On Two New Classes of B-q-bonacci Polynomials

1 2 3 47 6 Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.1 23 11 On Two New Classes of B-q-bonacci Polynomials Suchita Arolkar Department of Mathematics and Statistics Dnyanprassarak Mandal’s College and Research Centre Assagao Bardez Goa 403 507, India [email protected] Yeshwant Shivrai Valaulikar Department of Mathematics Goa University Taleigao Plateau Goa 403 206, India [email protected] Abstract In this paper we define two new classes of polynomials associated with generalized Fibonacci polynomials. We call them h(x)-B-q-bonacci polynomials and incomplete h(x)-B-q-bonacci polynomials. We present some identities for the two classes of polynomials, and the convolution property of h(x)-B-q-bonacci polynomials and its applications. 1 Introduction The Fibonacci sequence, polynomials associated with the Fibonacci sequence, and their extended forms produce interesting and fascinating properties. For details see [8, 14]. Arolkar and Valaulikar introduced the B-tribonacci sequence [2] and B-tribonacci polynomials [1]. 1 The B-tribonacci sequence [2] and B-tribonacci polynomials [1] are further extended to q th order recurrence relations in [5] and [6] respectively. Arolkar and Valaulikar extended and studied the h(x)-Fibonacci polynomials [9] to h(x)-B-tribonacci polynomials [3]. Filipponi [7] introduced the incomplete Fibonacci and Lucas numbers. Ramı́rez [11] studied various identities related to the incomplete k-Fibonacci and k-Lucas numbers. Ramı́rez [13] introduced interesting classes of polynomials, namely, the incomplete h(x)-Fibonacci and h(x)-Lucas polynomials. Arolkar and Valaulikar [4] extended the incomplete h(x)-Fibonacci and h(x)-Lucas polynomials. Ramı́rez and Sirvent [12] defined and studied identities related to the incomplete tribonacci numbers and polynomials. Yilmaz and Taskara [10] obtained identities for the incomplete tribonacci-Lucas numbers and polynomials. The aim of this paper is to extend two classes of polynomials, namely, the h(x)-Btribonacci polynomials [3] and incomplete h(x)-B-tribonacci polynomials of [4] to the q th order relations. We call them the h(x)-B-q-bonacci polynomials and incomplete h(x)-B-qbonacci polynomials. We study some properties of these polynomials. 2 h(x)-B-q-bonacci polynomials We first define the class of h(x)-B-q-bonacci polynomials. Definition 1. Let h(x) be a polynomial with real coefficients. The h(x)-B-q-bonacci polynomials, denoted by (q B)h,n (x), n ∈ N ∪ {0}, q ≥ 2, are defined by (q B)h,n+q−1 (x) = q−1 X (q − 1)r r=0 r! hq−1−r (x)(q B)h,n+q−2−r (x), ∀n ≥ 1, (1) with (q B)h,i (x) = 0, i = 0, 1, 2, 3, . . . , q − 2 and (q B)h,q−1 (x) = 1, where the coefficients of the terms on the right-hand side are the terms of the binomial expansion of (h(x) + 1)q−1 and (q B)h,n (x) is the nth polynomial. For simplicity, henceforth we denote (q B)h,n (x) by (q B)h,n and h(x) by h. We have the following identities for (q B)h,n . (1) The nth term (q B)h,n of (1) is given by q ( B)h,n = ⌋ ⌊ (q−1)(n−(q−1)) q X r=0 ((q − 1) (n − (q − 1) − r))r (q−1)(n−(q−1)−r)−r h , r! n ≥ q − 1, where ⌊·⌋ denotes the floor function. Proof. We prove the identity using induction on n. 2 (2) For n = q − 1, (2) implies q ( B)h,q−1 = 0 X ((q − 1) (−r))r r! r=0 h(q−1)(−r)−r = 1. Hence (2) is true for n = q − 1. Assume that (2) is true for n ≤ m. We divide the result into q cases, namely, m = qk, qk + 1, qk + 2, . . . , qk + (q − 1), for some k ≥ 1. k j . Then Case (i): Let m = qk and t = (q−1)(qk−s−(q−1)) q q−1 X (q − 1)s s=0 q−1 = = s! t X (q − 1)s X ((q − 1) (qk − (q − 1) − (r + s)))r (q−1)(qk+1−(q−1)−(r+s))−(r+s) h s! r! r=0 s=0 q−1 X (q − 1)s  (q − 1)0 0! (q − 1)1 + 1! + = (q−1)k−(q−2) X s! s=0 = hq−1−s (q B)h,qk−s p=s (q−1)k−(q−2) X p=0 (q−1)k−(q−2) X p=1 ((q − 1) (qk − (q − 1) − p))p p! ((q − 1) (qk − (q − 1) − p))p−1 + ··· (p − 1)! (q−1) (q−1)k−(q−2) X (q − 1) (q − 1)! p=q−1 ((q − 1) (qk − (q − 1) − p))p−s (q−1)(qk+1−(q−1)−p)−p h (p − s)! ((q − 1) (qk − (q − 1) − p))p−(q−1)  (q−1)(qk+1−(q−1)−p)−p h (p − (q − 1))! ((q − 1) (qk − (q − 1)))0 0! ! ((q − 1) (qk − (q − 1) − 1))1 (q − 1)1 ((q − 1) (qk − (q − 1) − 1))0 + ··· + + 1! 1! 0! ! (q−1)k−(q−2) q−1 X X (q − 1)s ((q − 1) (qk − (q − 1) − p))p−s + h(q−1)(qk+1−(q−1)−p)−p . s! (p − s)! p=q−1 s=0 3 Therefore, using (q−1)s np−s s=0 s! (p−s)! Pq−1 q−1 X (q − 1)s s! s=0 X p=0 q (n+(q−1))p , p! we have hq−1−s (q B)h,qk−s (q−1)k−(q−2) = = ((q − 1) (qk + 1 − (q − 1) − p))p (q−1)(qk+1−(q−1)−p)−p h p! = ( B)h,qk+1 . Thus, assuming the result for m = qk, we have proved it for m = qk + 1. Similarly, we can prove the other cases. Pq−1 (q−1)s q−1−s q We conclude that s=0 h ( B)h,m−s = (q B)h,m+1 . s! Hence, by induction, the result follows. (2) The sum of the first n + 1 terms of (1) is given by Pq−2 Pq−1 (q−1)r q−1−r q n X (q B)h,n+1 + i=0 h ( B)h,n−i − 1 r=1+i q r! ( B)h,r = , q−1 (h + 1) − 1 r=0 ( h 6= 0, if q is even; provided h 6= 0, −2, if q is odd. Proof. We obtain the result by induction on n. For n = q − 1, (q B)h,q−1 = 1. Also, Pq−2 Pq−1 (q−1)r q−1−r q (q B)h,q + i=0 h ( B)h,q−1−i − 1 r=1+i r! q−1 (h + 1) −1 P r q−1 (q−1) hq−1 + r=1 r! hq−1−r (q B)h,q−1 − 1 = 1. = (h + 1)q−1 − 1 Pq−1 r=0 ( (3) q Thus, (3) is true for n = q − 1. Assume that (3) is true for n ≤ m. Then m+1 X r=0 = = q ( B)h,r = m X (q B)h,r + (q B)h,m+1 r=0 (q B)h,m+1 + (q B)h,m+2 + Pq−2 Pq−1 i=0 (h + Pq−2 Pq−1 i=0 (q−1)r r! 1)q−1 − r=1+i r=1+i (h + hq−1−r (q B)h,m−i − 1 + (q B)h,m+1 1 (q−1)r hq−1−r r! 1)q−1 − 1 (q B)h,m+1−i − 1 . Thus, the result is true for n = m + 1. Hence by induction the result follows. 4 B)h,r = (3) The generating function for (1) is given by (q G(B) )h (z) = 1 , 1 − z(h + z)q−1 (4) provided |(z(h + z)q−1 )| < 1. Proof. Let t = j (q−1)(n−(q−1)) q k ∞ X ( G(B) )h (z) = (q B)h,n z n−(q−1) q n=0 q−2 ∞ X X q n−(q−1) ( B)h,n z + (q B)h,n z n−(q−1) = = n=0 ∞ X n=q−1 (q B)h,n z n−(q−1) , since (q B)h,i = 0, i = 0, 1, 2, 3, . . . , q − 2 n=q−1 ∞ X t X ((q − 1) (n − (q − 1) − r))r (q−1)(n−(q−1)−r)−r n−(q−1) = h z r! n=q−1 r=0   (q − 1)1 q−2 2 q−1 2(q−1) =1+h z+ h + z + ... h 1! = 1 + z(h + z)q−1 + z 2 (h + z)2(q−1) + . . . 1 = , provided |(z(h + z)q−1 )| < 1. q−1 1 − z(h + z) We now obtain the following property. Theorem 2. (Convolution property for (q B)h,n ) For all n ≥ q − 1, we have dh d q (( B)h,n ) = (q − 1) dx dx q−2 X (q − 2)r r=0 r! n+q−2−r h(q−2)−r X (q B)h,i (q B)h,n+q−2−r−i i=0 Proof. Equation (4) implies ∞ X (q B)h,n z n−(q−1) = n=0 5 1 . 1 − z(h + z)q−1 ! . (5) Differentiating both sides with respect to x, we get ∞ X dh d q 1 (( B)h,n ) z n−(q−1) = z(q − 1)(h + z)q−2 2 dx (1 − z(h + z)q−1 ) dx n=0  !2  q−2 ∞ r X X (q − 2) (q−2)−r r+1 dh = (q − 1) h z (q B)h,n z n−(q−1)  r! dx r=0 n=0  !2  q−2 ∞ r X X dh (q − 2) (q−2)−r −2(q−1)+r+1 h z (q B)h,n z n  = (q − 1) r! dx n=0 r=0 q−2 ∞ n dh X (q − 2)r (q−2)−r X X q h = (q − 1) ( B)h,i (q B)h,n−i z n−2(q−1)+r+1 dx r=0 r! n=0 i=0 ! . Comparing the coefficients of z n−(q−1) , we get q−2 X (q − 2)r dh d q (( B)h,n ) = (q − 1) dx dx r=0 r! n+q−2−r h(q−2)−r X (q B)h,i (q B)h,n+q−2−r−i i=0 ! . We now give an application of the convolution property. It is required to prove an identity in the next section. Theorem 3. For n ≥ q − 1, ⌋ ⌊ (q−1)(n−(q−1)) q X r r=0 ((q − 1)(n − (q − 1) − r))r (q−1)(n−(q−1))−qr h r! = (q − 1) (n − (q − 1)) q ( B)h,n q q−2 X h (q − 2)r (q−2)−r − (q − 1) h q r! r=0 n+q−2−r X i=0 (q B)h,i (q B)h,n+q−2−r−i ! . Proof. Equation (2) implies (q B)h,n = ⌋ ⌊ (q−1)(n−(q−1)) q X r=0 ((q − 1)(n − (q − 1) − r))r (q−1)(n−(q−1))−qr h . r! 6 (6) Differentiating both sides with respect to x and simplifying, we get
dh d q ( B)h,n h = ((q − 1)(n − (q − 1))) (q B)h,n dx dx (q−1)(n−(q−1)) ⌊ ⌋ q X dh ((q − 1)(n − (q − 1) − r))r (q−1)(n−(q−1))−qr h . −q r dx r! r=0 Thus, dh dx = ⌋ ⌊ (q−1)(n−(q−1)) q X r r=0 ((q − 1)(n − (q − 1) − r))r (q−1)(n−(q−1))−qr h r! dh h d q (q − 1)(n − (q − 1)) q ( B)h,n − (( B)h,n ) . q dx q dx Hence (5) implies ⌊ (q−1)(n−(q−1)) ⌋ q X r r=0 ((q − 1)(n − (q − 1) − r))r (q−1)(n−(q−1))−qr h r! (q − 1)(n − (q − 1)) q = ( B)h,n q q−2 X (q − 2)r (q−2)−r h h − (q − 1) q r! r=0 3 n+q−2−r X (q B)h,i (q B)h,n+q−2−r−i i=0 ! . Incomplete h(x)-B-q-bonacci Polynomials In this section we define the class of incomplete h(x)-B-q-bonacci polynomials and discuss some of its properties. Definition 4. The incomplete h(x)-B-q-bonacci polynomials are defined by ( q B)lh,n (x) = l X ((q − 1)(n − (q − 1) − r))r r! r=0  (q − 1)(n − (q − 1)) ∀0≤l≤ q 7  h(q−1)(n−(q−1)−r)−r (x), and n ≥ q − 1. (7) ⌋ ⌊ (q−1)(n−(q−1)) q (x) = (q B)h,n (x). Note that (q B)h,n For simplicity, we use (q B)lh,n (x) = (q B)lh,n , (q B)h,n (x) = (q B)h,n and h(x) = h. We prove identities related to the recurrence relation for (q B)lh,n . Theorem 5. The recurrence relation for (q B)lh,n is given by ( q l+q−1 B)h,n+q = q−1 X (q − 1)r r! r=0 h q−1−r q ( B)l+q−1−r h,n+q−1−r ,  (q − 1)(n − q) , ∀n ≥ q. 0≤l≤ q  Proof. Consider, q−1 X (q − 1)r r=0 q−1 = r! hq−1−r (q B)l+q−1−r h,n+q−1−r X (q − 1)r r=0 l+q−1−r r! hq−1−r X ((q − 1)(n + q − 1 − r − (q − 1) − i))i h(q−1)(n+q−1−r−(q−1)−i)−i i! i=0 = q−1 X (q − 1)r r=0 = h l+q−1−r q−1 X (q − 1)r r=0 = r! l+q−1−r X ((q − 1)(n − r − i))i h(q−1)(n−r)−qi i! i=0 X ((q − 1)(n − r − i))i h(q−1)(n+1)−qr−qi i! i=0 r! q−1 X (q − 1)r r=0 q−1−r l+q−1−r X ((q − 1)(n − (r + i)))i h(q−1)(n+1)−q(r+i) . i! i=0 r! Taking j = i + r, we get q−1 X (q − 1)r r=0 q−1 r! l+q−1−r (q B)h,n+q−1−r hq−1−r X (q − 1)r l+q−1 X ((q − 1)(n − j))j−r = h(q−1)(n+1)−qj r! (j − r)! r=0 j=r l+q−1 = X ((q − 1)(n + 1 − j))j h(q−1)(n+1)−qj j! j=0 l+q−1 = (q B)h,n+q . 8 (8) Theorem 6. For s ≥ 1, (q−1)s ( 0≤l≤ j (q−1)(n−s−(q−1)) q q k l+(q−1)s B)h,n+qs = X ((q − 1)s)i l+i (q B)h,n+i hi , i! i=0 (9) . Proof. Follows using induction. j k l+(q−1) l+q−1 Theorem 7. For n ≥ ql+2(q−1) , (q B)h,n+(q−1)+s − h(q−1)s (q B)h,n+q−1 q−1 = q−1 s−1 X X (q − 1)r r! i=0 r=1 l+(q−1)−r h(q−1)s−(q−1)i−r (q B)h,n+(q−1)+i−r . (10) Proof. Follows using induction. The next theorem is related to the sum of incomplete h(x)-B-q-bonacci polynomials ( B)lh,n . q Theorem 8. For all n ≥ q − 1, ⌊ (q−1)(n−(q−1)) ⌋ q X ( q B)lh,n l=0 =    (q − 1) (n − (q − 1)) q − (q − 1) (n − (q − 1)) q + ( B)h,n q q q−2 X h (q − 2)r (q−2)−r + (q − 1) h q r! r=0 Proof. P⌊ (q−1)(n−(q−1)) ⌋ q l=0 n+q−2−r X (q B)h,i (q B)h,n+q−2−r−i i=0 ! . (11) (q B)lh,n ⌋ ⌊ (q−1)(n−(q−1)) q = (q B)0h,n + (q B)1h,n + · · · + (q B)rh,n + · · · + (q B)h,n ((q − 1)(n − (q − 1)))0 (q−1)(n−(q−1)) h 0!   ((q − 1)(n − (q − 1)))0 (q−1)(n−(q−1)) (q − 1)(n − (q − 1) − 1))1 (q−1)(n−(q−1))−q h + h + 0! 1! + ···+   ((q − 1)(n − (q − 1)))0 (q−1)(n−(q−1)) ((q − 1)(n − (q − 1) − r))r (q−1)(n−(q−1))−qr h + ··· + h 0! r! + ··· = 9 ((q − 1)(n − (q − 1)))0 (q−1)(n−(q−1)) h + ··· 0! ((q − 1)(n − (q − 1) − r))r (q−1)(n−(q−1))−qr h + ··· + r! ⌊ (q−1)(n−(q−1))   ⌋ q (q−1)(n−(q−1))  ⌋ (q − 1) n − (q − 1) − ⌊ (q−1)(n−(q−1)) q (q−1)(n−(q−1))−q⌊ ⌋ q h + ⌋)! (⌊ (q−1)(n−(q−1)) q +  ⌊ (q−1)(n−(q−1)) ⌋  q X (q − 1)(n − (q − 1)) ((q − 1)(n − (q − 1) − r))r (q−1)(n−(q−1))−qr ⌋+1−r h ⌊ = q r! r=0 = ⌊ (q−1)(n−(q−1)) ⌋  q X r=0 ⌊ (q−1)(n−(q−1)) q   ((q − 1)(n − (q − 1) − r))r (q−1)(n−(q−1))−qr (q − 1)(n − (q − 1)) +1 h q r! ⌋ ((q − 1)(n − (q − 1) − r))r (q−1)(n−(q−1))−qr h r!  r=0   (q − 1)(n − (q − 1)) (q − 1)(n − (q − 1)) q = +1− ( B)h,n q q ! n+q−2−r q−2 X h (q − 2)r (q−2)−r X q ( B)h,i (q B)h,n+q−2−r−i . h + (q − 1) q r! r=0 i=0 − X r Using (6) of Theorem 3 in Section 2, the result follows. 4 Acknowledgments The authors thank the reviewers/referees for their comments and suggestions that helped to improve the article. References [1] S. Arolkar and Y. S. Valaulikar, Generalized bivariate B-tribonacci and B-tri-Lucas polynomials, Proc. CPMSED-2015, Krishi sanskriti publications, 2015, pp. 10–13. [2] S. Arolkar and Y. S. Valaulikar, On an extension of Fibonacci sequence, Bull. Marathwada Math. Soc. 17 (2016), 1–8. [3] S. Arolkar and Y. S. Valaulikar, h(x)-B-tribonacci and h(x)-B-tri Lucas polynomials, Kyungpook Math. J. 56 (2016), 1125–1133. 10 [4] S. Arolkar and Y. S. Valaulikar, Incomplete h(x)-B-tribonacci polynomials, Turkish Journal of Analysis and Number Theory 4 (2016), 155–158. [5] S. Arolkar and Y. S. Valaulikar, On a B-q bonacci sequence, International Journal of Advances in Mathematics 1 (2017), 1–8. [6] S. Arolkar and Y. S. Valaulikar, Identities involving partial derivatives of bivariate B-q bonacci and B-q Lucas polynomials, preprint. [7] P. Filipponi, Incomplete Fibonacci and Lucas numbers, Rend. Circ. Mat. Palermo 45 (1996), 37–56. [8] T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, 2001. [9] A. Nalli and P. Haukkane, On generalized Fibonacci and Lucas polynomials, Chaos, Solitons & Fractals 42 (2009), 3179–3186. [10] N. Yilmaz and N. Taskara, Incomplete Tribonacci-Lucas numbers and polynomials, Adv. Appl. Clifford Algebr 25 (2015), 741–753. [11] J. L. Ramı́rez, Incomplete k-Fibonacci and k-Lucas numbers, Chinese J. of Math. (2013), Article ID 107145. [12] J. L. Ramı́rez and V. F. Sirvent, Incomplete tribonacci numbers and polynomials, J. Integer Sequences 17 (2014), Article 14.4.2. [13] J. L. Ramı́rez, Incomplete generalized Fibonacci and Lucas polynomials, Hacet. J. Math. Stat. 42 (2015), 363–373. [14] S. Vajda, Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications, Dover, 2008. 2010 Mathematics Subject Classification: Primary 11B39; Secondary 11B83. Keywords: Fibonacci polynomial, h(x)-B-q-bonacci polynomial, incomplete h(x)-B-qbonacci polynomial. Received October 3 2017; revised versions received October 5 2017; March 16 2018; April 4 2018; April 6 2018. Published in Journal of Integer Sequences, May 7 2018. Return to Journal of Integer Sequences home page. 11