Annales Mathematicae et Informaticae 41 (2013) pp. 20 210 Proceedings of the 1 th International Conference on Fibonacci Numbers and Their Applications Institute of Mathematics and Informatics, Eszterházy Károly College Eger, Hungary, June 2 30, 2012 On second order non-homogeneous recurrence relation a C. N. Phadte, b S. P. Pethe a G.V.M s College of commerce & Eco, Ponda GOA, India b Flat No.1 Premsagar soc., Mahatmanagar D2, Nasik, India Abstract We consider here the sequence g n defined by the non-homogeneous recurrence relation g n+2 = g n+1 +g n + At n, n 0, A 0 and t 0, α, β where α and β are the roots of x 2 x 1 = 0 and g 0 = 0, g 1 = 1. We give some basic properties of g n.then using Elmore s technique and exponential generating function of g n we generalize g n by defining a new sequence G n. We prove that G n satisfies the recurrence relation G n+2 = G n+1 + G n + At n e xt. Using Generalized circular functions we extend the sequence G n further by defining a new sequence Q n(x). We then state and prove its recurrence relation. Finally we make a note that sequences G n(x) and Q n(x) reduce to the standard Fibonacci Sequence for particular values. 1. Introduction The Fibonacci Sequence {F n } is defined by the recurrence relation with F n+2 = F n+1 + F n, n 0 (1.1) F 0 = 0, and F 1 = 1. We consider here a slightly more general non-homogeneous recurrence relation which gives rise to a generalized Fibonacci Sequence which we call The Pseudo Fibonacci Sequence. But before defining this sequence let us state some identities for the Fibonacci Sequence. 20
206 C. N. Phadte, S. P. Pethe 2. Some Identities for {F n } Let α and β be the distinct roots of x 2 x 1=0, with Note that α = (1 + ) 2 Binets formula for {F n } is given by Generating function for {F n } is and β = (1 ). (2.1) 2 α + β = 1, αβ = 1 and α β =. (2.2) F (x) = F n = αn β n. (2.3) F n x n x = (1 x x 2 ). (2.4) Exponential Generating Function for {F n } is given by E(x) = F n x n 3. Elmores Generalisation of {F n } n! = eαx e βx. (2.) Elmore [1] generalized the Fibonacci Sequence {F n } as follows. He takes E 0 (x) = E(x) as in (2.) and then defines E n (x) of the generalized sequence {E n (x)} as the n th derivatives with respect to x of E 0 (x). Thus we see from (2.) that E n (x) = αn e αx β n e βx. Note that E n (0) = αn β n = F n. The Recurrence relation for {E n } is given by E n+2 (x) = E n+1 (x) + E n (x). 4. Definiton of Pseudo Fibonacci Sequence Let t α, β where α, β are as in (2.1). We define the Pseudo Fibonacci Sequence {g n } as the sequence satisfying the following non-homogeneous recurrence relation. g n+2 = g n+1 + g n + At n, n 0, A 0 and t 0, α, β (4.1)
On second order non-homogeneous recurrence relation 207 with g 0 = 0 and g 1 = 1. The few initial terms of {g n } are g 2 = 1 + A, g 3 = 2 + A + At. Note that for A = 0 the above terms reduce to those for {F n }.. Some Identities for {g n } Binet s formula: Let Then g n is given by p = p(t) = A t 2 t 1. (.1) g n = c 1 α n + c 2 β n At n + t 2 t 1 (.2) = c 1 α n + c 2 β n + pt n, (.3) where c 1 = 1 p(t)(t β), (.4) α β p(t)(t α) 1 c 2 =. (.) α β The Generating Function G(x) = g n x n is given by G(x) = Note from (.6) that if A = 0 x + x 2 (A t) (1 xt)(1 x x 2, 1 xt 0. (.6) ) G(x) = x 1 x x 2, which, as in section (2.4), is the generating function for {F n }. The Exponential Generating Function E (x) = g nx n n! is given by E (x) = c 1 e αx + c 2 e βx + pe xt, (.7) where c 1 and c 2 are as in (.4) and (.) respectively. Note that if A=0 we see from (.3), (.4) and (.) that p = 0, c 1 = 1, c 2 = 1, so that E (x) reduces to eαx e βx function for {F n }. which, as in (2.), is the Exponential generating
208 C. N. Phadte, S. P. Pethe 6. Generalization of {g n } by applying Elmore s Method Let E 0(x) = E (x) = c 1 e αx + c 2 e βx + pe xt be the Exponential Generating Function of {g n } as in (.7). Further, let G n (x) of the sequence {G n (x)} be defined as the n th derivative with respect to x of E 0(x), then G n (x) = c 1 α n e αx + c 2 β n e βx + pt n e xt. (6.1) Note that which, in turn, reduces to F n if A = 0. G n (0) = c 1 α n + c 2 β n + pt n = g n, (6.2) Theorem 6.1. The sequence {G n (x)} satisfies the non-homogeneous recurrence relation G n+2 (x) = G n+1 (x) + G n (x) + At n e xt. (6.3) Proof. R.H.S. = c 1 α n+1 e αx + c 2 β n+1 e βx + pt n+1 e xt + c 1 α n e αx + c 2 β n e βx + pt n e xt + At n e xt = c 1 α n e αx (α + 1) + c 2 β n e βx (β + 1) + pt n e xt (t + 1) + p(t 2 t 1)t n e xt. (6.4) Since α and β are the roots of x 2 x 1 = 0, α+1 = α 2 and β + 1 = β 2 so that (6.4) reduces to R.H.S = c 1 α n+2 e αx + c 2 β n+2 e βx + pt n+2 e xt = G n+2 (x). 7. Generalization of Circular Functions The Generalized Circular Functions are defined by Mikusinsky [2] as follows: Let Observe that N r,j = M r,j = t nr+j, j = 0, 1,..., r 1; r 1, (7.1) (nr + j)! ( 1) r t nr+j, j = 0, 1,..., r 1; r 1. (7.2) (nr + j)! N 1,0 (t) = e t, N 2,0 (t) = cosh t, N 2,1 (t) = sinh t, M 1,0 (t) = e t, M 2,0 (t) = cos t, M 2,1 (t) = sin t.
On second order non-homogeneous recurrence relation 209 Differentiating (7.1) term by term it is easily established that { N (p) r,0 (t) = N r,j p (t), 0 p j N r,r+j p (t), 0 j < j < p r (7.3) In particular, note from (7.3) 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 1. (7.4) N r,0 (0) = N (nr) r,0 (0) = 1. 8. Application of Circular functions to generalize {g n } Using Generalized Circular Functions and Pethe-Phadte technique [3] we define the sequence Q n (x) as follows. Let Q 0 (x) = c 1 N r,0 (α x) + c 2 N r,0 (β x) + pn r,0 (t x), (8.1) where α = α 1/r, β = β 1/r and t = t 1/r, r being the positive integer. Now define the sequence {Q n (x)} successively as follows: and in general Q 1 (x) = Q (r) 0 (x), Q 2(x) = Q (2r) 0 (x), Q n (x) = Q (nr) 0 (x), where derivatives are with respect to x. Then from (8.1) and using (7.4) we get Q 1 (x) = c 1 αn r,0 (α x) + c 2 βn r,0 (β x) + ptn r,0 (t x), Q 2 (x) = c 1 α 2 N r,0 (α x) + c 2 β 2 N r,0 (β x) + pt 2 N r,0 (t x), Q n (x) = c 1 α n N r,0 (α x) + c 2 β n N r,0 (β x) + pt n N r,0 (t x). (8.2) Observe that if r = 1, x = 0, A = 0, {Q n (x)} reduces to {F n }. Theorem 8.1. The sequence {G n (x)} satisfies the non-homogeneous recurrence relation Q n+2 (x) = Q n+1 (x) + Q n (x) + At n N r,0 (t x). (8.3)
210 C. N. Phadte, S. P. Pethe Proof. R.H.S. = c 1 α n+1 N r,0 (α x) + c 2 β n+1 N r,0 (β x) + pt n+1 N r,0 (t x) + c 1 α n N r,0 (α x) + c 2 β n N r,0 (β x) + pt n N r,0 (t x) + At n N r,0 (t x) = c 1 α n N r,0 (α x)(α + 1) + c 2 β n N r,0 (β x)(β + 1) + t n N r,0 (t x)(pt + p + A). (8.4) Using the fact that α and β are the roots of x 2 x 1 = 0 and (.1) in (8.4) we get R.H.S. = c 1 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) = Q n+2 (x). It would be an interesting exercise to prove 7 identities for Q n (x) similar to those proved in Pethe-Phadte with respect to P n (x) [3]. Acknowledgments. We would like to thank the referee for their helpful suggestions and comments concerning the presentation of the material. References [1] Elmore M., Fibonacci Functions, Fibonacci Quarterly, (1967), 371 382. [2] Mikusinski J. G., Sur les Fonctions, Annales de la Societe Polonaize de Mathematique, 21(1948), 46 1. [3] Pethe S. P., Phadte C. N., Generalization of the Fibonacci Sequence, Applications of Fibonacci Numbers, Vol., Kluwer Academic Pub. 1993, 46 472. [4] Pethe S. P., Sharma A., Functions Analogous to Completely Convex Functions, Rocky Mountain J. of Mathematics, 3(4), 1973, 91 617.