EXPONENTIAL GENERATING FUNCTIONS FOR FIBONACCI IDENTITIES

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1 EXPONENTIAL GENERATING FUNCTIONS FOR FIBONACCI IDENTITIES C A CHURCH Uiversity of North Carolia, Greesboro, North Carolia ad MARJORIE BICKNELL A. C. Wilco High Schl, Sata Clara, Califoria 1. INTRODUCTION Geeratig fuctios provide a startig poit for a appretice Fiboacci ethusiast who would like to do some research. I the Fiboacci Primer: P a r t VI, Hoggatt ad Lid [l] discuss ordiary geeratig fuctios for idetities relatig Fiboacci ad Lucas umbers. Also, Gould [2] has worked with geeralized geeratig fuctios. Here, we use epoetial geeratig fuctios to establish some Fiboacci ad Lucas idetities. 2. THE EXPONENTIAL FUNCTION AND EXPONENTIAL GENERATING FUNCTIONS The epoetial fuctio e appears i studyig radioactive decay, bacterial growth, compoud iterest, ad probability theory. The trascedetal costat e = is the base for atural logarithms. However, the particular property of e that iterests us is M\ i j., \A v3,, J. V^ The e - l + j? + - g j - + ~ ^ j J ad algebra shows that (2) e ^ - e ^ = ( l - l ) + ^ ^ + i 2 4 ^ + ^ - ^ To relate (2) to Fiboacci umbers, if F is the Fiboacci umber defied by Fi = F 2 = 1, F 1 = F + F f ad if a = (1 + Vl>)/2, p = (1 - N/"5)/2, the it is well 1 c +1-1 kow that (3) F = (<* - p )/(a - j3). Thus, dividig Eq. (2) by (a - p) gives at fit F ^ F 2 t 2 F 8 1? F 4 t^ * a - P 1! 2! 3! 4! 1! ~J! =l 275

2 276 EXPONENTIAL GENEBATING FUNCTIONS FOR FIBONACCI IDENTITIES [Oct. t sice F 0 = 0, we ca add the term F 0 -^ ad write the followig epoetial geeratig fuctio for Fiboacci umbers: (4) v at i3t, e - j = y F L. a - /3 Z-*! A elemetary compaio to the Fiboacci epoetial geeratig fuctio geerates Lucas umber coefficiets* The Lucas umbers are defied by Li = 1, L 2 = 3, L + L - J l * -1 = L -, ad have the property that (5) L = a + /3. If the power series for e ad e are calculated ad the added term-by-term, the result is (6) e + e r = > L -7. Z t! For a ovel use for these elemetary geeratig fuctios, the reader is directed to [3] for a prf that the determiat of eq is e, where Q = ( - 0 J. 3. PROPERTIES OF INFINITE SERIES We list without prf some properties of ifiite series ecessary to our developmet of epoetial geeratig fuctios. Give it follows that (7) ^ t ^ t A(t) = > a i,. ad B(t) = > b ^, L^J! ' Z i! A(t) B(t) = E ( EuKVklT ' \ X ' I A(t)B(-t)=E(L(-l) - k ( kkus \ ^ ' / Thus, if B(t) = e, the b = 1 for all, ad / A(t) e f ^ '-E s(jk)s \ X ' /

3 1973] EXPONENTIAL GENERATING FUNCTIONS FOR FIBONACCI IDENTITIES 277 To help the reader with the double summatio otatio, let The / A(t)B(t) = ~ A ( t T-^ t \ - ^ t > = E If «* B^ = E If \ k = 0 X ' / -(j>&*((!)-(i>)&*((!)-(0-(;>)i + - t 4t 2,. 2t v ^ t(2t) V * 2 t + 1 = 0 + ir + "2r + ' e ' +te = E -V- = E -sr V ( + D 2 t La ( + 1)! (a^v ^! where 1, I is the biomial coefficiet, ( " ) (A =! 1 k I k!( - k)f 4. EXPONENTIAL GENERATING FUNCTIONS FOR FIBONACCI IDENTITIES Geeratig fuctio (4) ad algebraic properties of a ad /3, the rts of = 0, give us a easy way to geerate Fiboacci idetities. Useful algebraic properties of a = (1 + is/t>)/2 ad 0 = (1 - \l"5)/2 iclude:. o?/3 = - 1 a 2 = a + 1 F = (a - /3 )/(a - /3) a - j3 = ^5 a m = a F + F L = a + ^ r j_ o -, HI m - 1 a + p = 1 Take B(t) = e t ad A(t) = (e a t - e^)/(a - 0). (See Eqs. (1) ad (4).) The their series product A(t) ad B(t) gives

4 278 EXPONENTIAL GENERATING FUNCTIONS FOR FIBONACCI IDENTITIES [Oct. (8) t e(a+i)t _ e(p+l)t Qah _ J$h ^ l i L r l k / ' k F, /! \ k = 0 X 7 a - p or - 0 t J zlr F 2! O the left, we used series property (7). O the right, we multiplied A(t) B(t) ad used algebraic properties of a ad p 9 ad the combied our kowledge of Eqs. (1) through (4). Lastly, equatig coefficiets of t /! gives us the idetity X ' If we follow the same steps with B(t) = e~ ad A(t) = (e^ - <s )/{a - p), the \ w / (9) -/3t -at ^, S^T/5 2 ^ (" 1} F T ' The idetity resultig from (9) is 2 <Wi) X ' F. = ( - l ) + 1 F k The techique, the, is this: Take B(t) ad A(t) as simple fuctios i terms of powers of e. Follow algebra as outlied i Eqs. (1) through (7), ad equate coefficiets of - t cp't j3^t t /! The reader is ivited to use B(t) = e" ad A(t) = (e - e H )/(a - p) to derive F O 1 2k = F For a idetity relatig Fiboacci ad Lucas umbers, let A(t) = (e at - e Pt )/(a - p), B(t) = e * + e Pt

5 1973] EXPONENTIAL GENERATING FUNCTIONS FOR FIBONACCI IDENTITIES 279 Sice B(t) is the geeratig fuctio for Lucas umber coefficiets (see Eq. (6)), computig the series product A(t) B(t) gives 00 / \ / / \ \, 2at 281 j i <"> 2 ( 2 ( 0 r * L» - t h - '-^r-- E **. r \ / yieldig. = 2 F. k Similarly, let A(t) = B(t) = (e at - ep t )/if - /3), leadig to / \ / V ' e2at + e2^t _ 2 e t } (11) = ^ l ( 2 L - 2 ) L, E ( k ) F k F - k = l ( 2 \ - 2 > ' at Bt The reader should use A(t) = B(t) = e + e K to derive i, i, To geeralize, try combiatios usig e ad e, such as Aft) = (e a fc - e^ *)/(«- /3), B(t) = e* t + e P l, which geeralize Eq. (10) as follows: o / \ _ m, 0 0m, E l X^fA \t _ e 2a t - e 2 / 3 t _ V t I Z - r l k J F m k L 2 m - m k ) T STTjg ^ F m T \ / By takig Aft) = B(t) = (e - e P )/(a - 0), Eq. (11) becomes

6 280 EXPONENTIAL GENERATING FUNCTIONS FOR FIBONACCI IDENTITIES [Oct. 00 / \ / r HiA 2 V[y/\ F F V - h a *_«f ' Z-/1 Z-* I k / mk m-mk I! 1 a - p \ X ' / \ - m, 0 0m, / m 1/a iiiv, ( l ) = i C e 2 0 t + e 2^ t - 2 e ( a ^ >*) = y ; i ( 2 L - 2 L ) ^. A / 5 m m! m, fim The geeralizatio of (12) foud by A(t) = B(t) = e^ + e^ is 00 / \ 2 / / \ \, m, 0m, * \ 0 m, o/3 m^. / m L O m u (12-) = e 2 " * + e 2^ * + 2e ( } t " ^ = y > ^ < L m + 2 L ) ^ m! The reader should ow eperimet with other simple fuctios ivolvig powers of e. A suggestio is to use some combiatios which lead to hyperbolic sies or cosies, which are defied i terms of e. 5. GENERATING FUNCTIONS FOR MORE GENERALIZED IDENTITIES To get idetities of the type F k+r = F 2+r Hi ote that the r derivative with respect to t of A(t) is Z f +r! so that if A(t) = (e at + e pt )/(a - p) t B(t) = e t, _ ^ / JL^ / \ \ t t r / e <*t _ /3t \ ar e (a+l)t ^ ( p + D t 2-/ ( Z ) ( k ) F k+r I ET = e D t ( 6 a - J ) = 2 ^ I i3 G X \ / ' (13) r " 2 t RT Ph A = a e - P e = V* F L a _ j3 Z-f 2+r!

7 1973] EXPONENTIAL GENERATING FUNCTIONS FOR FIBONACCI IDENTITIES 281 all of which suggests a whole family of idetities; e. g., for 4m, /D^m,, A(t) = (e a l - e P l )/(a - 0), B(t) = e l, \ k = 0, 4rm (a 4 m +l)t 04rm (/3 4m +l)t F I - = a - ]3 e ^ 4mk+r I! a - j3 A 2m, 2 m, 0 2 m u. 2m, 2 m I / D 2 m u 4rm a (a -f/3 )t p4rm a (a +/3 )t e (14) = 2 2 a "- Pp» J 2m 2m+4mr! From the other directio oe ca get idetities of the type (15) <*> m, Dm, (af +F )t (j3f +F )t v, a t t m m - 1 ' ^ m m - 1 ' E ^ t e - e r _ e - e m! ~ a - j3 a - j3,, <*F t j3f t \ /. \ Takig the r derivative of Eq. (15) leads to / \ ( 1 6 ) JLJ F m+rm itt = 2 ^ I 2 - M k I F m - 1 F m F k+rm J f = 0 \ k = 0 / Replace rm by s i Eq. (16) ad compare with Viso's result [4, p. 38], See also H. Leoard [5]. REFERENCES 1. V. E. Hoggatt, J r., ad D. A. Lid, A P r i m e r for the Fiboacci Numbers: P a r t V I, " Fiboacci Quarterly, 5 (1967), pp H. W. Gould, "Geeratig Fuctios for Products of Powers of Fiboacci Numbers," Fiboacci Quarterly, 1 (1963), No. 2, pp Joh L. Brow, solutio of Problem H-20 (proposed by V e r e r E. Hoggatt, J r., ad Charles H. Kig), Fiboacci Quarterly 2 (1964), pp Joh Viso, The Relatio of the Period Modulo to the Rak of Apparitio of m i the Fiboacci Sequece," Fiboacci Quarterly, 1(1963), pp Harold T. Leoard, J r., "Fiboacci ad Lucas Idetities ad Geeratig Fuctios," Master 1 s Thesis, Sa Jose State College, July 1969.