A new class of q-fibonacci polynomials

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1 A new class of -Fibonacci polynomials Johann Cigler Institut für Mathemati Universität Wien A Wien, Österreich Johann.Cigler@univie.ac.at Submitted: Mar 24, 2003; Accepted: May 2, 2003; Published: May 7, 2003 MR Subect Classifications: primary 05A30, 05A15; secondary: 15A15 Abstract We introduce a new -analogue of the Fibonacci polynomials and derive some of its properties. Extra attention is paid to a special case which has some interesting connections with Euler s pentagonal number theorem. 1 Introduction The Fibonacci polynomials f n (x, s are defined by the recursion f n (x, s =xf n 1 (x, s+ sf n 2 (x, s with initial values f 0 (x, s =0,f 1 (x, s = 1. They are given by the explicit n 1 2 formula f n (x, s = x n 1 2 s. L. Carlitz [3 has defined a -analogue, which ( n 1 has been extensively studied (cf. e.g. [6, [2, [8. In [7 I found that F n (x, s = n 1 2 [ n 1 ( +1 2 x n 1 2 s is another natural - analogue which satisfies the simple but rather unusual recursion (2.8. This recursion does not lend itself to the computation of special values. Therefore I was surprised as I n 1 learned that it has been shown in [9 and [13 that F n (1, 1= 2 ( 1 ( 2 [ n 1 has the simple evaluation (3.2. This fact led me to a thorough study of this -analogue via a combinatorial approach based on Morse code seuences. We show that these -Fibonacci polynomials satisfy some other recurrences too, generalize some well-nown facts for ordinary Fibonacci polynomials to this case, derive their generating function and study the special values F n (1, 1andF n(1, 1 which turn out to be intimately connected with Euler s pentagonal number series. Finally we show that the Hanel determinants n 1 det (F i++ (1, 1 can be explicitly evaluated. i, the electronic ournal of combinatorics 10 (2003, #R19 1

2 I want to than H. Prodinger for pointing out to me identity (4.7 in [1 and the paper [4, S.O. Warnaar for some helpful remars and drawing my attention to [9, and R. Chapman and C. Krattenthaler for providing another simple proof of (3.2. Morse code seuences are finite seuences of dots ( and dashes (. We assume that a dot has length 1 and a dash has length 2. The number of all such seuences of total length n 1 is the Fibonacci number F n, which is defined as the seuence of numbers satisfying the recursion F n = F n 1 + F n 2 with initial conditions F 0 =0andF 1 =1. Let MC be the set of all Morse code seuences. We interpret MC as a monoid with respect to concatenation whose unit element is the empty seuence ε. If we write a for a dot and b for a dash then MC consists of all words in a and b. Let P be the corresponding monoid algebra over C, i.e. the algebra of all finite sums λ v v with complex coefficients. An important element of P is the binomial v MC (a + b n = n C n (a, b. (1.1 Here C n (a, b is the sum of all words with dashes and n dots. It is characterized by the boundary values C 0(a, b =δ,0 and C0 n (a, b =a n and each of the two recursions C n+1 (a, b =bc 1 n (a, b+acn (a, b (1.2 or C n+1 (a, b =C 1(a, n bb + C n (a, ba. (1.3 It is clear that the image of C n (a, b under the homomorphism ϕ : P C, defined by ϕ(a =ϕ(b = 1, is the binomial coefficient ( n. Let R be the ring of linear operators on the vector space of polynomials C[x, s. We are interested in multiplication operators with polynomials and the operator ε in R defined by εf(s =f(s. Let now ϕ(a = xε, ϕ(b = sε. (1.4 Then ϕ(aϕ(b = ϕ(bϕ(a. The -binomial theorem (see e.g.[5 states that for n N (A + B n = [ n B A n if AB = BA. Here [ n = ( n 1 ( n +1 1 ( 1 ( 1 denotes the -binomial coefficient or Gaussian polynomial (cf. e.g. [1 or [5. Therefore we get the well-nown result (x + s(x + 2 s (x + n sε n =(xε + sε n = and as special case n [ n ( +1 2 s x n ε n (1.5 the electronic ournal of combinatorics 10 (2003, #R19 2

3 [ n [ n ϕ(c n (a, b = (sε (xε n = ( +1 2 s x n ε n. (1.6 2 A new class of -Fibonacci polynomials a To each Morse code seuence of n letters c 1 c 2...c n we associate the weight i 1+ +i s x n if the i are those indices for which c = b. This means that the weight w i at i is w i (a = x, w i (b = i s and the total weight satisfies w(c 1 c 2 c n =w 1 (c 1 w 2 (c 2 w n (c n. Then it is easy to see that w(c 1 c 2 c n =ϕ(c 1 c 2 c n 1. If the word c 1 c 2 c n has elements b then ϕ(c 1 c 2 c n = i 1+ i s x n ε n (2.1 In [6 we have defined polynomials F n (a, b as the sum of all monomials u MC of length l(u = n 1. There we have called them abstract Fibonacci polynomials. As the referee pointed out, it would be better to call them noncommutative Fibonacci polynomials. By classifying with respect to the first or last letter respectively we see that F n (a, b =af n 1 (a, b+bf n 2 (a, b and also F n (a, b =F n 1 (a, ba + F n 2 (a, bb with initial values F 0 (a, b =0,F 1 (a, b =1. If we apply the homomorphism ϕ we get ϕ(f n (a, b1 = F n (x, s with polynomials F n (x, s. These polynomials are the -Fibonacci polynomials which we will study in this paper. Theorem 2.1 The -Fibonacci polynomials satisfy each of the recurrences F n (x, s =xf n 1 (x, s+sf n 2 (x, s, (2.2 and F n (x, s =xf n 1 (x, s+ n 2 sf n 2 (x, s, (2.3 F n (x, s =xf n 1 (x, s+ n 2 sxf n 3 (x, s+ n 2 s 2 F n 4 (x, s (2.4 and are given by the explicit formula F n (x, s = n 1 2 [ n 1 (+1 2 x n 1 2 s. (2.5 Proof. The first euation follows by considering the first letter of the Fibonacci word. To prove (2.3 we get the first term if the last letter of the Fibonacci word is a. If the last letter is b we need the following result of [6: the electronic ournal of combinatorics 10 (2003, #R19 3

4 Lemma. The noncommutative Fibonacci polynomials are given by n 1 F n (a, b = C n 1 (a, b (2.6 This implies ϕ(f n (a, b = n 1 ϕ(c n 1 (a, b. If ub is a word with n 1 letters (2.1 gives ( ϕ(us = i 1+ +i s x n 2 3 ε n 3 s = i 1 +i s x n 2 3 n 3 s. Therefore ϕ(f n 2 (a, bsε = n 2 sf n 2 (x, s and (2.3 follows. Combining (2.2 and (2.3 we get recursion (2.4. F n(x,s s n From (2.6 and (1.6 we get the explicit formula. We can extend F n (x, s to negative n by assuming (2.2 for all n Z. Thisgives In order to verify this we must show that = x F n+1(x,s + s F n+2(x,s (s n+1 (s n+2 F n (x, s =( 1 n 1 F n(x, s s n. (2.7 or n+1 sf n (x, s = xf n+1 (x, s+f n+2 (x, s, which is (2.3. There is also another recursion of a different ind (cf.[7. Let D be the -differentiation operator defined by Df(x = f(x f(x x x.thenweget F n (x, s =xf n 1 (x, s+( 1sDF n 1 (x, s+sf n 2 (x, s. (2.8 For the proof it suffices to compare coefficients. This leads to the identity ( +1 2 [ n = ( +1 2 [ n 1 +( 1 ( 2 [ n 1 [n ( 2 [ n 1 1 which is easily verified. Here [n denotes [n = n 1 1. From this we may deduce a formula for the -derivative: ( 1DF n (x, s = n 1 F n 1 (x, s F n 1(x, s. We have also a precise -analogue of the Lucas polynomials (cf. [7. It satisfies the same recurrence Luc n (x, s =(x+( 1sDLuc n 1 (x, s+sluc n 2 (x, s but with initial values Luc 0 (x, 2 = 2,Luc 1 (x, 2 = x and is given by the explicit formula the electronic ournal of combinatorics 10 (2003, #R19 4

5 n 2 [n Luc n (x, s = [ n ( 2 [n x n 2 s. These polynomials are related to the -Fibonacci polynomials by the formulas Luc n (x, s = F n+1 (x, s+sf n 1 (x, s and DLuc n (x, s =[nf n (x, s. They also satisfy the rather ugly recursion Luc n+4 (x, s =xluc n+3 (x, s n+1 [2 sluc [n+1 n+2(x, s+ + n+1 [n+3 sxluc [n+1 n+1(x, s+ n+1 [n+3 [n+1 s2 Luc n (x, s. b For the usual -Fibonacci numbers G. Andrews [2 has obtained a bizarre generalization of the well nown formula for Fibonacci numbers F n = 1 2 ( n n For n 1 our -Fibonacci numbers a somewhat less bizarre generalization exists. Let A be the operator x +( 1sD on C[x, s. Since A and the multiplication operator s commute it is obvious that the same formulas hold as in the case = 1 of ordinary Fibonacci polynomials. E.g. F n (x, s = n 1 2 ( n 1 or the Binet formula (( 1 F n (x, s = A 2 +4s which is euivalent with n F n (x, s = 1 2 n 1 s A n A+ n ( A 2 +4s 2 A n A 2 +4s 1, 2 ( n 2+1 A n 2 1 (A 2 +4s 1. Let r(n,, x, s be the polynomial r(n,, x, s =A n 2 1 (A 2 +4s 1. It satisfies the recurrence r(n, +1,x,s=r(n,, x, s+4sr(n 2,,x,s, since A n 2(+1 1 (A 2 +4s +1 = A n 2 3 (A 2 +4s(A 2 +4s = = A n 2 1 (A 2 +4s +4sA (n (A 2 +4s. For the -Fibonacci numbers F n ( = F n ( = 1 2 n 1 n 1 2 n 1 2 [ n 1 ( +1 2 this implies ( n 2+1 r(n,, 1, 1, where r(n,, 1, 1 is a polynomial in with integer coefficients. Of course for =1wegetr(n,, 1, 1 = 5, which does not depend on n. From (2.4 it is easy to derive that the degree deg F n ( as polynomial in is given by deg F 3n ( = n(3n 1 2, deg F 3n+1 ( = n(3n+1 2, deg F 3n+2 ( = 3n(n+1 2. the electronic ournal of combinatorics 10 (2003, #R19 5

6 c The Fibonacci numbers satisfy F 2n = n generalizations and F 2n (x, s = F 2n (x, s = n ( n Fn. This property has two nice [ n (+1 2 s x n F n (x, n s (2.9 n [ n n (+1 2 s x n F n (x, s. (2.10 The first one follows immediately from the formula F 2n (a, b = n C n(a, bf n (a, b for noncommutative Fibonacci polynomials proved in [8. The second one follows from the companion formula F 2n (a, b = n F n (a, bc n (a, b. For the proof observe that ϕ(f n (a, b = n 1 ϕ(c n l 1 (a, b. l=0 Each word u ϕ(c n l 1 l (a, b has the form i 1+ +i l s l x n 2l 1 ε n l 1. Therefore us 1= i 1+ +i l s l x n 2l 1 (n l 1 s = i 1+ +i l ( s l n 2(+1 2 s x n 2l 1. This implies (2.10. d The formula F n+ = ( F n x, s = ( Fn can be generalized to [ ( s x (+1 2 Fn (x, s. (2.11 Observe that some F l (x, s will have negative subscripts l. This formula can be proved by induction starting from F n (x, s =xf n 1(x, s+sf n 2 (x, s. For we have s F n (x, = [ x +1 [ x ( s +1 (+1 2 Fn (x, s = ( s +1 (+1 2 (xfn 1 (x, s+sf n 2 (x, s = the electronic ournal of combinatorics 10 (2003, #R19 6

7 [ x =1 = +1 x +1 [ ( s +1 (+1 2 Fn 1 (x, s+ 1 x +1 [ +1 ( s +1 ( F n 1 (x, s = ( s +1 (+1 2 Fn 1 (x, s. Together with (2.10 this implies F 2n (x, s = [ n n ( +1 2 s x n [ x ( s (+1 2 Fn 2 (x, s = = [ n [ n (+1 2 +( +1 2 s + x n F n 2 (x, s., If in (2.11 we replace n by 2n, s by n s and by n we get again (2.9. In the other direction we have n 1 F n (x, s = ( 1 (s x 1 F n+1 (x, s. (2.12 This follows from the recursion (2.2 by induction. xf 1 (x, s =F 2 (x, s. If it is true for n 1weget x n 1 ( 1 (s x 1 F n+1 (x, s = = F n+1 (x, s s n 2 ( 1 (s x 1 F n (x, s = = F n+1 (x, s sf n 1 (x, s =xf n (x, s. It is true for n = 1 because e Another interesting formula is x F n (x, s= [ ( 1 2 s F n+ 2 (x, s. (2.13 We prove it by induction. The formula holds for =1andalln. If it is true for then we get x +1 F n (x, +1 s= [ ( 1 2 (s xf n+ 2 (x, +1 s= = [ ( 1 2 (s (F n++1 2 (x, s +1 sf n+ 1 2 (x, +1 s = = ([ [ ( s F 1 n++1 2 (x, s= 0 = [ ( s F n++1 2 (x, s. 0 For n =1wegetx = [ ( 1 2 s F +1 2 (x, s. the electronic ournal of combinatorics 10 (2003, #R19 7

8 This may be rewritten in the form +1 x 2 [ = ( 1 2 s F +1 2 (x, s+ 1 2 [ + ( s +1 F 2 1 (x, s. For n = 0 this reduces to [ ( 1 For 1 this is nontrivial identity. 1 2 s ( 2 F 2 (x, s 2 F 2 (x, s =0. f Define b n, by b 0, =[ =0,b 1, =[ = 0, the recursion b n+1, = b n, + b n, 1 for n 2,andb 2n+1,n+1 =0. Then we get x n = 2 n( 1 (s 2( 2 bn, F n+1 2 ( s. (2.14 Obviously (2.14 holds for n = 1. So we may assume that it holds for n. Since b n, does not depend on s the same formula holds for s in place of s. From (2.2 we therefore get x ( 1 ( 2 s 2( 2 bn, F n+1 2 ( s = = 2 n n 2 ( 1 ( 2 s 2( 2 bn, F n+2 2 ( s n 2 ( 1 ( 2 s 2( 2 bn, +1 sf n 2 ( +1 s= = n 2 ( 1 ( 2 s 2( 2 bn, F n+2 2 ( s+ n ( 1 (s 2( 2 bn, F n 2+2 ( s= = =1 ( 1 (s 2( 2 bn+1, F n+2 2 ( s. 2 n+1 There is no explicit formula for b n,. By comparing coefficients we get from (2.14 the following characterization [ ( 1 ( 2 bn, =[ =0. n Let now a n,n 2 = b n, or a n, = b n, n. Then we get a n, = n 2 a n 1, 1 + a n 1,+1 2 with a n, 1 = 0 for all n and initial values a 0, ( =[ =0anda 1, =[ =1. For = 1 it is well nown that a 2n,0 = 1 2n n+1 n is a Catalan number. Therefore a2n,0 is a -analogue of the Catalan numbers (cf. e.g. [6. the electronic ournal of combinatorics 10 (2003, #R19 8

9 3 Special values It turns out that for x =1,s= 1 orx =1,s= 1 we get very simple results. Theorem 3.1 Let r( = (3 1 2 denote a pentagonal number. Then F 3n (1, 1 = n = n+1 ( 1 r(,f 3n+1 (1, 1 = F 3n+2 (1, 1 = n ( 1 r(. (3.1 = n Remar. This is a curious result. It means that the values F n (1, 1 are ust the partial sums of Euler s pentagonal number series (1 n = = n 1 = ( 1 (3 1 2 = Z( 1 r(. = Theorem 3.2 For x =1,s= 1 we get F 3n (1, 1 =0,F 3n+1(1, 1 =( 1n r(n,f 3n+2 (1, 1 =( 1n r( n. (3.2 Remar. Formula (3.2 has been proved by Shalosh B. Ehad and D. Zeilberger [14 with a computer proof and by S. O. Warnaar [13 as a special case of a cubic summation formula in the form n 2 ( 1 ( 2 [ { n = ( 1 n 3 n(n 1 6,n 2 (mod 3 0,n 2 (mod 3. I learned from C. Krattenthaler that the cubic summation formula referred to by S. O. Warnaar is formula (5.22 of [10, and that from the hypergeometric view this seems to be the most natural approach to (3.2. Since I am not familiar with bibasic series I want to give some elementary proofs which do not need much theory. From (2.11 and (2.12 we see that Theorem 3.1 and Theorem 3.2 are euivalent. So it suffices to prove one of them. a We first give a direct proof of (3.2. For x =1,s= 1 formula (2.4 reduces to F n (1, 1=F n 1(1, 1 n 3 F n 3 (1, 1 +n 4 F n 4 (1, 1. From this we get F n+3 (1, 1 =(1 n F n (1, 1 +n 3 F n 3 (1, 1. (3.3 Since F 4 (1, 1 = F 1(1, 1,F 5(1, 1 = 2 F 2 (1, 1 weevenget F n+3 (1, 1 = n F n (1, 1. (3.4 the electronic ournal of combinatorics 10 (2003, #R19 9

10 This implies immediately F 3n (1, 1=0, F 3n+1 (1, 1 3i 2 =( 1n i=1 =( 1 n r(n, F 3n+2 (1, 1 3i 1 =( 1n i=1 =( 1 n r( n. np np b But there is a more illuminating combinatorial proof of (3.1, which is an adaptation of Franlin s proof of the pentagonal number theorem to this case (cf. [1 or [4. Let u beawordoflengthn 1,u a n 1. Then we can write it in the form u = a i 1 zb l a where i, l 1 are chosen to be maximal. In case i l we call the first b from the left b 1 and the i-th b from the right b 2. If b 1 b 2 we call u good of type b and define ψ(u by changing b 1 to a and b 2 to ab. Thenψ(u has the same length and the same weight as u but the number of b s is one less. If b 1 = b 2 we call u bad of type b. Itiseasytoseethat u is bad of type b if and only if u = a i 1 b i a,i 1. Then ψ(u has the form u = a i 1 zb l a with i>l 1. If u has this form let a 1 be the l-th element a from the left and a 2 the a immediately in front of b l.ifa 1 a 2 we call u good of type a and define ψ(u as follows. We change a 1 to b and drop a 2. Then it is clear that ψ maps the set of good words of type a biectivelyontothesetofgoodwordsoftypeb in such a way that words with an even number of b s are mapped onto words with an odd number of b s and vice versa. If a 1 = a 2 we call u bad of type a. Thenu is bad of type a if and ony if u = a l b l a for some l 1. Let for example n =10andu = ababba, thenu = ababba = ab 1 ab 2 ba and therefore ψ(u =aaaabba. The weight of these words is 11. If u = ababaa then u is good of type a with length 8 and weight 6, u = a 1 ba 2 ba. Here we get ψ(u =ψ(a 1 ba 2 ba =bbba. The weights of the bad words are w(a i 1 b i a = i(3i 1 and w(a l b l a = l(3l For a Morse code seuence of length 3n 1 the bad word with maximal weight is a n 1 b n, for the length 3n and 3n + 1 the corresponding words are a n b n and a n b n a. The weights of these words are r(n andr( n respectively. Therefore we get the desired result F 3n (1, 1 = n ( 1 r(,f 3n+1 (1, 1 = F 3n+2 (1, 1 = n ( 1 r(. = n+1 = n c Another proof of (3.2 has been communicated to me independently both by R. Chapman and C. Krattenthaler, which I reproduce in my notation. From (1 + s(1 + s (1 + n 1 s= n ( 2 s (cf.(1.5 follows [ n 1 ( 2 = i 1+ +i. 0 i 1 <i 2 < <i <n 1 In the sum ( 1 [ n 1 ( 2 all terms i 1 + i with i1 = 0 cancel, because ( 1 i 1+ i + ( 1 1 i 2+ i = 0. The only terms remaining are those with i1 > 0andi + = n 2. [ n the electronic ournal of combinatorics 10 (2003, #R19 10

11 The above proof or the original proof of Franlin lets i + invariant and gives immediately F 3n (1, 1 =0,F 3n+1(1, 1 =w(an 1 b n =( 1 n r(n, F 3n+2 (1, 1 =w(an 1 b n a=( 1 n r( n. 4 A generating function Now we want to calculate the generating function F (x, s, z = n 0 F n+1 (x, sz n of the -Fibonacci polynomials. We have F (x, s, z = F n+1 (x, sz n =1+ xf n (x, sz n + sf n 1 (x, sz n = n 0 n 1 n 1 =1+xzF (x, s, z+sz 2 F (x, s, z =1+xzεF (x, s, z+sz 2 εf (x, s, z i.e. F (x, s, z =1+(xz + sz 2 F (x, s, z (4.1 or (1 xzε z 2 sεf (x, s, z =1 and therefore F (x, s, z = n 0(xzε + z 2 sε n 1. (4.2 Now we have zε sz 2 ε = sz 2 ε zε. Therefore the -binomial theorem gives (xzε + z 2 sε n 1= n [ n (z 2 sε (xz n 1= n [ n ( +1 2 s x n z n+. Using the well nown formula 1 = [ n+ (1 z(1 z (1 z z n this implies F (x, s, z = n [ n x n (+1 2 z n+ s = n = (+1 0 i.e. 2 s z 2 n 0 [ n+ F (x, s, z = n 0 On the other hand we have x n z n = 0 (+1 2 s z 2 (1 xz(1 xz (1 xz, F n+1 (x, sz n = 0 From (4.3 and (4.2 we get 0 (+1 2 s z 2 (1 xz(1 xz (1 xz. (4.3 (xzε + z 2 sε n 1=z n (x + sz(x + 2 sz (x + n sz. (4.4 (+1 2 s z 2 (1 xz(1 xz (1 xz = z n (x + sz(x + 2 sz (x + n sz n 0 (4.5 the electronic ournal of combinatorics 10 (2003, #R19 11

12 If we set x =1,a= sz this reduces to ( 1 (az ( 2 (1 z(1 z (1 z = z n (1 a(1 a (1 n 1 a. ( From (4.6 and (4.3 we also get z (1 z(1 z (1 1 z= F +1 (1, 1 z. 0 0 Letting a = z this gives z (1 z(1 z (1 1 z= 0( 1 ( r( z 3 + r( z 3+1. (4.7 0 Remar. This identity which is euivalent with (3.2 is well nown (c.f. [1, p. 29, Example 10, [14, p. 951 or [4. In our notation the usual proof may be formulated in a very convincing way: The recursion (2.2 implies identity (4.1 for the generating function. In the same way the other recursion (2.3 gives (1 xzf (x, s, z =1+sz 2 F (x, s,z. (4.8 From (4.8 for x =1,s= 1 weget (1 zf (1, 1,z=1 z 2 F (1, 1,z and from (4.1 for x =1,s= 1 we get in the same way F (1, 1,z =1+z(1 zf (1, 1,z. This implies F (1, 1,z=1+z z3 F (1, 1,z, which is euivalent with ( F n+3 1, 1 ( = n F n 1, 1 (4.9 and therefore with (3.2. Formulae (3.2 and (3.1 are generalizations of the trivial fact that the recursion f n = f n 1 f n 2,f 0 =0,f 1 = 1 has the periodic solution 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, with period 6. For = 1 formulas (4.6 and (4.7 reduce to 1 = ( 1 z2 = z (1 z = 1+z = 1 z+z 2 (1 z +1 1+z 3 =1+z z 3 z 4 + z 6 + z 7 ++ the electronic ournal of combinatorics 10 (2003, #R19 12

13 5 A Hanel determinant Theorem 5.1 Let β(n, = n(n+ 2(n+ 3 and α(n, =β(n, if (n 2 1 (mod 3,α(n, =0 6 if n 2 1 (mod 6 and α(n, =β(n, + n if n 2 4 (mod 6. 6 ( n 1 Then the Hanel determinant d n, =det F i++ (1, 1/ satisfies i, n 2 d n, =( 1 ε(n 2 α(n, ( i 1 n 2i (5.1 where the sign ε(n is given by ε(2n =( 1 n,ε(6n +1=0,ε(6n +3=1,ε(6n +5= 1. Proof. We use the condensation method as outlined in Krattenthaler [11 or Zeilberger [15. In a similar context this method has been used in [12, which inspired our proof. Euation [11 (2.16 gives in this case euation [12 (2.3, which states that i=1 d n,+1 d n. 1 d n 1,+1 d n+1, 1 =(d n, 2 (5.2 holds for all n,. This identity lends itself to prove ( (5.1 by induction with respect to n for all. From F n+3 (1, 1 = n F n 1, 1 it is trivial that d n,+3 =( 1 n 2(n 2+n d n,. So it would suffice to prove (5.2 only for =0, 1, 2. But the general case is not harder to prove. From F 3n (1, 1 =0and F n (1, 1 =( 1 n 1 3 (n 1(n 2 6 for n 0 (mod 3 we conclude that d 1, = F (1, 1 =( 1 ε(1 2 α(1,. Inthesamewayweseebydirectcalculationthat d 2, =( 1 ε(2 2 α(2,. Now from (5.2 we immediately see that d n, for all n 3andall is uniuely determined. Thus all that remains is to verify formula (5.2 if we set n d n, =( 1 ε(n 2 α(n, 2 ( i 1 n 2i. i=1 This reduces to ε(n 2 2ε(n 2 +2 α(n,+1+α(n, 1 = = ε(n 2 3ε(n 2 +3 α(n 1,+1+α(n+1, 1 + 2α(n, ε(n 2 2 (5.3 for n 1 (mod 2 and to the electronic ournal of combinatorics 10 (2003, #R19 13

14 ε(n 2 2ε(n 2 +2 α(n,+1+α(n, 1 = = ε(n 2 3ε(n 2 +3 α(n 1,+1+α(n+1, 1 ( n α(n, ε(n 2 2 (5.4 for n 0 (mod 2. We consider several cases, where the resulting identity can easily be verified: 1 n 2 +1 (mod6: This implies d n, =0and (n 1 2( +1 (n +1 2( 1 4 (mod 6. Thus (5.3 reduces to β(n,+1+β(n, 1 n 1 β(n 1,+1+β(n+1, n+1 6 =0. 2n 2 +3 (mod6: Here (5.3 reduces to β(n 1,+1+β(n+1, 1 = 2β(n,. 3 The same is true for n 2 +5 (mod6. 4n 2 +4 (mod6: Here (5.4 reduces to β(n,+1+β(n, 1 = 2β(n,+ n 3. 5 For n 2 (mod 6 we have n 2( +1 4 (mod 6 and (5.4 reduces to (β(n,+1+ n 6 +β(n, 1 ( n 2 1 (β(n+1, 1+β(n 1,+1 = 2β(n,. 6 For n (mod 6 we have n 2( 1 4 (mod6and(5.4reducesto β(n,+1+(β(n, 1+ n 6 ( n 2 1 β(n+1, 1+β(n 1,+1 = 2β(n,. References [1 G. E. Andrews, The Theory of Partitions, Addison-Wesley 1976 [2 G. E. Andrews, Fibonacci numbers and Rogers-Ramanuan identities, To appear in Fibonacci Quaterly [3 L. Carlitz, Fibonacci notes 4, Fibonacci Quaterly 12(1975, [4 R. Chapman, Franlin s argument proves an identity of Zagier, Electron J. Comb. 7(2000, R 54 [5 J. Cigler, Operatormethoden für -Identitäten, Mh. Math. 88 (1979, [6 J. Cigler, -Fibonacci polynomials, Fibonacci Quarterly 41(2003, the electronic ournal of combinatorics 10 (2003, #R19 14

15 [7 J. Cigler, Einige -Analoga der Lucas- und Fibonacci-Polynome, to appear in Sitzungsberichte Österr. Aad. Wiss. [8 J. Cigler, Some algebraic aspects of Morse code seuences, to appear in DMTCS 2003 [9 Shalosh B. Ehad, D. Zeilberger, The number of solutions of X 2 = 0 in triangular matrices over GF(, Electron. J. Comb. 3(1996, R2 [10 G. Gasper, Summation, transformation, and expansion formulas for bibasic series, Trans. Amer. Math. Soc. 312(1989, [11 C. Krattenthaler, Advanced determinant calculus, Seminaire Lotharingien de Combinatoire B42, (1999 [12 U. Tamm, Some aspects of Hanel matrices in coding theory and combinatorics, Electron. J. Comb. 8(2001, A1 [13 S. O. Warnaar, -hypergeometric proofs of polynomial analogues of the triple product identity, Lebesgue s identity and Euler s pentagonal number theorem, arxiv:math. CO/ , 2002 [14 D. Zagier, Vassiliev invariants and a strange identity related to the Dedeind eta-function, Topology 40(2001, [15 D. Zeilberger, Dodgsons determinant-evaluation rule proved by TWO-TIMING MEN and WOMEN, Electron. J. Comb. 4(2, 1997, R22 the electronic ournal of combinatorics 10 (2003, #R19 15