On a Multisection Style Binomial Summation Identity for Fibonacci Numbers

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1 Int J Contep Math Sciences, Vol 9, 04, no 4, HIKARI Ltd, www-hiarico On a Multisection Style Binoial Suation Identity for Fibonacci Nubers Bernhard A Moser Software Copetence Center Hagenberg A-43 Hagenberg, Austria Copyright c 04 Bernhard A Moser This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original wor is properly cited Abstract A lattice path enueration approach is exploited in order to derive a binoial suation forula for the nuber of paths in the lattice 0,,, d with start 0 and n steps This approach reveals ultisection style binoial suation identitities and, particularly, a novel relationship between Fibonacci nubers and the Pascal triangle Matheatics Subect Classification: 05A0, B39 Keywords: Fibonacci nubers, Pascal triangle, lattice path enueration Introduction It is well nown that the generation rule F + F + + F, F, F of the Fibonacci nubers F,,, can be represented in atrix for by F+ F 0 F Equation leads to the relation F+ F F F F 0 for N, fro which a variety of Fibonacci properties can be derived, see, eg, [4, 5, 8]

2 76 Bernhard A Moser This paper is otivated by a different atrix approach, which is otivated by the following observation Let us denote by e i the i-th unit vector and by the vector,,, T The sequence a a, a, a 3, a 4 T N 4 0 given by a 0, 0, 0, 0 T and a + a, a + a + a 3, a + 3 a + a 4, a + 4 a 3 3 iplies a 3 a + a 4 a + a 3, and a + a + a 3 a 3 + a for N Hence, F a 3, F a 3 is equivalent to the atrix equation a + Qa, where Q q i, The eleents of the power Q {,, 3, 4} q i, obey the syetry relation i, q i, q 5 i,5 5 For a proof see the Appendix A Note, that a Q a 0 Particularly, we have a 0 e, a e This, together with 5 iplies 0 F 0 F Q F 0 F 0 0 F 0 F, F 0 F 0 Q F 0 F 0 0 F + 0 F F 0 F F 0 F Independently fro whether is even or odd, 6 iplies that T Q e F + As pointed out by [6, 0], the Toeplitz atrix 4 and its higher diensional versions Q d are adacency atrices of wals on bounded lattices L d 0,,, d [7] exploits the adacency atrix in order to derive enueration forulae for various classes of lattice paths, and relates these forulae to special nubers lie Catalan, Motzin, Fibonacci, Delannoy, Schröder and Pell nubers This paper continues the approach of [7] and focuses on paths with start at 0, length and arbitrary destinations in L d Due to [7] the total nuber of such paths equals F d, T Q d e While [7] stops with a trigonoetric representation, we go a step further by providing a discrete atheatical presentation in ters of binoial coefficients The resulting identity resebles ultisection suation and differs fro the binoial sus for Fibonacci nubers presented in the literature, see, eg, [,, 3]

3 Multisection style binoial suation identity 77 In Section we provide a cobinatorial interpretation of F d, in ters of lattice path enueration Linear recurrence relations, in Section 3, and a trigonoetric representation based on an eigenvalue decoposition, in Section 3, prepare the representation of the nubers F d, as binoial sus in Section 3 Lattice Path Enueration Consider a wal on the lattice L d 0,,, d, starting at 0 und proceeding with steps or A sequence of adissible steps x x,, x {, } generates the path γ x i, i x i,, with 0 i x d for i {,, }, can only be eleent of γ x if either, γ x or, + γ x This eans, that the total nuber of possible visits v, d, v, d #{x,, x {, } x i, r : 0 i r x i d}, 6 at lattice point after steps obeys the recurrence relation 0 < < d v, d v, d + v,+ d, 7 v,0 d v, d, v,d d v,d d Let d N and denote by Q d the d d tridiagonal atrix of type 4 This eans, Q d is tridiagonal with 0 in the ain diagonal and in the upper and sub diagonal Then, the recurrence relation 7 is represented by v d Q d v d, where v d v,0 d,, v,d d T Therefore, we obtain the interpretation that F d, equals the nuber of possible different paths on the lattice L d with start 0 and length, ie, F d, #{x,, x {, } r {,, } : 0 i i x i d} 3 Factorization of Q d Due to [] we get the factorization Q d V d Λ d Vd T, where V d v,, v d is an orthogonal and Λ d is a diagonal atrix with diagonal eleents λ,, λ d with d rows and d coluns given by λ cos v sin, 8,, sin d T

4 78 Bernhard A Moser 8 entails F d, + d+ cos cos 9 Due to cos/5 + 5/4, cos3/5 5/4, cos/ /8 and cos3/0 5 5/8 Equation 9 yields Binet s forula F 4, F / 5 for d 4 Therefore, 9 can be looed at as the analogon of Binet s forula for the sequences F d, As a direct consequence of 9 we obtain li F d, /λ 4/d+ cos /d+, hence, li F d,+ /F d, cos d+ 3 Representation of F d, by Recurrence Relations According to the Cayley-Hailton theore, the atrix Q d satisfies the characteristic polynoial equation, px U d x/ 0, where U d is the Chebychev polynoial of the second ind d/ U d x Bd, x d 0 0 The polynoial 0 is of order d and induces linear recurrence rules F d,+d d 0 α F d,+ with d constant coefficients α The su 9 gives rise to loo for less coplex recurrence forulae of only degree / For this, consider the following eigenvalues λ and eigenvectors ṽ, {,, / }: λ cos, ṽ cos,, cos n d T, where n d d+/ Note, that the vectors ṽ, {,, d+/ }, establish an orthonoral basis Consequently, we regain the su 9 by eans of e T ṼT d Λ dṽ de, where d /, Λ d denotes the diagonal atrix with λ as diagonal eleents, and Ṽ d ṽ,, ṽ d The eigenvalues lead to Chebychev polynoials T n of the first ind [] T n x n n x cos n For odd d, therefore, we obtain n/ 0 Bn, x x n χ dx T dx/ 3

5 Multisection style binoial suation identity 79 as characteristic polynoial of Λ d For even d we use the identity U n cosθ sinn+θ/ sinθ, see [] Because of sind/+/d+ sind//d+ 0, we obtain the characteristic polynoial for even d, d d/, d/ χ dx U dx/ U d x/ B d r/, r/ x d/ r 4 r0 For exaple, 3 and 4 yield the following characteristic polynoials for d,, 3, 4, 5: x 0, x, x 0, x x 0, x 3 0, inducing the recurrence relations : F, 0, F,, F 3, /, F 4, F +, F 5, 3, F 5, Representation by Binoial Coefficients Using 9 we represent 9 in the following way 0 + F d+,+ d + Σ,d + Σ +,d, 6 Σ,d d+ cos d + 7 Due to 3, 4, 7 and 9 we obtain for Equation 6 the following ultisection style suation forulae 8: F d+,+ + d+ q q + d+ q,q odd d+ q d+ q + + q d+ + + d+ q q d+ q,q odd q d+ qd qd q d+ q d+ 8, d, d, d, d

6 80 Bernhard A Moser Figure : Illustration of Pascal triangle identity for d with row sus on the left due to Equations 8 Note, that, in contrast to identities regarding sus of binoial coefficients that result fro series ultisection, the sus under consideration have alternating signs [9] With T Q e 0 and 8 we regain the well-nown binoial identity n 0 Bn, 0, where Bn, refers to the binoial coefficient n over For d we obtain novel Pascal identities See 5 and the Figures, and 3 4 Conclusion We have shown how the factorization of the adacency atrix of a wal on a bounded lattice leads to non-trivial identities in the Pascal triangle This illustrates the potential of a path enueration approach for revealing discrete atheatical identities In future, we will also consider wals on unbounded lattices and as for a cobinatorial characterization of the nuber of wals with a given range A Proof of Property 5 The proof is based on induction Obviously, Q, see 4, satisfies 5 Suppose that Q q i, satisfies 5 for soe > 0, that is q i, q 5 i,5 Fro this, by considering Q + QQ, it follows that q +, q, q 3,4 q + 4,4,

7 Multisection style binoial suation identity 8 Figure : Illustration of Pascal triangle identity for d 3 with row sus on the left due to Equations 8 Figure 3: Illustration of Pascal triangle identity, d 4, with Fibonacci sequence as row sus due to Equation 8

8 8 Bernhard A Moser q +, q, + q 3, q 4,4 + q,4 q + 3,4, q + 3, q, + q 4, q 3,4 + q,4 q +,4, q + 4, q 3, q,4 q +,4, q +, q, q 3,3 q + 4,3, q +, q, + q 3, q 4,3 + q,3 q + 3,3, q + 3, q, + q 4, q 3,3 + q,3 q +,3, and q + 4, q 3, q,3 q +,3 B Representation of 9 in Ters of Binoial Coefficients For the following transforations we need the well-nown trigonoetric forulae + 0 cos θ cos θ 0 cos θ 9 n cosθ sin n+ θ cos n θ sin, θ 0 n 0 sinθ sin n+ θ sin n θ sin 0 θ B Σ,d, Equation 7, in Ters of Binoial Coefficients for Even By eans of 9 Equation 7 can be represented for even by d + Σ,d + 0 ξ, where ξ d+ cos d+ With α, d+ β and applying cosα + β cosα cosβ sinα sinβ we d+ get ξ cos σ, d + + sin σ, d +, where σ, d+ cos and σ, d+ d+ sin Let us consider d odd and let us assue that d +, then d+ d+ d+, which, by setting θ, iplies d+ σ, + sin d+ + θ cos d+ θ sinθ

9 Multisection style binoial suation identity 83 Since sin d+3 θ cos d+ θ sin θ and cos d+ θ cos d+ θ cos θ we get σ, + sin θ cos θ/ sinθ, which iplies σ, Analogously, we get σ, sin d+ + θ sin d+ θ, sinθ which by sin d+3 d+ θ cos θ sin d+ d+ θ, sin θ cos θ sin θ leads to σ, sin θ / sinθ As a result for even and odd d, we, therefore, obtain σ, σ, { d + d+, d + { sin d+ d + sin d+ 0 d + This leads to ξ { cos sin d + d+ d+ d+ cos d + d+ { d + d+ d + d+ hence, Σ,d d + + 0,, d + + 0,, d + d + d+ For even and odd d we, therefore, obtain d + p N : pd + q N : qd + which is equivalent { } to q q,, Observe that / d+ d+ 0 and, therefore, / 0,, d + / / d+ q / qd +

10 84 Bernhard A Moser Fro this it follows that Σ,d d + 0,, d + + / d +, d+ q qd + hence, for even, d odd Σ,d + d + + / d+ q qd + 3 For d even we iediately obtain fro { ξ 0 d + d+ cos d+ d + Therefore, we get for and d even Σ,d d + d ,, d + d+ d+ d+ 4 B Σ,d, Equation 7, in Ters of Binoial Coefficients for Odd By eans of 9 Σ +,d, with even, can be represented by Σ +,d ξ, 5 where ξ d+ cos + d+ With α +, β + and applying cosα + β cosα cosβ d+ d+ sinα sinβ we get ξ cos + σ, d + + sin + σ, d +

11 Multisection style binoial suation identity 85 where σ, d+ cos + and σ, d+ d+ sin + For odd d we obtain d+ { σ, d + +, d + + σ, d+ { cos + d+ sin + d+ d d + + In analogy to Subsection B this leads to { ξ d + + d+ + d+ d + +, hence, Σ +,d + 0,, d ,, d d + + d+ 6 For even d the points + /d + are syetric wrt /, which iplies Σ +,d 0 7 Therefore, let us consider d odd, which iplies + q, q, N, q { d+,, + } d+ Observe that 0,, d This iplies that Σ +,d equals d + + d+ q,q odd + + q d+ hence, by taing 8 into account, Σ +,d d + + d+ q,q odd + + d+ q,q odd + + q d+ 0,, d q d+ 8 +, 9 Acnowledgeents This research is supported by the Austrian COMET progra

12 86 Bernhard A Moser References [] R H Anglin, Proble B-60, The Fibonacci Quarterly, Vol 8, No, 970, p 07 [] M K Azarian, Fibonacci Identities as Binoial Sus, International Journal of Conteporary Matheatical Sciences, Vol 7, No 38 0, [3] M K Azarian, Fibonacci Identities as Binoial Sus, International Journal of Conteporary Matheatical Sciences, Vol 7, No 4 0, [4] SL Basin and VE Hoggatt Jr, A Prier on the Fibonacci Sequence Part II, The Fibonacci Quarterly, Vol 963, 6 68 [5] J L Brenner, June Meeting of the Pacific Northwest Section Lucas Matrix, Aer Math Monthly, Vol 58 95, 0 [6] G M Cicuta, M Contedini and L Molinari, Enueration of siple rando wals and tridiagonal atrices, Journal of Physics A: Matheatical and General, Vol 35, No 5 00, 5 46 [7] S Felsner and D Heldt, Lattice Path Enueration and Toeplitz Matrices, in Proceedings of the 7th international conference on lattice path cobinatorics and applications Lattice 0, 00, Siena, Italy [8] R Honsberger, Matheatical Ges III, Math Assoc Aer, 985, [9] J Riordan, Cobinatorial identities, Wiley, New Yor, 968 [0] RP Stanley, Enuerative Cobinatorics: Volue, Cabridge University Press, New Yor, 0 [] WC Yueh, Eigenvalues of several tridiagonal atrices, Appl Math E- Notes, Vol [] D Zwillinger, CRC standard atheatical tables and forulae, CRC Press, Boca Raton, Florida, 996 Received: January 30, 04