Fn(k) are defined by (1)

RENDICONTI DEL CIRCOLO MATEMATICO DI PALERMO Serie II, Tomo XLVII! (1999), pp. 591-596

G E N E R A T I N G F U N C T I O N S OF T H E I N C O M P L E T E F I B O N A C C I

A N D LUCAS NUMBERS

A. PINTER - H. M. SRIVASTAVA

For the incomplete Fibonacci and incomplete Lucas numbers, which were introduced and studied recently by P. Filliponi [Rend. Circ. Math. Palermo (2) 45 (1996), 37-56], the authors derive two classes of generating functions in terms of the familiar Fibonacci and Lucas numbers, respectively.

1. Introduction and the Main Result.

The incomplete Fibonacci and incomplete Lucas numbers were intro- duced recently by Filipponi [1]. The incomplete Fibonacci numbers Fn(k) are defined by

(1) F n ( k ) = Z . n = 1 , 2 , 3 , . . . ; 0 < k <

j=O J

and the incomplete Lucas numbers Ln(k) are defined by

(2) Ln(k) = ~---~ n n - j n = 1 , 2 , 3 , . . . ' 0 < k <

j=o n - j j ' '

w h e r e [s] d e n o t e s the greatest i n t e g e r in s.

It is e a s y to see that

F ~ ( [ - ~ 2 1 ] ) = F n a n d L n ( [ 2 ] ) : L n ,

w h e r e Fn and Ln are the n t h F i b o n a c c i and L u c a s n u m b e r , r e s p e c t i v e l y . T h e p u r p o s e o f this n o t e is to d e r i v e the g e n e r a t i n g f u n c t i o n s for these classes o f n u m b e r s .

THEOREM L e t k be a natural number. T h e n

(3)

OO

Rk(t) : : Z Fk(j)tY : t 2 k + l ( F 2 k --}- F z g - l t ) ( 1 - t ) k + l - - t 2

j=o (1 - t) k+l (1 - t - t 2) '

and

(4) Sk(t) 9 Z Lk(J)tJ = t2k (L2k-1 -k- L2k-2t)(1 -- t) k+x -- t2(2 -- t)

j=o (1 - t ) k + l ( 1 -- t - t 2)

In the trivial case w h e n k = 0, w e h a v e

Fo(j) = L o ( j ) = 1 ( j E No : = {0, 1, 2 . . . . }), w h i c h i m m e d i a t e l y yields

Ro(t) : So(t) -- - - 1 - t

2. Proof of the Theorem.

T h e p r o o f o f o u r T h e o r e m is b a s e d u p o n the f o l l o w i n g

S O0

LEMMA Let { n}n=0 be a complex sequence satisfying homogeneous second-order recurrence relation:

the non-

(5) sn = ash-1 + bsn-z + rn, (n ~ 1~t\{1}; l~t : = No\{O}),

GENERATING FUNCTIONS OF THE INCOMPLETE FIBONACCI AND LUCAS NUMBERS 593 w h e r e a , b 9 C a n d rn : N ---> C is a g i v e n s e q u e n c e . T h e n t h e g e n e r a t i n g f u n c t i o n U ( t ) o f sn is

G ( t ) + so - ro + (Sl - soa - r l ) t

(6)

1 - a t - b t 2 w h e r e G ( t ) d e n o t e s t h e g e n e r a t i n g f u n c t i o n o f rn.

(7)

(8)

(9) and 0 o )

P r o o f Indeed, by using a fairly standard technique, we obtain

U (t) -~ so q- s i t "Jr" s2 t2 "k- ' ' ' -}- Sn tn -+- 9 "",

a t U ( t ) = s o a t + s l a t 2 + . . 9 + a s h - i t " + . . . ,

b t 2 U ( t ) = s o b t 2 + . . . + b s n - 2 t n h- " " ,

G ( t ) = ro + r l t + r 2 t 2 q- . . . + r n t n + . . . .

Therefore

(11) U(t)(1 - a t - b t 2) - G ( t ) = so - ro + ( s l - s o a - r l ) t , which completes the p r o o f of the L e m m a .

In the sequel, k is a fixed positive integer. It is k n o w n (see [1]) that F n ( k ) = O if 0 < n < 2 k + l , F z k + l ( k ) = F z k , and F 2 k + z ( k ) = F z k + l , and that

( 1 2 )

F~(k)

2k]

Set

so = F2k+l(k), Sl = F z k + 2 ( k ) , Also let

ro = rl = O and

and sn = F n + 2 k + l ( k ) ( n 9 1~\{1}).

n - 2 + k )

r n ~

n - 2

T h e generating function of the sequence rn is t z / ( 1 - t ) k+l (cf. [2, p.

349, Problem 216] or [3, p. 355, Equation 7.1(5)]). Thus the generating function Uk(t) of the sequence {s,}~__ 0 satisfies the equation:

t 2

(13) Uk(t)(1 -- t -- t 2) -4- -- F2k + (F2k+l -- F2k)t.

(1 - - t ) k + l

Finally, the generating function Rk(t) of {F(k)}~= 0 is t2k+lUk(t).

For the following facts we also refer to [1]:

L~(k) = 0 if n < 2k, L2k(k) = L 2 k - l , and L 2 k + l ( k ) = L2k , and (in general)

Ln(k) = Ln-~(k) + Ln-2(k) (14)

n - 2

n - 2 - k n - 2 - 2 k l ( n > 2 k + 2 ) . Put

so = L2k(k), sl = L2k+l(k), and Sn = L2~+n(k) (n ~ N\{1}).

Also set

n - 2 + 2 k ( n - 2 + k ) ( n ~ N \ { 1 } ) ' n - n - 2 (15) ro = rl = 0 and rn -- 2~r-'-k

Then the generating function of rn is (see [2, p. 348, Problem 212]

or [3, p. 355, Equation 7.1 (9)])

t2(2 - t)

(16) (1 - t) ~+1 '

and the generating function Uk(t) of the sequence { n}n-O satisfies the S

equation:

(17)

U~(t)(1 - t - t 2 ) +

t 2 ( 2 - - t )

( 1 - - t) k+l -- L2k-1 + (L2k -- L2k-1)t

= L2k-1 + L2k-2t.

GENERATING FUNCTIONS OF THE INCOMPLETE FIBONACCI AND LUCAS NUMBERS 595 Remark. Since (see [1, p. 47, Equation (3.8)]

( 1 8 )

Zn(k ) ~- Fn_l(k-1)--[- Fn+l(k ) (0 <k < [ 2 ] ) '

we can give an alternative proof for the generating function (4) of the incomplete Lucas numbers. From the equation (18) above, we have

oo oo oo

(19) y ~ Ln(k)t n = y ~ F n - l ( k - 1)t n + Z Fn+l(k)tn

n = 0 n = 0 n = 0

and

(20) Sk (t) = t Rk-1 (t) + - - Rk (t). 1 t

Applying the well-known identity:

(21) Fn-2 + Fn = Ln-1 (n ~ N\{1}) our alternative proof of (4) is complete again.

Acknowledgments.

The present investigation was carded out during the first-named author's visit to the University of Victoria in the second half of the academic year 1996-1997. This work was supported, in part, by E6tv6s Fellowship of Hungary (Grants 16975 and 19479 from HNFSR) and, in part, by the Natural Sciences and Engineering Research Council of Ca- nada under Grant OGP0007353.

REFERENCES

[1] Filipponi E, Incomplete Fibonacci and Lucas Numbers, Rend. Circ. Mat. Pa- lermo 45 (1996), 37-56.

[2] P61ya G., Szeg~i G., Problems and Theorems in Analysis, Vol. I (Translated from the German by D. Aeppli), Springer-Vedag, New York, Heidelberg and Berlin, 1972.

[3] Srivastava H. M., Manocha H. L., A Treatise on Generating Functions, Hal- sted Pres (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane, and Toronto, 1984.

Pervenuto il 15 maggio 1998.

A. Pint~r Mathematical Institute, Kossuth Lajos University H-4010 Debrecen, Hungary E-Mail: APINTER@ MATH.KLTE.HU H. M. Srivastava Department of Mathematics and Statistics University of ~rtctoria Victoria, British Columbia V8W 3P4, Canada

E-Mail: HMSRI@ UVVM. UVIC. CA