ELSEVIER Available online at www.sciencedirect.com MATHEMATICAL AND SCIENCE @DIRECT COMPUTER MODELLING Mathematical Computer Modelling 42 (2005) 1049-1056 www.elsevier.com/locate/mcm Incomplete Generalized Jacobsthal Jacobsthal-Lucas Numbers G. B. DJORDJEVI6 Department of Mathematics Faculty of Technology University of Nig YU-16000 Lesovad, Serbia Montenegro, Yugoslavia ganedj ~eunet. yu H. M. SRIVASTAVA Department of Mathematics Statistics University of Victoria Victoria, British Columbia V8W 3P4, Canada harimsri~math, uvic. ca (Received accepted October 200~) Abstract--In this paper, we present a systematic investigation of the incomplete generalized Jacobsthal numbers 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 Jacobsthal- Lucas numbers, Generating functions. 1. INTRODUCTION 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), their associated polynomials F~,.~(z) fi~,m(x). These polynomials are defined by the following recurrence relations (cf., [1-3]): (n _-->.~; ~, ~ ~ N; J0,~ (x) &,m (X) =,~n 1,m (x) -~ 2xol n... (x) j~,~ (x) (~ >=.~;.~, ~ ~ N; jo,~ (~) F,,,,,, (x) (,,~ > 'm; 'm, n C N; Fo,m (~) = 0, J,~,,~ (x) = 1, when n = 1,...,rn- 1), = jn--l,m (X) + 2xj... (X) = 2, j... (x) = 1, when n = 1,...,m-- 1), = F,~-I,~ (x) + 2xF,~... (x) + 3 (1.1) (1.2) = O, Fn,m (x) = 1, when n = 1,...,m- 1), (1.3) Tile present investigation was supported, in part, by the Natural Sciences 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. DJORDJEVIC AND H. M. SRIVASTAVA A,,z (*) = A-,,m (~) + 2~f~_,%,, (~) + 5 (n_>_m; rn, nen; fo,m(x) = 0; f,,,~(x)=l, whenn=l,...,m-1), N being the set of natural numbers (1.4) 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 (1.5) r=0 [~/m] n-- (rn - 2) (n- (m- 1))(2x) (1.6) J... (~)= ZX (.~ 1) ' =O F~,m (x) =Y~,~ (x) +3 [(rz--rn+ l ) /m] r=o r+l ' (I.7) f,... (.) =&,.,(.)+5 [(... +l)/rn] E (n-m+l-(m-1)r) (2z)~ r=o r+l ' respectively. Tables for Jn,m (x) jn,m (x) are provided in [2]. By setting x = 1 in definitions (1.1)-(1.4), we obtain the generalized Yacobsthal numbers (1.8) [(n-1)/rn] Jn,m := Orn,rn (1) = r=0 the generalized Yacobsthal-Lucas numbers (n-l-(m-l)r) 2 ~ ' r (1.9) their associated numbers -(m-2) r(n-(m-1)r) [~/m] j,,,m := j,,,,, (1) = ~ _n _ 2 ~, (1.10) ~=o n (m 1) r r F~,., :=F,,,m(1)=J~,m(1)+3 [(n--m+l)/m n- m + 1 - (m- 1) r'~2~ r+l J (i.ii) [(n--m+l)/rn] f,~,,~ := f7... (1)= J~,~ (1)+5 E /~n-m+l- (m-1) r)2 C" (1.12) ~=0 r+l Particular cases of these numbers are the so-called Jacobsthal numbers Yn the daeobsthal- Lucas numbers Jn, which were investigated earlier by Horadam [41. (See also a systematic investigation by Raina Srivastava [5], dealing with an interesting class of numbers associated with the familiar Lucas numbers.) Motivated essentially by the recent wors by Filipponi [6], Pint6r Srivastava [71, Chu Vicenti [8], we aim here at introducing ( investigating the generating functions of) the analogously incomplete version of each of these four classes of numbers.
Incomplete Generalized Jacobsthal 1051 2. GENERATING FUNCTIONS OF THE INCOMPLETE GENERALIZED JACOBSTHAL AND JACOBSTHAL-LUCAS NUMBERS We begin by defining the incomplete generalized Jacobsthal numbers J~,m by Jn, m := E(n-l-(m-1) r) r r=o 2 r (o_< [-~] ;,~,,~ c N), (2.1) so that, obviously, j[(n-1)/rn(n-1)/rn] = Jn rn, (2.2) J~,,~ = 0 (O<=n<m+l), (2.a) J~+t,.~ = Jm+/-1,.~ (l = 1,.., m). (2.4) The following nown result (due essentially to Pintdr Srivastava [7]) will be required in our investigation of the generating flmetions of such incomplete numbers as the incomplete generalized Jacobsthal numbers g~,m defined by (2.1). For the theory applications of the various methods techniques for deriving generating functions of special functions polynomials, we may refer the interested reader to a recent treatise on the subject of generating functions by Srivastava Manocha [9]. LEMMA 1. (See [7, p. 593].) Let {st~}n =0 be a complex sequence satisfying the following nonhomogeneous recurrence relation: Sn = Sn-1 n t- 2sn-rn + rn (n >= rr~; m, rt E N), (2.5) where {rn} is a given complex sequence. Then the generating function S(t) of the sequence {s,~} is S(t)= so-to+ E tz(sz-sl 1 rz)+g(t) (1-t-2tm) -', (2.6) /=1 where G(t) is the generating function of the sequence {rn}. Our first result on generating functions is contained in Theorem 1 below. THEOREM 1. The generating function of the incomplete generaiized 3acobsthaI numbers J~,,~ ( C No) is given by oo R~ (t) = ~ J;,,,,t ~ 7":0 rn--1 = tm+l Jm,m -~ E tl (Jrn+l,m l=l [(l-t-2t m)(1-t) +l]-'. --Jm+/-1,m)] (1 t) +l 2+lt m) (2.7)
1052 G.B. DJORDJEVIC AND H. M. SRIVASTAVA PROOF. From (1.1) (with x = 1) (2.1), we get ]~,m_j~_],m_2j~_~,m= E n- 1 - (rn- 1)r 2,. r=o r -E(n-2-(m-l)r)2r-E(n-l-m-(m-1)r) r r=o r=0 ~( P ) ( g ) r=o r=o +l -Z(~-~-(m-l/~) ~~-1 = n-l-(m-1)r 2~ - n-2-(m-1)r 2~ z (~-1-(~-1)~)~_~ ~ ~-~-(m-1)~)~ ~ - 1 r=o -E(n--2--(m--1)r)2r-- (n-2--(rn--1)(+l)) ------E[( n-2-(mr 1),')+ (n -- 2 -- 2~r-1 (m --1)r) ] Next, in view of (2.3) (2.4), we set -1- (n-2-(m-1)( l)2+l+e(n-l-(m-1)r)2~ ~=o r ----E(n-l-(m-1) r)2r+l-e(n-l-(m-1)r) r r'=l -1- <n- 2- (m- X) ( + l))2+l = _ (n-l-m-(m-1) 2+1 C~ 1 ~ Iv 1/~) n - 1 - m - m 2 +l (n > m + 1 + in; E No) SO = J~+l... Sl = J~+2,m,'", 8m-1 ~-~ J~n+m,m (2.8) Suppose also that Sn ~ J~+n+l,m" r0 =rl... r,~-l=0 r~ =2+l(n-m+~.-- \ n--m / Then, for the generating function G (t) of the sequence {rn}, we can show that c(t) - 2+lt m (I - t) +l Thus, in view of the above lemma, the generating function S~(t) of the sequence {sn} satisfies the following relationship: 2+ltm S~ (t) (1 -- t -- 2t m) + (1 -- t) +l ~--i -- Jm,m (]g) q- E tl (Jrn+l,m 2+ltm Jrn+l-l,m) + (1 -- t) +l' /=1
Incomplete Generalized Jacobsthal 1053 Hence, we conclude that This completes the proof of Theorem 1. = t sm (t). COROLLARY 1. The incomplete Jacobsthal numbers J~ ( E No) are defined by Jn::Jn'2:z(n--l--r) v=0 the corresponding generating function is given by (2.7) when m = 2, that is, by R2 (t)--t 2+1 [J2 Jr-t(J2+l--,/2) (1--t) +l-2+1t2] [(1--t--2t 2) (1--t)+l] -1. (2.9) 3. INCOMPLETE GENERALIZED JACOBSTHAL-LUCAS NUMBERS For the incomplete generalized Jacobsthal-Lucas numbers 3.... defined by [cf. equation (1.10)] 3n,m' := ~X--"(m2) - _ r(n(m--1) r 2~ ~=0n (m 1)r r (3.1) we now prove the following generating function. THEOREM 2. The generating function of the incomplete generalized gacobsthal-lucas numbers 3~,~' ( E No) is given by oo 7"=0 3,,~ t [( me ) 1 = t m jm-l,,~-t- E tl (jm+/-1,,~--jm+/-2,~) (1 -- t) +l-2+lt m (2 -- t) (3.2) l=l [(1 -- t -- 2t m) (1 -- t)+l] -1 PROOf. First of all, it follows from definition (3.1) that j[n/,q = j... (3.3) n,m. 3n,,~ = 0 (0 <= n < in), (3.4) (l 1,,,~) (3.5) ~mtl,m = ~m+l--l,m ~- Thus, just as in our derivation of (2.8), we can apply (1.2) (1.10) (with x - 1) in order to obtain... n-m+2[n-m+~2+l"" (3.6)
1054 Let G. B. DJORDJEVIC AND H. M. SRIVASTAVA SO : Jm-1... Sl ~-- jrn......, Srn 1 z jra+... Sn ---- jrn+n+l,rn. Suppose also that 7"0 : 7"1 = "'" ~" rm--1 ~" 0 ~'~ - -~-- n- m + 2 (n- m + )2+~ 7~-+-~ ~ -. ~ Then, the generating function G(t) of the sequence {rn} is given by o(t) = 2+lt m (2 -- t) (I - t) +l Hence, the generating function of the sequence {s~} satisfies relation (3.2), which leads us to Theorem 2. COROLLARY 2. For the incomplete 3acobsthal-Lucas numbers 3n,2, the generating function is given by (3.2) when m 2, that is, by W (t): t 2 [(J2-1-- t (j2 -- j2-1)) (1-- t) +l-2+lt 2 (2--t)]. [(Z--t-- ~t 2) (1--t)+l] -1.. 4. TWO FURTHER PAIRS OF INCOMPLETE NUMBERS For a natural number, the incomplete numbers in (1.11) are defined by F~,r~ corresponding to the numbers Fn,m j +3E(n--m l--(m--1)r)2 r (0<< [-~ml]" m, n cn), (4.1) Fr~,m := n,m r+l -- -- ' r=0 where V2,m =.]n~, rn = O, (n<m +in). THEOREM 3. The generating function of the incomplete numbers F~,m ( No) is given by rn+l t Srn (t), where ] S~m (t) = Fm,m + E tz (Fm+/,m - Fm+l-l,m) (1 -- t -- 2tin) -1 l=1 (4.2) 3t TM (1 - t) +l - 2+lt "~ (1 - t + 3P -1) (1 - t -- 2t m) (1 - t) +2 PROOF. Our proof of Theorem 3 is much ain to those of Theorems 1 2 above. Here, we let SO = F~n+l,rn ~ Frn, $1 = F~n+2, m frn-l,m,..., Sin-1 = fr~+m,m : ~m+m-l,m, Sn ~ ~r~-fn+l,m"
Incomplete Generalized Jacobsthal 1055 Suppose also that r0 =~"1 = "'' =rm-1 =-0 rn = 2 +l + 3 2 +l n-m n-m+ " Then, by using the stard method based upon the above lemma, we can prove that ~a 2+lt rn (1 - t + 3t m-l) a (t) = Z ~-tn = t)~+~,~=o (1 - Let S,~(t) be the generating fimction of F~,,~. Then, it follows that s,~ (t) = ~o + t~ +... + s~f ~ +..., tsar(t) = tso + t2sl + ''' q- tns~_l q- ''", 2tmS m (t) - 2t'~so + 2tin+is1 +... + 2t~sn_~, +..., G (t) = ro + rlt +... + rnt ~ +'". The generating function tm+ls~(t) asserted by Theorem 3 would now result easily. COROLLARY 3. For the incomplete numbers Fn,2 defined by (4.1) with rn = 2, the generating function is given by t2+xs~ (t) = t 2+1.([F2+t(F2+l--F2)](1--t)+2+3t2(1--t)+2--2+lt2(1--t+3t2)). (4.3) (1 - t- 2t 2) (1 - t) +~ Finally, the incomplete numbers f~,.~ ( C No) corresponding to the numbers f.~,m in (1.12) are defined by f,,,,:=d~,,+5~(n+l-m-(m-1)r)2r (0_<<_ I-n@]. m, n EN) (4.4) ' r 0 r + l ' " THEOREM 4. The incomplete numbers f~,,~ ( E No) have the following generating function: m-- I 7 W~ (t) = t "*+l fm,m + ~ t ~ (fm+l,m -- f~+l--l,-~)j (1 -- t -- 2t'~) -1 /=1 q_trn+l (5tm (1-- t)+l -- 2+ltm (2 ~ t~ 5tm--1) ) (1 t 2t "~) (1 - t) +2 (4.5) PROOF. Here, we set SO -- ft~q-l,m -- lin,m, 81 = fr~+2,m = frn+l,m, Sm--l,m : ~z +... ~- frn+m-l,rr~,
"" " 1056 G.B. DJORDJEVIC AND H. M. SRIVASTAVA We also suppose that Sn = fr~+n+l,m ---- fm+n,rn. ro = rl = = T'm-1 = 0 7"n = 2+IQTL--TTL-[-~7) 2_+_l(T~--27T~-~-2 -t- ~) +5- n--m n 2m+1 Then, by using the nown method based upon the above lemma, we find that a(t) = 2+lt m (1 - t + 5t m-t) (1 - t) +2 is the generating function of the sequence {rn}. Theorem 4 now follows easily. In its special case when m = 2, Theorem 4 yields the following generating function for the incomplete numbers investigated in [6,7]. COROLLARY 4. The generating function of the incomplete numbers f~,2 is given by (4.5) when m = 2, that is, by W2 (t) = t 2-l-1 (4.6) REFERENCES 1. G.B. Djordjevid, Generalized Jacobsthal polynomials, Fibonacci Quart. 38, 239 243, (2000). 2. G.B. Djordjevid, Derivative sequences of generalized Jacobsthal JacobsthM-Lucas polynomials, Fibonacci Quart. 38, 334-338, (2000). 3. A.F. Horadam, Jacobsthal representation polynomials, Fibonacci Quart. 35, 13~148, (1997). 4. A.F. Horadam, Jacobsthal representation numbers, Fibonacci Quart. 34, 40-54, (1996). 5. R.K. Raina H.M. Srivastava, A class of numbers associated with the Lucas numbers, Mathl. Comput. Modelling 25 (7), 15-22, (1997). 6. P. Filipponi, Incomplete Fibonacci Lucas numbers, Rend. Circ. Mat. Palermo (Set. 2) 45, 37-56, (1996). 7. A. Pintgr H.M. Srivastava, Generating functions of the incomplete Fibonacci Lucas numbers, Rend. Circ. Mat. Palermo (Set. 2) 48, 591-596, (1999). 8. W.-C. Chu V. Vicenti, Funzione generatrice e polinomi incompleti di Fibonacci e Lueas, Boll. Un. Mat. Ital. B (Ser. 8) 6, 289-308, (2003). 9. H.M. Srivastava H.L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley Sons, New Yor, (1984).