On second order non-homogeneous recurrence relation
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1 Annales Mathematicae et Informaticae 41 (2013) pp 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
2 206 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)
3 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
4 208 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.
5 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) = 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)
6 210 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), [2] Mikusinski J. G., Sur les Fonctions, Annales de la Societe Polonaize de Mathematique, 21(1948), [3] Pethe S. P., Phadte C. N., Generalization of the Fibonacci Sequence, Applications of Fibonacci Numbers, Vol., Kluwer Academic Pub. 1993, [4] Pethe S. P., Sharma A., Functions Analogous to Completely Convex Functions, Rocky Mountain J. of Mathematics, 3(4), 1973,
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