Binomial transforms of the modified k-fibonacci-like sequence

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1 Internatonal Journal of Mathematcs and Computer Scence, 14(2019, no. 1, M CS Bnomal transforms of the modfed k-fbonacc-lke sequence Youngwoo Kwon Department of mathematcs Korea Unversty Seoul, Republc of Korea emal: ywkwon81@korea.ac.kr (Receved May 22, 2018, Accepted June 20, 2018 Abstract Ths study apples the bnomal, k-bnomal, rsng k-bnomal and fallng k-bnomal transforms to the modfed k-fbonacc-lke sequence. Also, the Bnet formulas and generatng functons of the above mentoned four transforms are newly found by the recurrence relatons. 1 Introducton The Fbonacc sequence (F n n 0 s defned by the recurrence relaton F n+1 F n +F for n 1 wth the ntal condtons F 0 0 and F 1 1. Some authors have ntroduced new approaches to the Fbonacc sequence. In partcular, Falcón and Plaza [4] ntroduced the k-fbonacc sequence. Defnton 1.1 ([4]. For any postve real number k, the k-fbonacc sequence (F k,n n 0 s defned by recurrence relaton F k,n+1 kf k,n +F k, for n 1 wth the ntal condtons F k,0 0 and F k,1 1. Key words and phrases: modfed k-fbonacc-lke sequence, bnomal, k-bnomal, rsng k-bnomal, fallng k-bnomal. AMS (MOS Subject Classfcatons: 11B39, 11B65. ISSN , 2019,

2 48 Y. Kwon Also, Kwon [6] ntroduced the modfed k-fbonacc-lke sequence. Defnton 1.2 ([6]. For any postve real number k, the modfed k-fbonacclke sequence (M k,n n 0 s defned by the recurrence relaton M k,n+1 km k,n +M k, for n 1 wth the ntal condtons M k,0 2 and M k,1 2. The frst few modfed k-fbonacc-lke numbers are as follows: M k,2 2k +2, M k,3 2k 2 +2k +2, M k,4 2k 3 +2k 2 +4k +2, M k,5 2k 4 +2k 3 +6k 2 +4k +2. Kwon [6] studed denttes between the k-fbonacc sequence and the modfed k-fbonacc-lke sequence. M k,n 2(F k,n +F k, and F k,n 1 M k,n ( 1 2 Spvey and Stel [9] ntroduced varous bnomal transforms. (1 The bnomal transform B of the nteger sequence A {a 0,a 1,a 2,...}, whch s denoted by B(A {b n } and defned by b n a. 0 (2 Thek-bnomaltransformW ofthentegersequencea {a 0,a 1,a 2,...}, whch s denoted by W(A {w n } and defned by w n 0 k n a. (3 Thersngk-bnomaltransformRofthentegersequenceA {a 0,a 1,a 2,...}, whch s denoted by B(A {r n } and defned by r n k a. 0 0

3 Bnomal transforms of the modfed k-fbonacc-lke sequence 49 (4 Thefallngk-bnomaltransformF ofthentegersequencea{a 0,a 1,a 2,...}, whch s denoted by F(A {f n } and defned by f n 0 k n a. Other latest research [1, 5, 10] also examned varous bnomal transforms. Based on those precedng studes, ths study apples the four bnomal transforms namely, bnomal, k-bnomal, rsng k-bnomal and fallng k-bnomal transforms to the modfed k-fbonacc-lke sequence. Ths study also proves ther propertes. 2 The bnomal transform of the modfed k- Fbonacc-lke sequence The bnomal transform of the modfed k-fbonacc-lke sequence (M k,n n 0 s denoted by B k (b k,n n 0 where b k,n 0 M k,. The frst bnomal transforms ndexed n the On-Lne Encyclopeda of Integer Sequences(OEIS [8] are as follows: B 1 {2,4,10,26,68,178,...} : A {0} or A {1} B 2 {2,4,12,40,136,464,...} : A B 3 {2,4,14,58,248,1066,...} B 4 {2,4,16,80,416,2176,...} B 5 {2,4,18,106,652,4034,...} Lemma 2.1. The bnomal transform of the modfed k-fbonacc-lke sequence satsfes the relaton b k,n+1 b k,n 0 M k,+1. Proof. Note that ( ( n 0 1 and n+1 ( n ( + n 1.

4 50 Y. Kwon The dfference of the two consecutve bnomal transforms s the followng: b k,n+1 b k,n n+1 +1 M k, 0 0 [( +1 n 1 M k, +M k,n M k, M k, ] M k, +M k,n+1 M k,n+1 n 0 M k,+1 Note that b k,n+1 n 0 (Mk, +M k,+1. Theorem 2.2. The bnomal transform of the modfed k-fbonacc-lke sequence satsfes the recurrence relaton b k,n+1 (k +2b k,n kb k, for n 1 (2.1 wth the ntal condtons b k,0 2 and b k,1 4. Proof. By Lemma 2.1, snce b k,n+1 n n 0( (Mk, +M k,+1, then we have b k,n+1 M k,0 +M k,1 + (M k, +M k,+1 1 M k,0 +M k,1 + (M k, +km k, +M k, 1 [ 1 (k +1M k,0 +(k (k +1 0 (k +1b k,n + 1 ( ] n M k, M k, 1 +M k,1 km k,0 M k, M k, 1 +M k,1 km k,0 M k, k.

5 Bnomal transforms of the modfed k-fbonacc-lke sequence 51 On the other hand, n the case of ( n 0, we can obtan the followng: b k,n kb k, + M k, k. 1 Based on the above two denttes, ths study draws the below formulas. b k,n+1 (k +1b k,n b k,n kb k,, and so b k,n+1 (k +2b k,n kb k,. Takng the above nto consderaton, ths study further suggests that Bnet s formula for the bnomal transform of the modfed k-fbonacc-lke sequence s: Theorem 2.3. Bnet s formula for the bnomal transform of the modfed k-fbonacc-lke sequence s gven by 1 r2 b k,n 4 rn 1 r2 n 2k r, r 1 r 2 r 1 r 2 where r 1 and r 2 are the roots of the characterstc equaton x 2 (k+2x+k 0, and r 1 > r 2. Proof. Thecharacterstc polynomal equatonofb k,n+1 (k+2b k,n kb k, s x 2 (k + 2x + k 0, whose soluton are r 1 and r 2 wth r 1 > r 2. The general term of the bnomal transform may be expressed n the form, b k,n C 1 r n 1 +C 2 r n 2 for some coeffcents C 1 and C 2. (1 b k,0 C 1 +C 2 2 (2 b k,1 C 1 r 1 +C 2 r 2 4 Then Therefore, C 1 4 2r 2 r 1 r 2 and C 2 2r 1 4 r 1 r 2. b k,n 4 2r 2 r1 n + 2r 1 4 r2 n 4 rn 1 rn 2 2k r 1 r2. r 1 r 2 r 1 r 2 r 1 r 2 r 1 r 2

6 52 Y. Kwon The bnomal transform B k can be seen as the coeffcents of the power seres whch s called the generatng functon. Therefore, f b k (x s the generatng functon, then we can wrte b k (x b k, x b k,0 +b k,1 x+b k,2 x 2 +. And then, 0 (k +2xb k (x (k +2b k,0 x+(k +2b k,1 x 2 +(k +2b k,2 x 3 +, kx 2 b k (x kb k,0 x 2 +kb k,1 x 3 +kb k,2 x 4 +. Snce b k,n+1 (k +2b k,n +kb k, 0, b k,0 2 and b k,1 4, then we have (1 (k +2x+kx 2 b k (x b k,0 +(b k,1 (k +2b k,0 x+(b k,2 (k +2b k,1 +kb k,0 x 2 + b k,0 +(b k,1 (k +2b k,0 x 2+(4 (k +22x 2 2kx. Hence, the generatng functon for the bnomal transform of the modfed k-fbonacc-lke sequence (b k,n n 0 s b k (x 2(1 2kx 1 (k +2x+kx 2. 3 The k-bnomal transform of the modfed k-fbonacc-lke sequence Thek-bnomaltransformofthemodfedk-Fbonacc-lkesequence(M k,n n 0 s denoted by W k (w k,n n 0 where { n 0 k n M k,, for k 0 or n 0; w k,n 0, f k 0 and n 0. The frst k-bnomal transforms are as follows: W 1 {2,4,10,26,68,178,...} : A {0} or A {1} W 2 {2,8,96,320,1088,3712,...} W 3 {2,12,378,1566,6696,28782,...} W 4 {2,16,1024,5120,26624,...} W 5 {2,20,2250,13250,81500,...}

7 Bnomal transforms of the modfed k-fbonacc-lke sequence 53 Note that the 1-bnomal transform W 1 concdes wth the bnomal transform B 1. Note that and so w k,n from Lemma k n M k, k n 0 w k,n+1 k n+1 0 M k, k n b k,n, (M k, +M k,+1 Theorem 3.1. The k-bnomal transform of the modfed k-fbonacc-lke sequence W k (w k,n n 0 satsfes the recurrence relaton w k,n+1 k(k +2w k,n k 3 w k, for n 1 (3.2 wth the ntal condtons w k,0 2 and w k,1 4k. Proof. By Theorem 2.2, we can easly obtan the followng: w k,n+1 k n+1 b k,n+1 k n+1 [(k +2b k,n kb k, ] k n+1 (k +2b k,n k n+2 b k, k(k +2w k,n k 3 w k, Smlarly, Bnet s formula for the k-bnomal transform of the modfed k-fbonacc-lke sequence s the followng: Theorem 3.2. Bnet s formula for the k-bnomal transform of the modfed k-fbonacc-lke sequence s gven by 1 s2 w k,n 4 sn 1 s n 2 2k s, s 1 s 2 s 1 s 2 where s 1 and s 2 are the roots of the characterstc equaton x 2 k(k+2x+ k 3 0, and s 1 > s 2. Proof. The proof s same as that of the bnomal transform, whch s n Theorem 2.3.

8 54 Y. Kwon Smlarly, the generatng functon for the k-bnomal transform of the modfed k-fbonacc-lke sequence s w k (x 2(1 k 2 x 1 k(k +2x+k 3 x 2. 4 The rsng k-bnomal transform of the modfed k-fbonacc-lke sequence The rsng k-bnomal transform of the modfed k-fbonacc-lke sequence (M k,n n 0 s denoted by R k (r k,n n 0 where { n n r k,n 0( k M k,, for k 0 or n 0; 0, f k 0 and n 0. The frst rsng k-bnomal transforms are as follows: R 1 {2,4,10,26,68,178,...} : A {0} or A {1} R 2 {2,6,34,198,1154,6726,...} R 3 {2,8,86,938,10232,...} R 4 {2,10,178,3194,57314,...} R 5 {2,12,322,8682,234092,...} Lemma 4.1. For any nteger n 0 and k 0, r k,n 0 k M k, M k,2n. Proof. Ths dentty concdes wth Theorem 4.10 n [6]. Theorem 4.2. The rsng k-bnomal transform of the modfed k-fbonacclke sequence R k (r k,n n 0 satsfes the recurrence relaton r k,n+1 (k 2 +2r k,n r k, for n 1 (4.3 wth the ntal condtons r k,0 2 and r k,1 2k +2.

9 Bnomal transforms of the modfed k-fbonacc-lke sequence 55 Proof. From the defnton of the modfed k-fbonacc-lke sequence, we obtan M k,2n+2 km k,2n+1 +M k,2n k(km k,2n +M k,2 +M k,2n (k 2 +1M k,2n +km k,2 (k 2 +1M k,2n +M k,2n M k,2n 2 (k 2 +2M k,2n M k,2n 2. By Lemma 4.1, snce r k,n M k,2n, then we have r k,n+1 (k 2 +2r k,n r k,. Smlarly, Bnet s formula for the rsng k-bnomal transform of the modfed k-fbonacc-lke sequence s the followng: Theorem 4.3. Bnet s formula for the rsng k-bnomal transform of the modfed k-fbonacc-lke sequence s gven by 1 t2 r k,n (2k +2 tn 1 tn 2 2 t, t 1 t 2 t 1 t 2 where t 1 and t 2 are the roots of the characterstc equaton x 2 (k 2 +2x+1 0, and t 1 > t 2. Proof. The proof s same as that of the bnomal transform, whch s n Theorem 2.3. Smlarly, the generatng functon for the rsng k-bnomal transform of the modfed k-fbonacc-lke sequence s r k (x 2 (2k2 2k +2x 1 (k 2 +2x+x 2. 5 The fallng k-bnomal transform of the modfed k-fbonacc-lke sequence The fallng k-bnomal transform of the modfed k-fbonacc-lke sequence (M k,n n 0 s denoted by F k (f k,n n 0 where { n n f k,n 0( k n M k,, for k 0 or n 0; 0, f k 0 and n 0.

10 56 Y. Kwon The frst fallng k-bnomal transforms are as follows: F 1 {2,4,10,26,68,178,...} : A {0} or A {1} F 2 {2,6,22,90,386,1686,...} F 3 {2,8,38,206,1208,7370,...} F 4 {2,10,58,386,2834, } F 5 {2,12,82,642,5612, } Lemma 5.1. The fallng k-bnomal transform of the modfed k-fbonacclke sequence satsfes the relaton f k,n+1 kf k,n 0 k n M k,+1. Proof. The proof s smlar to the proof of Lemma 2.1. And, we obtan f k,n+1 kf k,n n+1 +1 k n+1 M k, 0 0 [+1 1 k n+1 M k, +M k,n k n M k, k n+1 M k, ( ] n k n+1 M k, +M k,n+1 M k,n+1 n 0 k n M k,+1. Note that f k,n+1 n 0 (k n+1 M k, +k n M k,+1. Theorem 5.2. The fallng k-bnomal transform of the modfed k-fbonacclke sequence F k (f k,n n 0 satsfes the recurrence relaton f k,n+1 3kf k,n (2k 2 1f k, for n 1 (5.4 wth the ntal condtons f k,0 2 and f k,1 2k +2.

11 Bnomal transforms of the modfed k-fbonacc-lke sequence 57 Proof. By Lemma 5.1, snce f k,n+1 n n 0( (k n+1 M k, +k n M k,+1, then we have f k,n+1 k n (km k, +M k, k 1 2k k n (2kM k, +M k, 1 +k n (km k,0 +M k,1 k n M k, + k n M k, k n M k, 1 +k n (km k,0 +M k,1 k n M k, 1 +k n (km k,0 +M k,1 2kM k,0 2kf k,n + k n M k, 1 +k n (M k,1 km k,0. On the other hand, n the case of ( n 0, we can obtan the followng: 1 kf k,n 2k 2 f k, + k n M k, 1 +k n (M k,1 km k,0 1 [ ( ] 2k 2 f k, f k, k M k, 0 n k M k, +k n (M k,1 km k, ( 2k 2 1 [( ( ] f k, + + k M k, k n (M k,1 km k,0 ( 2k 2 1 f k, + k M k, +k n (M k,1 km k, ( 2k 2 1 f k, + k n M k, 1 +k n (M k,1 km k,0. 1 Based on the above two denttes, ths study draws the below formulas. f k,n+1 2kf k,n kf k,n ( 2k 2 1 f k,,

12 58 Y. Kwon and so f k,n+1 3kf k,n (2k 2 1f k,. Smlarly, Bnet s formula for the fallng k-bnomal transform of the modfed k-fbonacc-lke sequence s the followng: Theorem 5.3. Bnet s formula for the fallng k-bnomal transform of the modfed k-fbonacc-lke sequence s gven by 1 u2 f k,n (2k +2 un 1 u n 2 2 u, u 1 u 2 u 1 u 2 where u 1 and u 2 are the roots of the characterstc equaton x 2 3kx+(2k 2 1 0, and u 1 > u 2. Proof. The proof s same as that of the bnomal transform, whch s n Theorem 2.3. Smlarly, the generatng functon for the fallng k-bnomal transform of the modfed k-fbonacc-lke sequence s 6 Concluson f k (x 2+(2 4kx 1 3kx+(2k 2 1x 2. Ths paper apples the four transforms the bnomal, k-bnomal, rsng k-bnomal and fallng k-bnomal transforms to the modfed k-fbonacclke sequence. Ths study examnes new approaches and methods to obtan the recurrence relatons (2.1, (3.2, (4.3 and (5.4. Ths study, furthermore, nvestgates Bnet s formulas and generatng functons of the abovementoned transforms. 7 Acknowledgments Ths research was supported by Basc Scence Research Program through the Natonal Research Foundaton of Korea(NRF funded by the Mnstry of Educaton(NRF-2018R1D1A1B The author thanks Ms. Juhee Son for proofreadng ths manuscrpt.

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