Jacobsthal polynomials Jn,m(x), the generalized Jacobsthal-Lucas polynomials jn .... Our first result on generating functions is contained in Theorem 1 below.
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ELSEVIER
MATHEMATICAL AND COMPUTER MODELLING
@DIRECT •
Mathematical and Computer Modelling 42 (2005) 1049-1056 www.elsevier.com/locate/mcm
I n c o m p l e t e G e n e r a l i z e d J a c o b s t h a l and Jacobsthal-Lucas Numbers G. B. DJORDJEVI6 Department of Mathematics Faculty of Technology University of Nig YU-16000 Leskovad, Serbia and Montenegro, Yugoslavia ganedj ~eunet. yu
H. M. SRIVASTAVA Department of Mathematics and Statistics University of Victoria Victoria, British Columbia V8W 3P4, Canada harimsri~math, uvic.
ca
(Received and accepted October 200~) Abstract--In this paper, we present a systematic investigation of the incomplete generalized Jacobsthal numbers and the incomplete generalized Jacobsthal-Lueas numbers. The main results, which we derive here, involve the generating functions of these incomplete numbers. @ 2005 Elsevier Ltd. All rights reserved. Keywords--Incomplete generalized Jacobsthal numbers, Incomplete generalized JacobsthalLucas numbers, Generating functions.
1. I N T R O D U C T I O N
AND
DEFINITIONS
Recently, Djordjcvid [1,2] considered four interesting classes of polynomials: the generalized Jacobsthal polynomials Jn,m(x), the generalized Jacobsthal-Lucas polynomials jn,m(x), and their associated polynomials F~,.~(z) and fi~,m(x). These polynomials are defined by the following recurrence relations (cf., [1-3]):
&,m (X) = ,~n 1,m (x) -~ 2xol n . . . . .
(x)
(n _-->.~; ~ , ~ ~ N; J0,~ (x) = 0, J,~,,~ (x) = 1, when n = 1 , . . . , r n -
1),
j~,~ (x) = jn--l,m (X) + 2xj . . . . . . . (X) (~ >= .~; .~, ~ ~ N; jo,~ (~) = 2, j ..... (x) = 1, when n = 1 , . . . , m - -
1),
F,,,,,, (x) = F,~-I,~ (x) + 2xF,~ . . . . . . (x) + 3 (,,~ > 'm; 'm, n C N; Fo,m (~) = O, Fn,m (x) = 1, when n = 1 , . . . , m -
1),
(1.1)
(1.2)
(1.3)
Tile present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353. 0895-7177/05/$ - see front matter @ 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2004.10.026
Typeset by Aj~S-'I~X
1050
C.B.
(n_>_m; rn, n e N ;
DJORDJEVICAND H. M.
SRIVASTAVA
A,,z (*) = A - , , m (~) + 2~f~_,%,, (~) + 5 fo,m(X) = 0; f , , , ~ ( x ) = l , w h e n n = l , . . . , m - 1 ) ,
(1.4)
N being the set of natural numbers and No := N u {o} = {o, 1, m . . . } .
Explicit representations for these four classes of polynomials are given by [(,~-l)/m] dn,m(X' : E (n-l-r(m-i)r)(2x)r r=0 [~/m] n - - (rn - 2) k ( n -
J.... (~)= Z X
(.~
k=O
1) k
(1.5)
(m- 1)k)(2x) k
k
(1.6) '
[(rz--rn+ l ) /m]
(I.7)
F~,m (x) =Y~,~ (x) + 3 r=o and
[(. . . . +l)/rn]
E
f, .... (.) = & , . , ( . ) + 5
r+l
(n-m+l-(m-1)r)
r=o
'
(2z)~
r+l
(1.8) '
respectively. Tables for Jn,m (x) and jn,m (x) are provided in [2]. By setting x = 1 in definitions (1.1)-(1.4), we obtain the generalized Yacobsthal numbers [(n-1)/rn]
( n - l - ( m - l2) r )~ ' r
Jn,m := Orn,rn (1) =
(1.9)
r=0
and the
generalized Yacobsthal-Lucasnumbers [~/m] - ( m - 2 ) r ( n - ( m - 1 ) r ) j,,,m := j,,,,, (1) = ~ _n _ ~=o n (m 1) r r
2 ~,
(1.10)
and their associated numbers [(n--m+l)/m
F~,., : = F , , , m ( 1 ) = J ~ , m ( 1 ) + 3 and f,~,,~ := f7.... ( 1 ) = J~,~ ( 1 ) + 5
n - m + 1 - ( m - 1) r'~2~ r+l J
[(n--m+l)/rn] E /~n-m+l( m - 1 ) r )C2" ~=0 r+l
(i.ii)
(1.12)
Particular cases of these numbers are the so-called Jacobsthal numbers Yn and the daeobsthalLucas numbers Jn, which were investigated earlier by Horadam [41. (See also a systematic investigation by Raina and Srivastava [5], dealing with an interesting class of numbers associated with the familiar Lucas numbers.) Motivated essentially by the recent works by Filipponi [6], Pint6r and Srivastava [71, and Chu and Vicenti [8], we aim here at introducing (and investigating the generating functions of) the analogously incomplete version of each of these four classes of numbers.
Incomplete Generalized Jacobsthal
1051
2. G E N E R A T I N G FUNCTIONS OF THE INCOMPLETE GENERALIZED JACOBSTHAL AND JACOBSTHAL-LUCAS NUMBERS We begin by defining the incomplete generalized Jacobsthal numbers J~,m k by k
Jkn,m : = E ( n - l - ( m - 1 ) r=o
r) 2 r
r
(o_< k __ [-~]
; ,~,,~ c N),
(2.1)
so that, obviously, j[(n-1)/rn(n-1)/rn] = Jn rn, J~,,~ = 0
(2.2) (2.a)
(O
