR ASSOCIATED DIAGONAL POLYNOMIALS M. N. S. Swamy Concordia University, Montreal, Quebec H3G 1M8, Canada (Submitted August 1997-Final Revision November 1997)

1. INTRODUCTION Horadam [7], in a recent article, defined two sequences of polynomials Jn(x) and j„(x), the Jacobsthal and Jacobsthal-Lucas polynomials, respectively, and studied their properties. In the same article, he also defined and studied the properties of the rising and descending polynomials i^(x), rn(x), Dn(x)y and dn(x), which are fashioned in a manner similar to those for Chebyshev, Fermat, and other polynomials (see [2], [3], [4], [5], and [6]). The purpose of this article is to extend these results to the generalized Fibonacci and Lucas polynomials defined by U„(x,y) = xUn_l(x,y)+yU„-2(x,y)

(»>2),

(1.1a)

with U0(x,y) = 0, Ul(x,y) = l,

(l.lb)

and V„(x,y) = xV„_l(x,y)+yV„_2(x,y)

(n>2),

(1.2a)

with V0(x,y) = 2, Vl{x,y) = x:

(1.2b)

In Section 2, we will give some basic properties of the polynomials Un(x,y) and Vn(x, y), most of which are generalizations of those given in [7] for Jn(x) and jn(x). In Section 3, we will derive some new properties of Un(x, y) and Vn(x, y) concerning their derivatives, as well as the differential equations they satisfy. In the remaining sections, we will define and study the properties of the rising and descending diagonal polynomials associated with Un(x, y) and Vn(x, y), thus generalizing the results already known for Fibonacci, Lucas, Chebyshev, Fermat, and Jacobsthal polynomials. 2. BASIC PROPERTIES OF UH(x, y) AND Vn(x, y) Binet Forms: (2.1) Vn{x,y) = an+(3\

(2.2)

where a + / ? = x,

aj3 = -y,

(2.3a)

a-ft = jA,

A = x2 + 4y,

(2.3b)

2a = x + VA,

2/? = J C - V A .

(2.3c)

Simson Formulas: t/„+1(x, y)U„_x{x, y) - U2n{x, y) = (-l)"/*" 1 , K+l(x, y)Vn_x{x, y) - V„\x, y) = (-\)"y"-lA.

(2.4) (2.5) 213

1999]

GENERALIZED FIBONACCI AND LUCAS POLYNOMIALS AND THEIR ASSOCIATED DIAGONAL POLYNOMIALS

Summation Formulas:

t u y)=7T7ri[t/"+i(x'y)+yU"(x>y) i,^y) o

~1]>

= —^Z\lVnH(x,y)+yV„{x,y) x-i-y

+ {x-2)l

(26)

(2.7)

A

Important Interrelations: K(x, y) = Un+l(x, y)+yU„_1(*, y), Vn(x,y) + xUn(x,y) = 2Un+l(x,y), V„(x,y)-xU„(x,y) = 2yUn_l(x,y), AU„(x,y) = Vn+l(x,y)+yVn_i(x,y),

(2-8) (2.9) (2.10) (2.11)

AUn(x,y) = 2Vn+1(x,y)-xV„(x,y), U2n(x,y) = Un(x,y)V„(x,y),

(2-12) (2.13)

V2n{x,y) = VZ{x,y)-2{-y)», 2 F 2 „(X,JO = AU „(x,y) + 2(-y)",

(2.14) (2.15)

At/„2(x, j)+V„2(x, y) = 2Vln(x, y), 2Um+„(x, y) = t/m(x, ^)F„(x, j)+F ffl (x, j;)t/„(x, y),

(2.16) (2.17)

2^m+„(x, 7) = VJx, y)V„(x, y) + AUm(x, y)Un{x, y).

(2.18)

All the above results from (2.4)-(2.18) may be derived using the Binet forms (2.1) and (2.2) or, alternately, using the earlier results of Horadam [8]. Most of these results are to be found in Lucas ([10], Ch. 18). Now we let X = a and Y = (3 in the following identities, where a and /? are given by (2.3), X and Y arbitrary: yn _yn

i(n-l)/2]

^Y~T=

/^

_ ~ i \

yXY)r{X

E H ) ^ r

Xn + Yn= Y(~lY-Z-[

+ Y)"~^ («>0),

/ )(XY)r(X + Y)n-2r

(n>0).

(2.19) (2.20)

We can then easily establish the following expressions for Un(x, y) and Vn(x, y). Closed Form Expressions: [(w-l)/2] /

Um(x,y)=

I

.x

r~r)x"-2r-y,

(2.21)

r=0

F

[if/2]

«^^)=S^:("/J^-V (it >0).

(2.22)

It is seen from (2.21) and (2.22) that U2n(x,y) and V2n_l(x9y) are odd polynomials in x of degree (2n-l) and polynomials my of degree (w-1), while ^w+iC^j) a n ^ ^2n(x^y) a r e e v e n P°'y~ nomials in a: of degree 2n and polynomials in j ; of degree n. It may be mentioned that expression (2.21) for Un(x,y) has already been established by Hoggatt and Long [3]; however, the expression for Vn(x, y) is new. By letting x = 1 and y - 2x, we obtain the results of Horadam [7] for the polynomials Jn{x) and jn(x). 214

[AUG.


Hoggatt and Long [3] have shown that Un(x,y) = flU-2^coI^7^

(n>2).

(2.23)

Using a similar procedure, or by using the technique used by Swamy [11] in obtaining the zeros of Morgan-Voyce polynomials, we can show that Vn(x,y) = T\^-2^co{^7^

(«>2).

(2.24)

We may now rewrite expressions (2.23) and (2.24) to express the polynomials Un(x,y) and Vn(x, y) in the product form. Product Form: [(»-l)/2]r

fh

Un(x,y) = x** n {

x2+4 co

VI

^ %^Jj ^>2),

F„(x,j) = x 1 - ^ n | x 2 + 4 j c o s 2 ^ ^ ^ |

(n>2),

(2.25) (2.26)

where fl if?? is even, S„ = \ 0 if n is odd.

(2.27)

By letting x = 1 and y - 2x in (2.26) and (2.27), we get the zeros of the Jacobsthal polynomials J„(x) and jn(x) to be, respectively, x=-|sec2fej,

k = l,2,...,(n-l\

(2.28)

and X=

~bQCi^br7r^

k = l,2,...,n.

(2.29)

The generating functions for U„(x9 y) and Vn(x, y) are given below. Generating Functions: U(x, y, t) = £ U,(x, y)rl = {1 - t(x+yt)yi, V(x,y,t)=fiV,(x,yy=(2-xt){l-t(x+yt)}-1

(2.30) (2.31)

/=1

= l + (l+ytz){l-t(x+yt)}-1

(2.32)

3. DERIVATIVE PROPERTIES From (2.30), (2.31), and (2.32), a number of relations involving the derivatives of Un(x,y) and Vn(x,y) may be derived. However, only the following derivative relations are listed here. Throughout this section, where not explicitly mentioned, U, V, U„, and Vn stand for U(x,y,t), V(x, y, 0 y Un(x, y), and Vn(x, y), respectively. We can prove that 1999]

215


f=,

(3.0

f-'f-

C3.3,

f"'f^-

\),

d0(x,y) = 2.

(5.2)

(n>l).

(5.3)

Now consider

Hence dn(x,y) = Dn+l(x,y)+yDH(x,y) Thus, dn(x,y) = (x + 2y)(x+yy-1

(n>l).

(5.4)

We also have, from (5.1) and (5.4),

^A=d^x^=x D„(x,y)

}

(55)

dn(x,y)

d„+1(x,y)+yd„(x,y) = (x + 2y)2Dn(x,y)

(«>1).

(5.6)

We may also formulate the following generating functions for the descending polynomials Dn(x, y) and dn(x, y) by following the usual procedure:

D(x, yj) = T Di(x> yy~l = il~ (x+yWl,

(5-7)

1=1

d(x,y,t) = fjdi{x,y)t>-' = {x + 2y){\-{x+y)ty\ (5.8) From the above generating functions, we may deduce the following relations for the derivatives of D(x, y, t) and d(x, y, i) where, for the sake of convenience, D and d are used for D(x, y, t) and d(x,y,t): dD = dD (5.9) dy~ dx'

f = f + Z>, ay

dD

(5.10)

ox

+ x — +y +y

dD

dD

=t

dx -& dy ir' ' dt ' -*

dd dd dta dy , .dD .dD dD dD_ = , (x+y)^ dy ={x+}j (x+y)— = t—, dx 'a dd_ dd_ (x+y)^-x X = t ^ + (x+y)D, dy ~ a

,_„.

(511)

dd dx

dd

dd

dy

a

y)D. (x+y)™= t™=+2(x+ i — t-—

1999]

,. . . . (5.13) (5.14) (5.15)

221


Using the above relations, we may write the corresponding interrelations for the derivatives of the polynomialsZ)n(x, y) and dn(x, y) with respect to x andy as has been done for i^(x, y) and 6. CONCLUDING REMARKS We have generalized all the known results concerning the diagonal functions associated with Fibonacci, Lucas, Chebyshev, Fermat, Pell, and Jacobsthal polynomials to the case of diagonal functions associated with the generalized polynomials given by (1.1) and (1.2). We have also derived a number of new interesting results concerning the derivatives of Un(x, y) and Vn(x, y) with respect to y, the differential equations satisfied by these polynomials, as well as the interrelations between their derivatives with respect to x and y. Similar results have also been derived for both the rising and the descending diagonal polynomials associated with Un{x, y) and Vn(x, y); however, we have not been able to find the differential equations satisfied by i^(x, j/), rn(x,y), and dn(x,y) with respect to either x or y. It may also be observed that the descending (rising) polynomials associated with the rising (descending) polynomials of Un(x, y) and Vn(x, y) are, respectively, Un(x,y) and Vn(x,y). This answers one of the questions raised by Horadam [7] regarding the rising polynomials of the descending polynomials of J„(x) and jn(x) as well as the descending polynomials of the rising polynomials of Jn(x) and jn(x). ACKNOWLEDGMENT The author would like to thank the anonymous referee for many valuable comments and suggestions that have enhanced the quality and presentation of this article REFERENCES 1. R. Andre-Jeannin. "Differential Properties of a General Class of Polynomials." The Fibonacci Quarterly 33.5 (1995):453-58. 2. D. V. Jaiswal. "On Polynomials Related to Tchebichef Polynomials of the Second Kind." The Fibonacci Quarterly 12.3 (1974):263-65. 3. V. E. Hoggatt, Jr., & C. T. Long. "Divisibility Properties of Generalized Fibonacci Polynomials." The Fibonacci Quarterly 12.2 (1974): 113-20. 4. A. F. Horadam. "Diagonal Functions." The Fibonacci Quarterly 16.1 (1978):33-36. 5. A. F. Horadam. "Chebyshev and Fermat Polynomials for Diagonal Functions." The Fibonacci Quarterly 17.4 (1979):328-33. 6. A. F. Horadam. "Extensions of a Paper on Diagonal Functions." The Fibonacci Quarterly 18.1 (1980):3-8. 7. A. F. Horadam. "Jacobsthal Representation Polynomials." The Fibonacci Quarterly 35.2 (1997): 137-48. 8. A. F. Horadam. "Basic Properties of a Certain Generalized Sequence of Numbers." The Fibonacci Quarterly 3.3 (1965): 161-76. 9. A. F. Horadam. "Rodrigues' Formula for Jacobsthal-Type Polynomials." The Fibonacci Quarterly 35.4 (1997):361-70. 10. E.Lucas. Theorie desNombres. Paris: Blanchard, 1961. 11. M. N. S. Swamy. "Further Properties of Morgan-Voyce Polynomials." The Fibonacci Quarterly 6.2 (1968): 167-75. AMS Classification Numbers: 11B39, 33C25

222

[AUG.