DEDICATED TO THE MEMORY OF R.J. WEINSHENK 1. INTRODUCTION

Transcription

1 CONVOLUTION TRIANGLES FOR GENERALIZED FIBONACCI NUMBERS VERNER E. HOGGATT, JR. San Jse State Cllege, San Jse, Califrnia DEDICATED TO THE MEMORY OF R.J. WEINSHENK. INTRODUCTION The sequence f integers Fj =, F 2 =, and F 2 = F t + F are called the Fibnacci numbers. The numbers Fj and F 2 are called the starting pair and F 2 = F n + i + F i s c a^ec^ n t n e recurrence relatin. The lng divisin prblem /( - x - x 2 ) yields - X - X' = F t + F 2 x + F 3 x F n + x n + This expressin is called a generating functin fr the Fibnacci numbers. The generating functin yielding = F ( k ) + F, ( k ) x F ( k U n + - " ( - x - x 2 ) K + n + is the generating functin fr the k Fr k =, we get just the Fibnacci numbers. ways t get the cnvlved Fibnacci numbers. cnvlutin f the Fibnacci numbers. We nw shw tw different 2. CONVOLUTION OF SEQUENCES If aj, a 2, a 3, 9 a, and b l 9 b 2, b 3,, b n, are tw sequences, then the cnvlutin f the tw sequences is anther sequence cj, c 2, c 3,, c n, whse terms are calculated as shwn: c j = a j b j c 2 = ajb 2 + a 2 bi c 3 = ajb 3 + a 2 b 2 + a 3 bj c = a^b + a b - + a n b a. b,,., + + a b- n I n 2 n- 3 n-2 k n-k+ n 58

2 CONVOLUTION TRIANGLES M a r. 97 FOR GENERALIZED FIBONACCI NUMBERS 59 This l a s t expressin m a y als be written n c ii = E a k b n - k + l k=l Let us cnvlve the Fibnacci number sequence with itself. These n u m b e r s we call the F i r s t Fibnacci Cnvlutin Sequence: Fi ( ) = F J F J = F 2 ( ) = F i F 2 + F 2 Fi = + = 2 F! ) = F 4 F 3 + F 2 F 2 + F 3 F 4 = = 5 F J P = F j F 4 + F 2 F 3 + F 3 F 2 + F 4 F t = = *P = ke F k F 5 _ k+ = 2 F { } = E F k F 6 _ k + = 38 k=l (i) F * = Si FkVk+i = 7 ' Nw let us "cnvlve" the first Fibnacci cnvlutin sequence with the Fibnacci sequence t get the Secnd Fibnacci Cnvlutin Sequence: F< 2 ) = F i r i ( ) = F 2 ( 2 ) F 3 ( 2 ) = F 2 F l ( ) = F 3 F l ( ) + FiFU = 3 + F 2 F 2 ( > + F ^ ' = 9 F 4 ( 2 ) = F 4 Fi ( ) + F a F 2 ( ) + F 2 F 3 ( ) + F F 4 ( ) = = 22 F 2) = s ^K k+ i = 4l k=l Fi 2) = EFfF 6 _ k + = k=l

3 6 CONVOLUTION TRIANGLES [March The Fibnacci sequence i s btained frm n - X - X 4 = F j + F 2 x + F 3 x F n+ -x + The first Fibnacci cnvlutin sequence i s btained frm ( - x - x 2 ) = Fi,(D + F,(D. 'x + F (). V + + F ( > x» + n+ The secnd Fibnacci cnvlutin sequence is btained frm ( x 2 ),(2) + F,(2). (2). x + Fi 'x i + + F ( 2 > x n +.. n+ These culd all have been btained by lng divisin and cntinued t find a s (k) many F a s desired r ne culd have fund the cnvluted sequence by the methd f this sectin. In the next sectin we shall see yet anther way t find the cnvlved Fibnacci sequences. 3. THE FIBONACCI CONVOLUTION TRIANGLE Suppse ne w r i t e s dwn a clumn f z e r s. T the right and ne space dwn place a ne. T get the elements belw the ne we add the elements ne abve and the ne directly left. This i s, f curse., the rule f frmatin fr Pascal s arithmetic triangle. Such a rule generates a cnvlutin triangle. Next suppse instead we add the ne abve and then diagnally left. Nw the rw sums a r e the Fibnacci numbers. We illustrate: Clumn: " N* 3 \ s l X ^ 3 \ 5 \ 6 6 \ s <N

4 97] FOR GENERALIZED FIBONACCI NUMBERS 6 Hwever, if we add the tw elements abve and diagnally left, we generate the Fibnacci cnvlutin triangle as fllws. Please nte these are the same numbers we gt in Sectin 2 The zer-th clumn are the Fibnacci numbers, F ; the first clumn are the first cnvlutin Fibnacci numbers, F (), etc. 2 T 5 \ ~~\ 2V U-^ ten Clumn: triangle are 4. COLUMN GENERATORS OF CONVOLUTION TRIANGLES It is easily established that the clumn generating functins fr Pascal s g k (x) = ( - x) k+ n= when the triangle is generated nrmally as the expansin f ( + x), n =,, 2, and as we said t d in the first part f Sectin 3. The clumn generatrs becme g k (x) = 2k ( - x) ETi k =,, 2,

5 62 CONVOLUTION TRIANGLES [March if we fllw the secnd set f instructins. These clumn generatrs are such that the elements acrss the rws each are multiplied by the same pwer f x. We make the clumn mve up r dwn by changing the pwer f x inv^e. YXQWLYeratr f the clumn generating functin. If we nw sum k = k= U " X) X / k= X > - X 2 i 2 X^ - X - XT " - x Thus the rw sums acrss the specially psitined (Psitin 2) Pascal triangle are Fibnacci numbers. These are, f curse, the numbers in the zer-th clumn f the Fibnacci cnvlutin triangle. If we multiply the clumn generatrs f Pascal* s triangle by a special set f cefficients, we may btain ther clumns f the Fibnacci cnvlutin triangle. Recall that the k clumn generatr f PascaPs triangle is g k ( x ) = (kv n= \ / k = \k-ri ( - x)* Replace x by - x in the abve t btain t *> ) (l-a)'

6 97] FOR GENERALIZED FIBONACCI NUMBERS 63 while \ - x/ g k^l - x) Z^ \k/ g k / x2 \k+l 2k ( - x - x 2 2>k+l ) th Thus the rw sums a r e the k that i s the clumn generatr we have btained. cnvlutin f the Fibnacci n u m b e r s since We illustrate: Multipliers: (First clumn f Pascal) Rw Sums: * = = 2 = 5 = = 2 = = =3 ) & GO a :tf F 4 g Q ih O rh m u r-4 (Secnd clumn f Multipliers: 3 6 Rw Sums: ) a a a$ a ci ) CO

7 64 CONVOLUTION TRIANGLES [March Thus if we use the numbers in the k clumn f Pascal* s arithmetic triangle (Psitin ) as a set f multipliers with the clumns f Pascal's triangle (Psitin 2), we get rw sums which frm the k Fibnacci sequence. 5. EXTENSION TO GENERALIZED FIBONACCI NUMBERS CONVOLUTION TRIANGLES The Fibnacci numbers are the sums f the rising diagnals f Pascal's triangle which is generated by expanding ( + x). The generalized Fibnacci numbers are defined as the sums f the diagnals f generalized Pascal's t r i - angles which are generated by expanding ( + x + x x "" ). i-2 The sequences can be shwn t satisfy u 4 =, u. = 2 J fr j = 2,3, 8, r, and u, = 7 u,., n >, n+r / JI n+r-j and the generating functins are r - x X A + I X I n= The simplest instance is the Tribnacci sequence, where Tj =, T 2 =, T 3 = 2, and T + 3 = T T - + T, and these sums are the rising diagnal sums f the expansins f ( + x + x 2 ) fr n =,, 2, 3,. The first few terms are,, 2, 4, 7, 3, 24, 44, 8. If we return t ur Fibnacci cnvlutin triangle at the end f Sectin 3, we nte the rw sums are the Tribnacci numbers. the Fibnacci cnvlutin are 3k g k ( x ) = " 2^+ ' ( - X - X 4 ) The clumn generatrs f

8 97] FOR GENERALIZED FIBONACCI NUMBERS 65 where the numbers n each rw in the triangle all multiply the same pwers f x in the clumn generatrs. Adding, we get fv. (_*_) /_*_.. ^ ^ \ l - X - W ^ V. l - X - W - X - X 2 - X 3 which is the Tribnacci sequence generating functin. multipliers [, J as befre, we get If we use the special b±%p>uj (=)- 3k x ( - X - X2 - v3 X 3 ) > k + l and this is the k Tribnacci cnvlutin sequence generatr and the cefficients appear in the k clumn f the Tribnacci cnvlutin triangle. Thus we can btain all the clumns f the Tribnacci cnvlutin triangle frm the Fibnacci cnvlutin triangle in the same way we btained the Fibnacci cnvlutin triangle frm Pascal s arithmetic triangle. We can thus generate a sequence f cnvlutin triangles whse zer-th clumns are the rising diagnal sums taken frm generalized Pascal triangles induced frm expansins f ( + x + x x ). The clumn generatrs fr the r case rk >k ( T - x - x ~ X r - l ) k + St can easily be seen t generate the clumn generatrs fr the (r+) case g k (x) = x(r+l)k ( - X - X^ - - X ) ^ using the preceding methds.

9 66 CONVOLUTION TRIANGLES [March Referring back t the Fibnacci cnvlutin triangle f sectin three, each number in the triangle is the sum f the ne number abve and the number diagnally left. Because the clumn generatrs must bey that law and multiplying by pwers f x s that the prper cefficients will be added, we culd write a recurrence relatin fr the clumn generatrs f the Pascal cnvlutin triangle as fllws: G k (x) = xg k (x) + x 2 G k _ (x) r G k «= ^ % ^ ) By similar reasning, each number f the Fibnacci cnvlutin triangle is the sum f the tw terms abve it and ne diagnally left. Prceeding t clumn generatrs, then, G k (x) = xg k (x) + x 2 G k (x) + x 3 G k _ (x) r G k ( x ) =, - X - X^ G k - i ( x ) 6. THE REVERSE PROCESS One can retrieve the Fibnacci cnvlutin triangle frm the Tribnacci cnvlutin triangle quite simply. First recall k x ( k ) x n = a-d k+ Replace x by -x; then it becmes k/. < - i ) V = /, A xk+ ' ( + x ) n= ' '

10 97] FOR GENERALIZED FIBONACCI NUMBERS 67 r <- -vn+k n x ) x = n= " ( + x ) With these multipliers, a (-D n + k. we can return frm Tribnacci t Fibnacci. Let the clumn generatrs f the Tribnacci case be 3n g n vx) = zrrr» ( - X - X2 - X 3 ) n + and multiplying thrugh by and summing, yields n \, -vn+k C< n= n= \ - X - X 2 - X 3 / + /-, 2 \ k + ( - X - X^) which are the clumn generatrs f the Fibnacci cnvlutin triangle. The st same thing applies, in general, t return frm the (r+) cnvlutin triangle t the r cnvlutin triangle.

11 68 CONVOLUTION TRIANGLES [March 7. SPECIAL PROBLEMS are. Assuming Pascal's triangle in Psitin and the clumn generatrs k Sk (x) = X,k-KL ( - x) ' then shw the rw sums f Pascal's triangle are the pwers f 2. Hint: V* n n inns = Z ^ 2 n= x s that 2. Assuming the Fibnacci cnvlutin triangle has its clumns psitined k gtt = ( - X - X 4 ) then shw the rw sums are the Pell numbers P 4 =, P 2 = 2, P + 2 = 2 P n + i + P. Hint: l _ 2 x - X 2 *-H n= = V L n + /. 3. Shw that the cnvlutin triangle fr the sequence, 3, 3 2,, 3, can be btained frm the cnvlutin triangle fr the sequence, 2, 2 2, 2 3,, 2, using the techniques discussed in this paper. 4. By using the cefficients in J ( ) <-»»* *",

12 97] FOR GENERALIZED FIBONACCI NUMBERS 69 as multipliers, shw hw t get the cnvlutin triangle fr the alternate Fibnacci numbers frm the cnvlutin triangle fr the pwers f three. Hint:! _ 3 i + x 2 = Z F 2 n + l X n n= 5. By using the multipliers frm n= L W J n= n the Fibnacci cnvlutin triangle with clumn generatrs g k (x) k - X - X 4 btain the cnvlr" Hint: A. ;riangle fr every third Fibnacci number sequence, F«,- x n 2 LJ " 3n+l - 4x + x- n = Q Let, 8. OTHER CONVOLUTION TRIANGLES k^r-i-"**--,th be the k cnvlutin f the sequence u(n; p,q), where the sequences u(n; p,q) are the generalized Fibnacci numbers f Harris and Styles []. (Als see [2]..)

13 7 CONVOLUTION TRIANGLES [March Let Jp+q)n ( - x) n( * + n= n= V x > / ( - x ) ^ ( - x) q - x P + q But, U k &-^ Thus, / v t v n d - A '.'M- ( - x) q - x P ^ J. k + l ( - x) q - XP +( ( - x ) ^ ^ ^ ( k ),, n ^r = > u '(n; p,q)x ( - x) q - X P ^ K= (k) and the g (x) are the crrespnding clumn generatrs in the Pascal' s triangle with the first k clumns trimmed ff.

14 97] FOR GENERALIZED FIBONACCI NUMBERS 7 9. REVERSING THE PROCESS, AGAIN If we cnsider the cnvlutin triangles whse clumn generatrs are g n (x) - ( x > ( ( l - x ^ - x ^ J and if we sum these with alternating signs, (-l) n g (x) n- ^ & " ^ - x P + q l + * P + q ( - x) q ( - x) q - x p + q while.( - x ) 4 - x J H k S(^k) ( """ s w = (i->^ Thus, we can recver the clumns f Pascal s triangle frm the abve cnvlutin triangle. This may be extended in many ways. Thus, we can btain the cnvlutin triangles fr all the sequences u(n; p,q) by using multipliers frm Pascal s triangle n the clumn generatrs f Pascal s triangle and taking rw sums. REFERENCES. V. C. Harris and Carlyn C. Styles, "A Generalizatin f Fibnacci Numbers," Fibnacci^^ Vl. 2, N. 4, D e c, 964, pp V. E. Hggatt, J r., M A New Angle n PascaPs Triangle," Fibnacci Quarterly, Vl. 6, N. 4, Oct., 968, pp H. T. Lenard, Jr., f 'Fibnacci and Lucas Identities and Generating Functin," San Jse State Cllege* Master's Thesis, January? David Zeitlin, "On Cnvluted Numbers and Sums," American Mathematical Mnthly, March, 967, pp