Incomplete Generalized Jacobsthal and Jacobsthal-Lucas Numbers

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1 ELSEVIER Available online at MATHEMATICAL AND COMPUTER MODELLING Mathematical Computer Modelling 42 (2005) Incomplete Generalized Jacobsthal Jacobsthal-Lucas Numbers G. B. DJORDJEVI6 Department of Mathematics Faculty of Technology University of Nig YU 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 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 OGP /05/$ - see front 2005 Elsevier Ltd. All rights reserved. doi: /j.mcm Typeset by Aj~S-'I~X

2 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 (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.

3 Incomplete Generalized Jacobsthal 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)

4 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 ~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 m - m 2 +l (n > m 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

5 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 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)

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] 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"

7 Incomplete Generalized Jacobsthal 1055 Suppose also that r0 =~"1 = "'' =rm-1 =-0 rn = 2 +l 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+is t~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~,

8 "" " 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, , (2000). 2. G.B. Djordjevid, Derivative sequences of generalized Jacobsthal JacobsthM-Lucas polynomials, Fibonacci Quart. 38, , (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, , (1999). 8. W.-C. Chu V. Vicenti, Funzione generatrice e polinomi incompleti di Fibonacci e Lueas, Boll. Un. Mat. Ital. B (Ser. 8) 6, , (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).

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