New Hypergeometric Connection Formulae Between Fibonacci And

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In order to study Fibonacci polynomials, one may consider linking Fibonacci polynomials to other well-studied polynomials, such as Chebyshev polynomials. Chebyshev polynomials of the ﬁrst and second kinds, Tn(x) and Un(x) respectively, are subfamilies of the larger class of Jacobi polynomials. They are of crucial importance from both the theoretical and practical points of view. The interested reader in these polynomials may consult [16].

Given two sets of polynomials Pi i 0 and Qj j 0, the so-called connection { } ≥ { } ≥ problem between these polynomials is to determine the coeﬃcients Ai,j in the expression

Pi(x)= Ai,j Qj(x). j=0 X

The connection coefficients Ai,j play an important role in many problems in pure and applied mathematics and in mathematical physics. The problem of finding connection coefficients between two sets of orthogonal polynomials has been inves- tigated by many authors, see for instance [3,6,7,10,15,17,18]. In fact, most of the formulae for the connection coefficients between orthogonal polynomials are given in terms of terminating hypergeometric series of various types. For example, the connection formula of Jacobi-Jacobi polynomials is given in terms of a terminating hypergeometric series of the type 3F2(1), see [7]. In this article, we solve the connection problem between Fibonacci polynomials and the orthogonal polynomials Tn(x) and Un(x). We exhibit the Chebyshev expression form of Fibonacci polynomials Fn(x). In fact, the connection coefficients turn out to be terminating hypergeometric series of type 2F1(λ) where λ is either 4 or 1/4. Furthermore, we tackle the inverse connection coefficients problem. − − In other words, we express Tn(x) and Un(x) in terms of Fn(x). Again, the latter coefficients involve the terminating hypergeometric series 2F1(λ). Based on the new connection formulae that we derive, we display several identities satisfied by Fibonacci numbers, and we evaluate some finite sums. More- over, we find relations between terminating hypergeometric series of type 2F1(λ) for certain values of λ and specific parameters. Also, some identities involving the derivatives sequences of Fibonacci numbers are given. Finally, we express weighted definite integrals of products of Fibonacci and Chebyshev polynomials, and products of Fibonacci polynomials as sums involving hypergeometric series. FIBONACCI AND CHEBYSHEV POLYNOMIALS 3

2. Some relevant properties of Fibonacci and Chebyshev polynomials

In this section, we recall the properties of Fibonacci and Chebyshev polynomials that we are going to use throughout the note.

2.1. Fibonacci polynomials. Fibonacci polynomials are generated via the following recurrence relation

F (x)= x F (x)+ F (x), n 0, where F (x)=0, F (x)=1. n+2 n+1 n ≥ 0 1 The n-th Fibonacci polynomial can be described explicitly as follows

n n x + √x2 +4 x √x2 +4 Fn(x)= − − . n 2 2 √x +4 The latter expression has the following explicit power form representation

n−1 2 ⌊ ⌋ n j 1 n 2j 1 F (x) = − − x − − n j j=0 X where z represents the largest integer less than or equal to z. Now one can ⌊ ⌋ deﬁne the n-th Fibonacci number as follows (1 + √5)n (1 √5)n Fn = Fn(1) = − − . 2n √5 The corresponding derivatives sequences of the Fibonacci numbers are denoted by (q) Fn . They are deﬁned via

(q) q Fn = D Fn(x) x=1.

Some of the identities relating Fibonacci and Lucas numbers to complex values of Chebyshev polynomials of the ﬁrst and second kinds are as follows, see [4,22],

1 i (1) F = U , n+1 in n 2 ( i)n (2) U ( 2 i)= − F . n − 2 3(n+1) There are several articles that study relations and identities satisﬁed by Fibonacci numbers and their derivatives sequences, see for example [5,12,14]. 4 W. M. ABD-ELHAMEED,Y. H.YOUSSRI,N.EL-SISSI,AND M. SADEK

2.2. Chebyshev polynomials of the first and second kinds. Chebyshev polynomials Tn(x) and Un(x) of the first and second kinds, respectively, are polynomials in x, which can be defined by (see, Mason and Handscomb [16]):

Tn(x) = cos(n θ), and sin(n + 1)θ U (x)= , n sin θ where x = cos θ. The polynomials T (x) and U (x) are orthogonal on ( 1, 1) with n n − respect to the weight functions 1 and √1 x2, that is √1 x2 − − 1 1 0, m = n, (3) Tn(x) Tm(x) dx = 6 2 π 1 √1 x ( cn, m = n, Z− − 2 and 1 0, m = n, √ 2 (4) 1 x Un(x) Um(x) dx = π 6 1 − m = n, Z− ( 2 where 2 n =0, cn = (1, n> 0.

The polynomials Tn(x) and Un(x) may be generated, respectively, by means of the two recurrence relations:

Tn(x)=2x Tn 1(x) Tn 2(x), n =2, 3,..., − − − with

T0(x)=1, T1(x)= x, and

Un(x)=2x Un 1(x) Un 2(x), n =2, 3,..., − − − with

U0(x)=1, U1(x)=2x. They also have the following explicit power forms:

n 2 r n ⌊ ⌋ ( 1) n r n 2 r T (x)= − − (2 x) − , n 2 n r r r=0 X − and n 2 ⌊ ⌋ r n r n 2 r U (x)= ( 1) − (2 x) − , n − r r=0 X FIBONACCI AND CHEBYSHEV POLYNOMIALS 5

The following special values are of important use later:

(5) Tn(1) = 1, Un(1) = n +1,

q 1 − (n i)(n + i) (6) DqT (1) = − , q 1, n 2 i +1 ≥ i=0 Y and q 1 − (n i)(n + i + 2) (7) DqU (1) = (n + 1) − , q 1. n 2 i +3 ≥ i=0 Y 3. New connection formulae between Chebyshev polynomials of the first and second kinds and Fibonacci polynomials

The following two theorems establish two new connection formulae between Fibonacci polynomials and Chebyshev polynomials of the ﬁrst and second kinds. The formulae are given in terms of values of hypergeometric functions.

Theorem 3.1. For every j 1, the following connection formula holds: ≥ j 2 j 2m 1 ⌊ ⌋ m j m 2 − − m, j m Tj(x)= j ( 1) − 2F1 − − 4 Fj 2m+1(x). − j 2m j m  −  − m=0 j 2m +2 X − − −   Theorem 3.2. For every j 1, the following connection formula holds: ≥ j 2 j ⌊ ⌋ m+1 j j +2m 1 m, j + m 1 1 Uj(x)=2 ( 1) − − 2F1 − − − − Fj 2m+1(x). − m j m +1  4  − m=0 j X − −   We will prove Theorem 3.1. The proof of Theorem 3.2 is similar.

Proof: The connection coeﬃcients are written using the following terminating hypergeometric series

m k m, j m ( m)k (j m)k ( 4) 2F1 − − 4 = − − − ,  −  (2 + j 2m)k k! 2+ j 2m k=0 − X −   Therefore it suﬃces to show that the following identity holds

j 2 m m 1+j+2k 2m ⌊ ⌋ ( 1) 2− − j (1 + j 2m)(j + k m 1)! Tj(x)= φj(x) := − − − − Fj 2m+1(x). k!(m k)!(1 + j + k 2m)!(m k)! − m=0 k X X=0 − − − 6 W. M. ABD-ELHAMEED,Y. H.YOUSSRI,N.EL-SISSI,AND M. SADEK

In other words, we want to prove that the function φj(x) satisﬁes the recurrence relation deﬁning T (x). It is easy to see that φ (x)= x and φ (x)=2x2 1. We j 1 2 − now make use of the recurrence relation

x Fj(x)= Fj+1(x) Fj 1(x) − − satisﬁed by Fibonacci polynomials

j 2 m m j+2k 2m ⌊ ⌋ ( 1) 2 − j (1 + j 2m)(j + k m 1)! 2xφj(x) = − − − − Fj 2m+2(x) k!(m k)!(1+ j + k 2m)!(m k)! − m=0 k=0 − − − Xj X 2 m m+1 j+2k 2m ⌊ ⌋ ( 1) 2 − j (1 + j 2m)(j + k m 1)! + − − − − Fj 2m(x). k!(m k)!(1+ j + k 2m)!(m k)! − m=0 k X X=0 − − − Some algebraic manipulations will yield

j−1 2 m m 2+j+2k 2m ⌊ ⌋ ( 1) 2− − (j 1)(j 2m)(j + k m 2)! 2xφj(x) = − − − − − Fj 2m(x) k!(m k)!(j + k 2m)!(m k)! − m=0 k=0 − − − Xj+1 X 2 m m j+2k 2m ⌊ ⌋ ( 1) 2 − (j +1)(2+ j 2m)(j + k m)! + − − − Fj 2m+2(x) k!(m k)!(2 + j + k 2m)!(m k)! − m=0 k X X=0 − − − = φj 1(x)+ φj+1(x), − Since Chebyshev polynomials T (x), j 1, are uniquely determined by the recur- j ≥ rence relation

2 2 x Tj(x)= Tj 1(x)+ Tj+1(x), T1 = x, T2(x)=2 x 1, − − it follows that φ (x) is the j-th Chebyshev polynomial T (x) for j 1. This j j ≥ completes the proof of Theorem 3.1. ✷

4. Inversion formulae between Fibonacci and Chebyshev polynomials of the first and second kinds

In this section we are concerned with deriving the inversion formulae to those given in the previous section. We will express Fibonacci and Lucas polynomials in terms of Chebyshev polynomials. Again the connection coeﬃcients turn out to be expressions involving values of hypergeometric functions of the type 2F1. We will need the following lemma in order to proceed. FIBONACCI AND CHEBYSHEV POLYNOMIALS 7

m, j + m 1 Lemma 4.1. We set d = F − − − 4 . The following recur- j,m 2 1  −  j − rence relation holds  

j 2 j 1 4 − (j 2m +1) (j m + 1) dj 2,m 1 + − (j 2m +2) (j m) dj 1,m 1 m 1 − − − − m 1 − − − − − − j 1 j + − (j 2m) (j m + 1)dj 1,m (j m) (j 2m + 1) dj,m =0. m − − − − m − − Proof: Let j e = (j 2m + 1) d . j,m m − j,m In order to prove that the recurrence relation above is satisﬁed, it suﬃces to show that

4(j m + 1) ej 2,m 1 +(j m) ej 1,m 1 +(j m + 1) ej 1,m =(j m) ej,m. − − − − − − − − − Now, ej,m, can be written in the form m k j ( m)k (m j 1)k ( 4) ej,m = (j 2m + 1) − − − − . m − ( j)k k! k X=0 − Using the identity ( 1)k m! ( m) = − , − k (m k)! − ej,m can be written equivalently as m (j k)!4k e =(j m + 1)(j 2m + 1) − . j,m − − k!(m k)!(j k m + 1)! k X=0 − − − It follows that

4(j m + 1) ej ,m + (j m) ej ,m + (j m + 1) ej ,m = − −2 −1 − −1 −1 − −1 m−1 m (j k)! 4k (j k 1)!4k (j m) 4(1 + j 2m) − + (j 2m) − − − 2 − k! (m k)! (j k m)! − k! (m k)! (j k m)! k k X=0 − − − X=0 − − − m−1 (j k 1)!4k +(2 + j 2m) − − . − k! (m k 1)! (j k m + 1)! k X=0 − − − − Taking a common denominator yields the following simpliﬁcation

4(j m + 1) ej 2,m 1 +(j m) ej 1,m 1 +(j m + 1) ej 1,m = − − − − − − m− − (j k)! (j m) (1 + j 2m) − = (j m) e . − 2 − k!(m k 1)!(j k m + 1)! − j,m k X=0 − − − − Lemma 4.1 is now proved. ✷ 8 W. M. ABD-ELHAMEED,Y. H.YOUSSRI,N.EL-SISSI,AND M. SADEK

In the following two theorems, we exhibit the connection relation between Fi- bonacci polynomials and Chebyshev polynomials of the ﬁrst and second kinds. Theorem 4.2. For every nonnegative integer j, the following connection formula holds j 2 ⌊ ⌋ 1 j m j+2m+1 m, j m +1 1 Fj+1(x)= − 2− 2F1 − − − Tj 2m(x), cj 2m j 2m  j 2m +1 4  − m=0 − X − −   where 2 j =0, cj = (1, j> 0. Theorem 4.3. For every nonnegative integer j, the following connection formula holds j 1 ⌊ 2 ⌋ j (j 2m + 1) m, j + m 1 Fj+1(x)= − 2F1 − − − 4 Uj 2m(x). 2j m j m +1  −  − m=0 j X − −   The proofs of Theorems 4.2 and 4.3 are similar, so it suﬃces to prove Theorem

4.3. Proof: We will proceed by induction. Assume that the above identity is valid for any k j. We know that ≤ Fj+1(x)= x Fj(x)+ Fj 1(x), − therefore using the induction hypothesis to write Fj 1(x) and Fj(x) in terms of − Chebyshev polynomials, together with making use of the recurrence relation 1 x Uj(x)= [Uj 1(x)+ Uj+1(x)] , 2 − yield

j−1 1 ⌊ 2 ⌋ j 1 j 2m m, j + m Fj (x) = − − F − − 4 Uj m (x)+ Uj m(x) +1 2j m j m 2 1  −  −2 −2 −2 m=0 1 j X − − j −1   2 1 ⌊ ⌋ j 2m 1 j 2 m, j + m +1 + − − − F − − 4 Uj m (x). 2j−2 j m 1 m 2 1  −  −2 −2 m=0 2 j X − − −   In fact the latter identity can be simpliﬁed and rewritten as follows

j−1 ⌊ 2 ⌋ 1 Fj+1(x)= gj,mUj 2m(x)+ gj 1,µj Uj 2 µj 2(x)+ gj 2,νj 1 Uj 2νj (x) θj, − 2 − − − − − − m=0 X FIBONACCI AND CHEBYSHEV POLYNOMIALS 9 where

1 j 1 (j 2m + 2) 1 m, j + m 1 gj,m = − − F − − − 4 + 2j  m 1 j m +1 2 1  −  1 j  − − −    j 2 (j 2m + 1) 1 m,m j 4 − − F − − 4 + m 1 j m 2 1  −  2 j − − −   j 1 (j 2m) m,m j − − F − − 4 , m j m 2 1  −  1 j − −   

and  j 1 j 1, if j even, µj = − , νj = , θj = 2 2  0, if j odd.  Making use of Lemma 4.1 together with some manipulations imply that gj,m can be reduced to take the form

j (j 2m + 1) m, 1+ m j g = m − F − − − 4 . j,m 2j(j m + 1) 2 1  −  j − −  

This yields that j 2 j 1 ⌊ ⌋ m (j 2m + 1) m, j + m 1 Fj+1(x)= − 2F1 − − − 4 Uj 2m(x), 2j j m +1  −  − m=0 j X − −   ✷ which completes the proof of Theorem 4.3.

5. Some applications In this section, we introduce three applications to the new derived connection formulae and their inversion ones: (i) We display some new expressions involving Fibonacci and Lucas numbers. (ii) Some new expressions for the derivatives sequences of Fibonacci numbers are given. (iii) We evaluate some deﬁnite integrals involving certain products of Fibonacci and Chebyshev polynomials.

5.1. New expressions involving Fibonacci and Lucas numbers. In this section we use the results we developed in §3 and §4 to evaluate ﬁnite sums involving certain values of hypergeometric functions and Fibonacci numbers. 10 W. M. ABD-ELHAMEED, Y. H. YOUSSRI,N. EL-SISSI,AND M. SADEK

Corollary 5.1. For every nonnegative integer j, the following two identities hold:

j 2 j 2m 1 ⌊ ⌋ m j m 2 − − m, j m ( 1) − 2F1 − − 4 Fj 2m+1 =1, − j 2m j m  −  − m=0 j 2m +2 X − − −   and

j 2 j ⌊ ⌋ m+1 j ( j +2m 1) m, j + m 1 1 2 ( 1) − − 2F1 − − − − Fj 2m+1 = j +1. − m j m +1  4  − m=0 j X − −   Corollary 5.2. For every nonnegative integer j, the following two expressions for

Fibonacci numbers are valid

j 2 ⌊ ⌋ 1 j m j+2m+1 m, j m +1 1 Fj+1 = − 2− 2F1 − − − , cj 2m j 2m  j 2m +1 4  m=0 − X − −  

j 1 ⌊ 2 ⌋ j (j 2m + 1)2 m, j + m 1 F = − F − − − 4 . j+1 2j m j m +1 2 1  −  m=0 j X − −   Proof: The proof of Corollaries 5.1 and 5.2 are immediately obtaine d from The- orems 3.1, 3.2, 4.2 and 4.3, respectively, by setting x =1. ✷ The fact that Fibonacci numbers themselves can be expressed as values of terminating hypergeometric series can be exploited in order to find a linear relation between hypergeometric series of type 2F1 with different parameters at a specific value. The n-th Fibonacci number can be written as a hypergeometric series itself. In fact one knows that

1 n 2 n 1 n 2 n n −2 , −2 −2 , −2 Fn = 2F1 5 = 2F1 4 , n 1 3 2 −    −  2 1 n −     see [5] for example. One of the linear transformations of hypergeometric series is given by the following identity

a, b a a, c b z F z = (1 z)− F − . 2 1   − 2 1  z 1 c c −     Putting these together, one can rewrite Corollary 5.2 as fol lows.

FIBONACCI AND CHEBYSHEV POLYNOMIALS 11

Corollary 5.3. For every nonnegative integer j, the following two expressions identities hold true

j j 1 j 2 −2 , −2 ⌊ ⌋ 1 j m j+2m+1 m, j m +1 1 2F1 4 = − 2− 2F1 − − −  j −  cj 2m j 2m  j 2m +1 4  m=0 − − X − −   j   2 1 ⌊ ⌋ j (j 2m + 1)2 m, j + m 1 = − F − − − 4 , 2j m j m +1 2 1  −  m=0 j X − − j   j 1 j 2 m −2 , −2 2 ⌊ ⌋ 5 j m m, m 1 2F1 5 = − 2F1 − − ,  3  j +1 cj 2m j 2m  j 2m +1 5 2 m=0 − X − −   j   2 1 ⌊ ⌋ j (j 2m + 1)2 m, 1 m 4 = 5m − F − − . 2j m j m +1 2 1  5 m=0 j X − −   Furthermore, one may obtain similar identities to the ones above relating values of the hypergeometric series 2F1 evaluated at 1/5 or 4/5 and several other values, see §4 in [5], using diﬀerent linear transformations. Now, and based on the identities (1) and (2), along with the connection formulae obtained in §4, the following identities follow. Corollary 5.4. For every nonnegative integer j, and i2 = 1, the following identities hold: −

j 2 m+1 j i j j ( 1) m ( j +2m 1)Fj−2m+1 2 m, j + m 1 1 i Fj = 2 − − − F − − − − , +1 j m +1 2 1  4  m=0 j X − − j   2 m+1 j j j+1 ( 1) m ( j +2m 1) Fj−2m+1( 2i) m, j + m 1 1 ( i) F j = 2 − − − − F − − − − . − 3 +3 j m +1 2 1  4  m=0 j X − −  

One may also obtain some new trigonometric identities making use of the fact that Tn(cos θ) = cos(nθ). In fact one has

j 2 ⌊ ⌋ 1 j m j+2m+1 m, j m +1 1 Fj+1(cos θ)= − 2− 2F1 − − − cos((j 2m)θ) , cj 2m j 2m  j 2m +1 4  − m=0 − X − −   where

2 j =0, cj = . (1, j> 0. 12 W. M. ABD-ELHAMEED, Y. H. YOUSSRI,N. EL-SISSI,AND M. SADEK

x + x 1 xn + x n Another interesting identity is obtained by observing that T − = − , n 2 2 whence j −1 2 x + x ⌊ ⌋ 1 j m −j+2m m, j m +1 1 j−2m 2m−j Fj+1 = − 2 2F1 − − − x + x . 2 cj m j 2m  4  m=0 −2 j 2m +1 X − −  

5.2. New derivatives sequences identities. Based on the connection formulae introduced in Theorems 3.1, 3.2, 4.2, and 4.3, we may obtain new formulae for the derivatives sequences of Fibonacci numbers by using the identities (6) and (7). Corollary 5.5. For all q 1, the following two formulae are valid ≥ j 2 j 2m 1 ⌊ ⌋ m j m 2 − − m, j m (q) ( 1) − 2F1 − − 4 Fj 2m+1 = − j 2m j m  −  − m=0 j 2m +2 X − − − q+1 ( 1) √π j (1 j)q 1(j+ 1)q 1  − − , − q − 1 2 Γ q + 2 and j 2 ⌊ ⌋ m+1 j ( j +2m 1) m, j + m 1 1 (q) ( 1) − − 2F1 − − − − Fj 2m+1 = − m j m +1  4  − m=0 j X − − q+1 ( 1) √π (j)3 (1 j)q 1(j + 3)q 1  − − . − j+q+1 − 3 2 Γ q + 2 Corollary 5.6. For all q 1, the following two formulae are valid ≥ j q+1 2 (q) ( 1) √π ⌊ ⌋ 1 j m j+2m q+1 2 Fj+1 = − 1 − 2− − (j 2m) (j 2m + 1)q 1 cj m j 2m − − − × Γ q + 2 m=0 2 X − − m, j m +1 1 ( j +2m + 1)q 1 2F1 − − − , − −  4  j 2m +1 −   and j ( 1)q+1 √π ⌊ 2 ⌋ j (j 2m)(j 2m + 1)2 (j 2m + 2) F (q) = − − − − j+1 j+q+1 3 m j m +1 × 2 Γ q + 2 m=0 X − m, j + m 1 (j 2m + 3)q 1 2F1 − − − 4 . − −  −  j −  

FIBONACCI AND CHEBYSHEV POLYNOMIALS 13

Proof: In order to prove Corollaries 5.5 and 5.6, one needs to diﬀerentiate the connection formulae in Theorems 3.1, 3.2, 4.2, and 4.3, then one sets x =1. Now the identities follow from (6) and (7). ✷

5.3. Some integrals formulae involving Chebyshev and Fibonacci polynomials. The following two integrals formulae are direct consequences of Theo- rems 4.2 and 4.3.

Corollary 5.7. For all j k, the following two integrals formulae hold: ≥ j+k k−j 1 1 π 2 , (j + k + 2) ( k ) 2 2 −1 Fj+1(x)Tk(x) 2k c 2F1 4 , if (k + j) even, dx =  k k +1 ! √1 x2 Z  −1 − 0, otherwise,

 and 

j k−j −1 1 π ( j−k ) (k+1) , (j + k + 2) 2 2 2 j 2F1 4 , if (k + j) even, 2 2 (j+k+2) 1 x Fj+1(x) Uk(x) dx =  j − ! − − Z  −1 p 0, otherwise.

 Now one can ﬁnd an explicit description for weighted deﬁnite integrals of products of Fibonacci polynomials in terms of hypergeometric series.

Corollary 5.8. For all j k, the following two integrals formulae hold: ≥ 1 F (x)F (x) j+1 k+1 dx = √1 x2 −Z1 − ⌊ k ⌋ j−m k−m π 2 m, k m +1 1 m, j m +1 1 4md j−2m k−2m F − − F − − k+j−1 2 m 2 1 − 2 1 − 2 ck mcj m  4  ×  4  m=0 −2 −2 k 2m +1 j 2m +1 X − −    

and

1 2 1 x Fj (x)Fk (x) dx = − +1 +1 −Z1 p ⌊ k ⌋ π 2 j k (k 2m + 1)(j 2m + 1) m, k + m 1 − − F − − − 4 2k+j+1 m m (k m + 1)(j m + 1) 2 1  −  m=0 k X − − −   m, j + m 1 F − − − 4 . × 2 1  −  j −  

14 W. M. ABD-ELHAMEED, Y. H. YOUSSRI,N. EL-SISSI,AND M. SADEK References

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Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt

E-mail address: walee−[email protected]

Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt E-mail address: [email protected]

American University in Cairo, Mathematics and Actuarial Science Department, AUC Avenue, New Cairo, Egypt E-mail address: [email protected]