Trigonometric Pseudo Fibonacci Sequence

Transcription

1 Notes on Number Theory and Discrete Mathematics ISSN Vol. 2, 205, No. 3, Trigonometric Pseudo Fibonacci Sequence C. N. Phadte and S. P. Pethe 2 Department of Mathematics, Goa University Taleigao Goa, India dbyte09@gmail.com 2 Dr. Flat No. Premsagar Society Mahatmanagar, Road D-2, Nasik , India Abstract: In this paper, we establish some results about second order non homogeneous recurrence relation containing extended trignometric function. Earlier 4, we proved some properties of recurrence relation g n+2 = g n+ + g n + At n, n = 0,,... with g 0 = 0, g =, where both A 0 and t 0, and also t α, β, where α, β are the roots of x 2 x = 0. Using the properties of generalised circular functions and Elmore s method, we define a new sequence {H n } which is the extension of Pseudo Fibonacci Sequence, given by recurrence relation H n+2 = ph n+ qh n + Rt n N r,0 (t x), where N r,0 (t x) is extended circular function. We state and prove some properties for this extended Pseudo Fibonacci Sequence {H n }. Keywords: Pseudo Fibonacci Sequence, Non-homogeneous recurrence relation. AMS Classification: B39. Introduction The second order linear recurrence sequence {w n } = {w n (a, b; p, q)} was defined by A. F. Horadam 3, given by the relation w n+2 = pw n+ + qw n, n 0 with w 0 = a and w = b. 70

2 This sequence was used to generalise many sequences such as Fibonacci sequence, Lucas sequence, Pell sequence, Pell Lucas sequence, Jacobsthal sequence. In this paper we generalise the Pseudo Fibonacci Sequence defined in 4. We now extend this Sequence by taking the recurrence relation as G n+2 = pg n+ qg n + At n, where both A 0 and t 0, and also t α, β, where α, β are the roots of x 2 px + q = 0. We state and prove some identites for the Sequence {G n }. Further we extend this sequence to get more generalise sequence using circular functions. Definition: We define Generalized Pseudo Fibonacci Sequence {G n } as the sequence satisfying the following non-homogeneous recurrence relation G n+2 = pg n+ qg n + At n, n 0, A 0 and t 0, α, β. (.) with G 0 = a and G = b. Here a, b, p, q are arbitrary integers. First few initial terms of {G n } are given below: G 2 = pb qa + A G 3 = p 2 b pqa qb + A(p + t) G 4 = p 3 b p 2 qa 2pqb + q 2 a + A(p 2 + pt q + t 2 ). 2 Some fundamental identities of G n (x) (i) Binets Formula: Let B = B(t) = Then the explicit Binet form of G n is given by A t 2 pt + q. (2.) where G n = c α n + c 2 β n + Bt n, (2.2) and where c = c 2 = (b aβ) B(t β), (2.3) α β (aα b) B(α t), (2.4) α β α = p + p 2 4q 2 and β = p p 2 4q 2 (2.5) 7

3 are the roots of x 2 px + q = 0. From (2.2) and (2.3), we deduce that, (2b ap) B(2t p) c + c 2 = a B and c c 2 =. α β (ii) Generating function: Generating function G (x) for G n is given as G (x) = G n x n. ( ) Ax 2 = + (a + bx apx). (2.6) ( px + qx 2 ) tx (iii) Exponential Generating function: Exponential Generating function E (x) for G n is given as E (x) = G n x n. n! = c e αx + c 2 e βx + Be tx. (2.7) where c and c 2 are the as in (2.3) and (2.4) respectively. E (x) redues to exponential generating function for Fibonacci Sequence {W n } 6, if B = 0, p =, q = ; a = 0 and b =. G (iv) lim n n G n = α, if t/α <. G (v) lim n n G n k = α k, if t/α <. (vi) G k = (p q ) G n+ qg n b + (p )a A n t. k (vii) n ( ) k G k (viii) = (p q + ) G k t k = ( ) n+ G n+ + b (p + )(a + ( ) n+ G n ) + A( t) n ( ) k t 2k. a( pt) + bt G ( pt + qt 2 n+ t n+ + qg n t n+2 + A n t 2k. ) We prove identity (viii). Proof: By recurrence relation, we have 72

4 and on summing both sides we get Therefore, G n t n = pg n qg n 2 + At n 2 t n G n t n = pg n 2 qg n 3 + At n 3 t n. G 2 t 2 = pg qg 0 + At 0 t 2 n n 2 G k t k = p G k t k+ q G k t k+2 + A k= n n 2 G k t k = G 0 + G t + pt G k t k qt 2 G k t k + A k= Hence, G k t k ( pt + qt 2 ) = G 0 + G t + pt( G 0 G n t n ) qt 2 ( G n t n G n t n ) + A Therefore, G k t k = Hence, ( pt + qt 2 ) G k t k = G 0 ( pt) + G t (G n+ At n )t n+ + qg n t n+2 + A = ( pt + qt 2 ) G 0 ( pt) + G t G n+ t n+ + qg n t n+2 + A n t 2k. a( pt) + bt G ( pt + qt 2 n+ t n+ + qg n t n+2 + A n t 2k. ) This completes the proof. 3 Extension using Elmore s Method We generalise sequence {G n } by applying Elmore s Method as follows. Let E 0 (x) = E (x) = c e αx + c 2 e βx + Be tx, as in (2.7). On differentiating E (x) n times with respect to x, we get a sequence {E n (x)} of E n (x) where E (n) 0 (x) = E n (x) = c α n e αx + c 2 β n e βx + Bt n e tx. (3.) It is interesting to note that E n (x) satisfy the non-homogeneous recurrence relation E n+2 = pe n+ qe n + Ae tx t n. 73

5 4 Extended Circular Functions The Generalized Circular Functions defined by Mikusinsky 2 are as follows. Let: t nr+j N r,j (t) =, j = 0,,..., r ; r, (4.) (nr + j)! M r,j (t) = ( ) r t nr+j, j = 0,,..., r ; r. (4.2) (nr + j)! Observe that N,0 (t) = e t, N 2,0 (t) = cosht, N 2, (t) = sinht M,0 (t)= e t, M 2,0 (t)= cost, M 2, (t)= sint. and One obtains following result by differentiating (4.) term by term with respect to t: N (p) r,j (t) = N r,j p (t), 0 p j N r,r+j p (t), 0 j < j < p r (4.3) In particular, note from (4.3) that N (r) r,0 (t) = N r,0 (t), so that in general Further note that N (nr) r,0 (t) = N r,0 (t), r. (4.4) N r,0 (0) = N (nr) r,0 (0) =. 5 Generalisation of {G n } using Extented Circular Functions Using Generalized Circular Functions and the technique of extension used in 5 we define the sequence {H n (x)} as follows. Let H 0 (x) = c N r,0 (α x) + c 2 N r,0 (β x) + RN r,0 (t x), (5.) where α = α /r, β = β /r and t = t /r, r being the positive integer. α, β are the roots of x 2 px + q = 0. we have α + β = p, αβ = q. (5.2) Now, we define the sequence {H n (x)} successively as follows: H (x) = H (r) 0 (x), H 2 (x) = H (2r) 0 (x), and in general H n (x) = H (nr) 0 (x), where derivatives are with respect to x. Then from (5.) and using (4.4) we get H (x) = c αn r,0 (α x) + c 2 βn r,0 (β x) + BtN r,0 (t x), 74

6 H 2 (x) = c α 2 N r,0 (α x) + c 2 β 2 N r,0 (β x) + Bt 2 N r,0 (t x), and in general H n (x) = c α n N r,0 (α x) + c 2 β n N r,0 (β x) + Bt n N r,0 (t x). (5.3) Theorem. The sequence {H n (x)} satisfies the non-homogeneous recurrence relation H n+2 (x) = ph n+ (x) qh n (x) + At n N r,0 (t x). (5.4) Proof. R.H.S. = p{c α n+ N r,0 (α x) + c 2 β n+ N r,0 (β x) + Bt n+ N r,0 (t x)} q{c α n N r,0 (α x) + c 2 β n N r,0 (β x) + Bt n N r,0 (t x)} + At n N r,0 (t x) = c α n N r,0 (α x){pα q} + c 2 β n N r,0 (β x){pβ q} + t n N r,0 (t x){bt B + A}. (5.5) Using the fact that α and β are the roots of x 2 px + q = 0 and (2.) in (5.2) we get, R.H.S. = c α n+2 N r,0 (α x) + c 2 β n+2 N r,0 (β x) + pt n+2 N r,o (t x). = H n+2 (x). 6 Reduction to Fibonacci Sequence Observe that, for r =, α = α, β = β and N r,0 (t) = N,0 (t) = e t, (5.3) becomes H n (x) = c α n e αx + c 2 β n e βx + Bt n e tx = E n (x). In addition to above particular value of r, p =, q =, a = 0, b =, B = 0 and x = 0, we use (6.), to obtain H n (0) = E n (0) = F n. (6.) References Elmore, M. (967) Fibonacci functions, Fibonacci Quarterly 4, 5, Mikusinski, J. G. (948) Sur les Fonctions, Annales da la Societe Polonaize de Mathematique, 2, Horadam, A. F. (965) Basic Property of a certain Generalized Sequence of Numbers, Fibonacci Quarterly, 3(3),

7 4 Phadte, C. N., & Pethe S. P. (203) On Second Order Non-Homogeneous Recurrence Relation, Annales Mathematicae et Informaticae 4, Pethe, S. P., & Phadte C. N. (993) Generalization of the Fibonacci Sequence, Applications of Fibonacci Numbers, Kluwer Academic Pub., 5, Walton, J. E., & Horadam A. F. (974) Some Aspect of Fibonacci Numbers. The Fibonacci Quarterly, 2(3),