(p,q)− deformed Fibonacci and Lucas polynomials: characterization and Fourier integral transforms
aa r X i v : . [ m a t h - ph ] J u l ( p, q ) − deformed Fibonacci and Lucas polynomials:characterization and Fourier integral transforms Mahouton Norbert Hounkonnou and Sama Arjika
International Chair of Mathematical Physics and Applications (ICMPA-UNESCOChair), University of Abomey-Calavi, 072 B. P.: 50 Cotonou, Republic of BeninE-mail: [email protected] , [email protected] Abstract.
A full characterization of ( p, q )-deformed Fibonacci and Lucas polynomi-als is given. These polynomials obey non-conventional three-term recursion relations.Their generating functions and Fourier integral transforms are explicitly computed anddiscussed. Relevant results known in the literature are examined as particular cases.
1. Introduction
The classical orthogonal polynomials (COPs) and the quantum orthogonal polynomials(QOPs), also called q − orthogonal polynomials, constitute an interesting set of specialfunctions with potential applications in physics, in probability and statistics, inapproximation theory and numerical analysis, to cite a few domains where they areinvolved. Since the birth of quantum mechanics, the COPs also made their appearancein the bound-state wavefunctions of exactly solvable potentials.Depending on the set of parameters, each family of orthogonal polynomials occupiesdifferent levels within the Askey hierarchy [16]. For instance, the classical Hermitepolynomials H n ( x ) are the ground level, the Laguerre L ( α ) n ( x ) and Charlier C n ( x ; a )polynomials are one level higher, the Jacobi P ( α,β ) n ( x ), the Meixner M n ( x ; β, c ), theKrawtchouk K n ( x ; p, N ) and the Meixner/Pollaczek P ( λ ) n ( x ; φ ) polynomials are twolevels higher, the Hahn Q n ( x ; α, β, N ), the dual Hahn R n ( λ ( x ); γ, δ, N ) polynomials,etc. are three levels higher, and so on. Besides, all orthogonal polynomial families inthis Askey scheme are characterized by a set of properties: • they are solutions of second order differential or difference equations, • they are generated by three-term recurrence relations, • they are orthogonal with respect to weight functions, • they obey the Rodrigues-type formulas.The other polynomial families which do not obey the above characteristic properties,do not belong to the Askey q -scheme. p, q ) − deformed Fibonacci and Lucas polynomials: characterization and Fourier integral transforms p, q ) − Fibonacci and ( p, q ) − Lucaspolynomials characterized by non-conventional recurrence relations. Their q − analogswere recently introduced by Atakishiyev et al [3].The paper is organized as follows. In Section 2, we give a formulation of the( p, q ) − deformed Fibonacci and ( p, q ) − deformed Lucas polynomials. Their generatingfunctions are computed and discussed. We perform the computation of the associatedFourier integral transforms in Section 3 and end with some concluding remarks in theSection 4. ( p, q ) − deformed Fibonacci and Lucas polynomials In this section, we study in details the ( p, q ) − deformed Fibonacci and Lucaspolynomials. The corresponding generating functions are computed and discussed. ( p, q ) − deformed Fibonacci polynomials Start with the following definition of ( p, q ) − analogs of Fibonacci polynomials [3, 7, 8], F n +1 ( x, s ) = ⌊ n/ ⌋ X k =0 (cid:18) n − kk (cid:19) s k x n − k = x n F − n , − n − n (cid:12)(cid:12)(cid:12)(cid:12) − sx ! , n ≥ , (1)given by: Definition 2.1 F n +1 ( x, s | p, q ) := ⌊ n/ ⌋ X k =0 ( pq ) k ( k +1) / (cid:20) n − kk (cid:21) p,q s k x n − k , (2) where the ( p, q ) − binomial coefficients (cid:2) nk (cid:3) p,q are given by [15] (cid:20) nk (cid:21) p,q := (( p, q ); ( p, q )) n (( p, q ); ( p, q )) k (( p, q ); ( p, q )) n − k , (3) and ( p, q ) − shifted factorial (( a, b ); ( p, q )) n is defined as (( a, b ); ( p, q )) n := n − Y k =0 ( ap k − bq k ) for n ≥ , (( a, b ); ( p, q )) := 1 . (4)When n → ∞ , (( a, b ); ( p, q )) ∞ := ∞ Y k =0 ( ap k − bq k ) . (5) Remark 2.2
As expected, when the parameter p → , the ( p, q ) − Fibonacci polynomials(2) are reduced to their q − version [3], i.e F n +1 ( x, s | q ) = ⌊ n/ ⌋ X k =0 q k ( k +1) / (cid:20) n − kk (cid:21) q s k x n − k , (6) p, q ) − deformed Fibonacci and Lucas polynomials: characterization and Fourier integral transforms where the q − binomial coefficient (cid:2) nk (cid:3) q is given by (cid:20) nk (cid:21) q := ( q ; q ) n ( q ; q ) k ( q ; q ) n − k , (7) and ( z ; q ) n is the q -shifted factorial defined as ( z ; q ) := 1 , ( z ; q ) n := n − Y k =0 (1 − zq k ) , n ≥ , ( z ; q ) ∞ := ∞ Y k =0 (1 − zq k ) . (8)Further, the following statement holds. Proposition 2.3
The ( p, q ) − Fibonacci polynomials (2) can be rewritten in terms of thehypergeometric function ϕ as follows: F n +1 ( x, s | p, q ) = x n ϕ ( p − n , q − n ) , ( p − n , q − n ) , ( p − n , − q − n ) , ( p − n , q − n ) , ( p, , ( p, , (9)( p − n , − q − n ) , , , , p, , ( p, (cid:12)(cid:12)(cid:12)(cid:12) ( p, q ); − sq n p n x , where the ( p, q ) − hypergeometric function r ϕ s is defined as [15] r ϕ s ( a , b ) , ( a , b ) , . . . , ( a r , b r )( c , d ) , ( c , d ) , . . . , ( c s , d s ) (cid:12)(cid:12)(cid:12)(cid:12) ( p, q ); x ! := ∞ X n =0 (( a , b )( a , b ) . . . ( a r , b r ); ( p, q )) n (( c , d )( c , d ) . . . ( c s , d s ); ( p, q )) n h ( − n ( q/p ) ( n ) i s − r (( p, q ); ( p, q )) n x n . (10) Proof.
By using the ( p, q ) − identities:(( p, q ); ( p, q )) n − k = (( p, q ); ( p, q )) n (( p − n , q − n ); ( p, q )) k ( − k ( pq ) ( k ) − nk , (( p n , q n ); ( p, q )) k = (( p n , q n ) , ( p n , q n ) , ( p n , − q n ) , ( p n , − q n ); ( p, q )) k , we get F n +1 ( x, s | p, q ) = ⌊ n/ ⌋ X k =0 ( pq ) k ( k +1) / (( p, q ); ( p, q )) n − k (( p, q ); ( p, q )) k (( p, q ); ( p, q )) n − k s k x n − k = x n ⌊ n/ ⌋ X k =0 ( p − q ) − k ) (( p − n , q − n ); ( p, q )) k (( p − n , q − n ) , ( p, q ); ( p, q )) k p k +12 ) (cid:18) − sq n p n +4 x (cid:19) k , with p ( k +12 ) = (( p, p, q )) k . (cid:3) Remark 2.4
In the limit when p → , (10) is reduced to q − hypergeometric functioncharacterizing the q − Fibonacci polynomials investigated in [3], i. e. F n +1 ( x, s | q ) = x n φ q − n/ , q (1 − n ) / , − q − n/ , − q (1 − n ) / q − n (cid:12)(cid:12)(cid:12)(cid:12) q ; − q n sx ! , n ≥ . (11) p, q ) − deformed Fibonacci and Lucas polynomials: characterization and Fourier integral transforms Lemma 2.5
The ( p, q ) − binomials coefficients (cid:20) n − kk (cid:21) p,q = (( p, q ); ( p, q )) n − k (( p, q ); ( p, q )) k (( p, q ); ( p, q )) n − k , (12) where ≤ k ≤ n, n ∈ N satisfy the following identities: (cid:20) n − kk (cid:21) p,q = q k (cid:20) n − − kk (cid:21) p,q + p n − k (cid:20) n − − kk − (cid:21) p,q , (13) (cid:20) n − kk (cid:21) p,q = p k (cid:20) n − − kk (cid:21) p,q + q n − k (cid:20) n − − kk − (cid:21) p,q , (14) (cid:20) n − kk (cid:21) p,q = p k (cid:20) n − − kk (cid:21) p,q + p n − k q − k (cid:20) n − − kk − (cid:21) p,q − ( p n − k +1 − q n − k +1 ) q − k (cid:20) n − kk − (cid:21) p,q . (15) Proof.
Using the relations [7]: (cid:20) n − kk (cid:21) q = q k (cid:20) n − − kk (cid:21) q + (cid:20) n − − kk − (cid:21) q , (cid:20) nk (cid:21) q/p = p − k ( n − k ) (cid:20) nk (cid:21) p,q , (16) (cid:20) n − kk (cid:21) q = (cid:20) n − − kk (cid:21) q + q n − k (cid:20) n − − kk − (cid:21) q , (17)and q k (cid:20) n − kk (cid:21) q = q k (cid:20) n − − kk (cid:21) q + (cid:20) n − − kk − (cid:21) q − (1 − q n − k +1 ) (cid:20) n − kk − (cid:21) q , (18)yields the required identities: (cid:20) n − kk (cid:21) p,q = q k (cid:20) n − − kk (cid:21) p,q + p n − k (cid:20) n − − kk − (cid:21) p,q , (19) (cid:20) n − kk (cid:21) p,q = p k (cid:20) n − − kk (cid:21) p,q + q n − k (cid:20) n − − kk − (cid:21) p,q , (20)and (cid:20) n − kk (cid:21) p,q = p k (cid:20) n − − kk (cid:21) p,q + p n − k q − k (cid:20) n − − kk − (cid:21) p,q − ( p n − k +1 − q n − k +1 ) q − k (cid:20) n − kk − (cid:21) p,q . (21) (cid:3) Proposition 2.6
The ( p, q ) − deformed Fibonacci polynomials satisfy the following non-standard three-term recursion relations: F n +1 ( x, s | p, q ) = xF n ( x, qs | p, q ) + sqp n − F n − ( x, qp − s | p, q ) , (22)= xF n ( x, sp | p, q ) + spq n − F n − ( x, spq − | p, q ) , (23)= ( x + sp ( q − p ) D ( p,q ) ) F n ( x, sp | p, q ) + sp n F n − ( x, s | p, q ) , n ≥ , (24) p, q ) − deformed Fibonacci and Lucas polynomials: characterization and Fourier integral transforms with the initial conditions F ( x, s | p, q ) = 0 , F ( x, s | p, q ) = 1 and the ( p, q ) − Jackson’sderivative D ( p,q ) given by [15] D ( p,q ) f ( x ) = f ( px ) − f ( qx )( p − q ) x . (25) Proof.
By multiplying the equations (13), (14) and (15) of the Lemma 2.5 by( pq ) k ( k +1) / s k x n − k and summing from k = 0 to ⌊ n/ ⌋ , we get ⌊ n/ ⌋ X k =0 ( pq ) k ( k +1) / (cid:20) n − kk (cid:21) p,q s k x n − k = ⌊ n/ ⌋ X k =0 ( pq ) k ( k +1) / (cid:20) n − − kk (cid:21) p,q ( qs ) k x n − k + p n ⌊ n/ ⌋ X k =1 ( pq ) k ( k +1) / (cid:20) n − − kk − (cid:21) p,q ( p − s ) k x n − k , ⌊ n/ ⌋ X k =0 ( pq ) k ( k +1) / (cid:20) n − kk (cid:21) p,q s k x n − k = ⌊ n/ ⌋ X k =0 ( pq ) k ( k +1) / (cid:20) n − − kk (cid:21) p,q ( ps ) k x n − k + q n ⌊ n/ ⌋ X k =1 ( pq ) k ( k +1) / (cid:20) n − − kk − (cid:21) p,q ( q − s ) k x n − k , and ⌊ n/ ⌋ X k =0 ( pq ) k ( k +1) / (cid:20) n − kk (cid:21) p,q s k x n − k = ⌊ n/ ⌋ X k =0 ( pq ) k ( k +1) / (cid:20) n − − kk (cid:21) p,q ( ps ) k x n − k + p n ⌊ n/ ⌋ X k =1 ( pq ) k ( k +1) / (cid:20) n − − kk − (cid:21) p,q ( p − q − s ) k x n − k + p n ⌊ n/ ⌋ X k =1 ( pq ) k ( k +1) / (cid:20) n − kk − (cid:21) p,q ( p n − k +1 − q n − k +1 )( q − s ) k x n − k , which are equivalent to F n +1 ( x, s | p, q ) = xF n ( x, qs | p, q ) + sqp n − F n − ( x, qp − s | p, q ) , (26) F n +1 ( x, s | p, q ) = xF n ( x, sp | p, q ) + spq n − F n − ( x, spq − | p, q ) , (27)and F n +1 ( x, s | p, q ) = ( x + sp ( q − p ) D ( p,q ) ) F n ( x, sp | p, q ) + sp n F n − ( x, s | p, q ) , n ≥ , (28)respectively, with F ( x, s | p, q ) = 0, F ( x, s | p, q ) = 1. (cid:3) Remark 2.7
In the limit case when p → , the equations (22)-(24) are reduced to their q − analogs [3, 7], i.e F n +1 ( x, s | q ) = xF n ( x, qs | q ) + sqF n − ( x, qs | q ) , (29)= xF n ( x, s | q ) + sq n − F n − ( x, sq − | q ) , (30)= xF n ( x, s | q ) + s ( q − D q F n ( x, s | q ) + sF n − ( x, s | q ) , n ≥ , (31) with the initial values F ( x, s | q ) = 0 and F ( x, s | q ) = 1 , where D q is the q − Jacksondifferential operator defined by D q f ( x ) := f ( x ) − f ( qx )(1 − q ) x . (32) p, q ) − deformed Fibonacci and Lucas polynomials: characterization and Fourier integral transforms Definition 2.8
The generating function f F ( x, s ; t | p, q ) associated with the ( p, q ) -Fibonacci polynomials is defined as follows: f F ( x, s ; t | p, q ) := ∞ X n =0 F n ( x, sp − n | p, q ) t n . (33) Proposition 2.9
The generating functions (33) can be re-expressed in terms of thehypergeometric function ϕ : f F ( x, s ; t | p, q ) = t − xt ϕ ( p, q ) , p, xtq ) , ( p, (cid:12)(cid:12)(cid:12)(cid:12) ( p, q ); − qst ! , | t | < . (34) Proof.
From (2), we have f F ( x, s ; t | p, q ) : = ∞ X n =0 F n ( x, sp − n | p, q ) t n = ∞ X n =0 ⌊ n − ⌋ X k =0 ( pq ) k ( k +1) / (cid:20) n − kk (cid:21) p,q s k p − nk x n − k t n = ∞ X k =0 ( pq ) k ( k +1) / s k (( p, q ); ( p, q )) k ∞ X n − k (( p, q ); ( p, q )) n − − k (( p, q ); ( p, q )) n − − k x n − − k t n p − nk = ∞ X k =0 ( pq ) k ( k +1) / s k (( p, q ); ( p, q )) k ∞ X m =0 (( p, q ); ( p, q )) m + k (( p, q ); ( p, q )) m x m t m +1+2 k p − ( m +1+2 k ) k = t ∞ X k =0 ( pq ) k ( k +1) / p − k − k ( st ) k (( p, q ); ( p, q )) k × ∞ X m =0 (( p, q ); ( p, q )) k (( p k , q k ); ( p, q )) m (( p, q ); ( p, q )) m ( xtp − k ) m = t ∞ X k =0 ( pq ) k ( k +1) / p − k − k ( st ) k ∞ X m =0 (( p k , q k ); ( p, q )) m (( p, q ); ( p, q )) m ( xtp − k ) m = t ∞ X k =0 ( pq ) k ( k +1) / p − k − k ( st ) k ϕ ( p k , q k ) − (cid:12)(cid:12)(cid:12)(cid:12) ( p, q ); xtp − k ! = t ∞ X k =0 ( pq ) k ( k +1) / p − k − k ( st ) k (( p, xtp − k q k ); ( p, q )) ∞ (( p, xtp ); ( p, q )) ∞ . (35)By using the equation (50) of [15], the expression (35) is transformed into f F ( x, s ; t | p, q ) = t ∞ X k =0 ( pq ) k ( k +1) / p − k − k ( st ) k p ( k +22 ) (( p, xtp ); ( p, q )) k +1 = t − xt ∞ X k =0 ( p − q ) k ( k − / ( st q ) k (( p, xtq ) , ( p, p, q )) k , which achieves the proof. (cid:3) Remark 2.10
In the limit when p → , the generating function (34) is reduced to its q − version provided by Atakishiyev et al [3]: f F ( x, s ; t | q ) := ∞ X n =0 F n ( x, s | q ) t n = t − xt φ qqxt (cid:12)(cid:12)(cid:12)(cid:12) q ; − qst ! , | t | < . (36) p, q ) − deformed Fibonacci and Lucas polynomials: characterization and Fourier integral transforms Remark 2.11
In the limit(i) When p, q → , the ( p, q ) − deformed Fibonacci polynomials (2) are reduced to theclassical case F n ( x, s ) given by the explicit sum formula [7, 8] F n +1 ( x, s ) = ⌊ n/ ⌋ X k =0 (cid:18) n − kk (cid:19) s k x n − k = x n F − n , − n − n (cid:12)(cid:12)(cid:12)(cid:12) − sx ! , n ≥ , (37) where (cid:0) nk (cid:1) := n ! / [ k !( n − k )!] is the binomial coefficient and F is a hypergeometricfunction [17]. They obey the following three-term recursion relation F n +1 ( x, s ) = xF n ( x, s ) + sF n − ( x, s ) , n ≥ , (38) with initial values F ( x, s ) = 0 and F ( x, s ) = 1 . Their generating function f F ( x, s ; t ) is given by [17] f F ( x, s ; t ) := ∞ X n =0 F n ( x, s ) t n = t − x t − s t , | t | < . (39) The polynomials (37) are monic and normalized so that for s = 1 , one recovers theFibonacci polynomials f n ( x ) = F n ( x, introduced by Catalan ([18], eq. (37.1) p.443).(ii) When x = s = 1 , the ( p, q ) − Fibonacci polynomials (2) furnish the ( p, q ) − Fibonaccinumber F n (1 , | p, q ) := F n ( p, q ) given by F n ( p, q ) = ⌊ n/ ⌋ X k =0 ( pq ) k ( k +1) / (cid:20) n − kk (cid:21) p,q , (40) which is the ( p, q ) − extension of the q − Fibonacci number [8] F n ( q ) = ⌊ n/ ⌋ X k =0 q k ( k +1) / (cid:20) n − kk (cid:21) q . (41) From the limit ( p, q ) → (1 , , we recover the classical Fibonacci number, i.e. F n = 12 n − ⌊ n − ⌋ X k =0 (cid:18) n k + 1 (cid:19) k . (42) The ( p, q ) − deformed generating function associated with the ( p, q ) − Fibonaccinumber is given by f F ( t | p, q ) = t − t ϕ ( p, q ) , p, tq ) , ( p, (cid:12)(cid:12)(cid:12)(cid:12) ( p, q ); − qt ! , | t | < which is reducible to the following corresponding q − generating function by passingto the limit p → f F ( t | q ) = t − t φ qqt (cid:12)(cid:12)(cid:12)(cid:12) q ; − qt ! , | t | < . (44) p, q ) − deformed Fibonacci and Lucas polynomials: characterization and Fourier integral transforms Finally, the limit case when ( p, q ) → (1 , yields the classical version of thegenerating function f F ( t ) of the Fibonacci polynomials (see [10, 20, 22, 5] for moredetails), i.e. f F ( t ) = ∞ X n =0 F n t n = t − t − t , | t | < . (45) ( p, q ) − deformed Lucas polynomials Let us introduce the ( p, q ) − analogs of the Lucas polynomials: L n ( x, s ) := ⌊ n/ ⌋ X k =0 nn − k (cid:18) n − kk (cid:19) s k x n − k = x n F − n , − n − n (cid:12)(cid:12)(cid:12)(cid:12) − sx ! , n ≥ , (46)as follows: Definition 2.12 L n ( x, s | p, q ) := ⌊ n/ ⌋ X k =0 ( pq ) ( k ) [ n ] p,q [ n − k ] p,q (cid:20) n − kk (cid:21) p,q s k x n − k , (47) where the ( p, q ) − number [ n ] p,q is given by [ n ] p,q := p n − q n p − q . (48)In the limit when p →
1, the ( p, q ) − Lucas polynomials are reduced to the q − Lucaspolynomials introduced in [7] L n ( x, s | q ) := ⌊ n/ ⌋ X k =0 q ( k ) [ n ] q [ n − k ] q (cid:20) n − kk (cid:21) q s k x n − k , (49)where the q − number [ n ] q is defined as[ n ] q := 1 − q n − q . (50)The following proposition holds. Proposition 2.13
The ( p, q ) − Lucas polynomials (47) can be defined as follows: L n ( x, s | p, q ) = x n ϕ ( p − n , q − n ) , ( p − n , q − n ) , ( p − n , − q − n ) , ( p − n , q − n ) , ( p, , ( p, , (51)( p − n , − q − n ) , , , , p, , ( p, (cid:12)(cid:12)(cid:12)(cid:12) ( p, q ); − sq n p n x . Proof.
The proof is the same as in the Proposition 2.3. (cid:3) p, q ) − deformed Fibonacci and Lucas polynomials: characterization and Fourier integral transforms Lemma 2.14
The ( p, q ) − coefficients [ n ] p,q [ n − k ] p,q (cid:20) n − kk (cid:21) p,q (52) satisfy the following identities: [ n ] p,q [ n − k ] p,q (cid:20) n − kk (cid:21) p,q = q k (cid:20) n − kk (cid:21) p,q + p n − k (cid:20) n − − kk − (cid:21) p,q (53) and [ n ] p,q [ n − k ] p,q (cid:20) n − kk (cid:21) p,q = p k (cid:20) n − kk (cid:21) p,q + q n − k (cid:20) n − − kk − (cid:21) p,q . (54)Besides, in analogous way as for the Fibonacci polynomials, we can prove the followingresult. Proposition 2.15
The ( p, q ) − Lucas polynomials (47) satisfy non-standard recursionrelations for n ≥ L n ( x, s | p, q ) = F n +1 ( x, p − s | p, q ) + sp n − F n − ( x, sp − | p, q ) , (55) L n ( x, sqp − | p, q ) = F n +1 ( x, sp − | p, q ) + sp − q n F n − ( x, p − s | p, q ) (56) with L ( x, s | p, q ) = 1 , L ( x, s | p, q ) = x . Remark 2.16
In the limit when p → , the polynomials (47) are reduced to the wellknown q − Lucas polynomials L n ( x, s | q ) studied by Atakishiyev et al [3]: L n ( x, s | q ) : = ⌊ n/ ⌋ X k =0 q k ( k − / [ n ] q [ n − k ] q (cid:20) n − kk (cid:21) q s k x n − k = x n φ q − n/ , q (1 − n ) / , − q − n/ , − q (1 − n ) / q − n (cid:12)(cid:12)(cid:12)(cid:12) q ; − q n sx ! , n ≥ , (57) where the q − number [ n ] q is defined as [ n ] q := (1 − q n ) / (1 − q ) , satisfying the followingrecursion relations L n ( x, s | q ) = F n +1 ( x, s | q ) + sF n − ( x, s | q ) , (58) L n ( x, sq | q ) = F n +1 ( x, s | q ) + sq n F n − ( x, s | q ) (59) with L ( x, s | q ) = 1 , L ( x, s | q ) = x. The proof is similar to that previously performed for the Fibonacci polynomials.
Definition 2.17
The generating function f L ( x, s ; t | p, q ) associated with the ( p, q ) − Lucaspolynomials L n ( x, s | p, q ) is defined by f L ( x, s ; t | p, q ) := ∞ X n =0 L n ( x, sp − n | p, q ) t n . (60) Proposition 2.18
The generating functions (60) is explicitly given by f L ( x, s ; t | p, q ) = 1 + spt − xpt ϕ ( p, q ) , p, xtpq ) , ( p, (cid:12)(cid:12)(cid:12)(cid:12) ( p, q ); − qst ! , | t | < . (61) p, q ) − deformed Fibonacci and Lucas polynomials: characterization and Fourier integral transforms Proof.
The proof is immediate from the definition: f L ( x, s ; t | p, q ) : = ∞ X n =0 L n ( x, sp − n | p, q ) t n = ∞ X n =0 F n +1 ( x, p − − n s | p, q ) t n + s ∞ X n =0 F n − ( x, sp − − n | p, q ) p n − t n . (62)and the use of the Proposition 2.9. (cid:3) Remark 2.19
In the limit,(i) When ( p, q ) → (1 , the polynomials (47) are reduced to the well known Lucaspolynomials L n ( x, s ) given by the following formula [3]: L n ( x, s | q ) := ⌊ n/ ⌋ X k =0 nn − k (cid:18) n − kk (cid:19) s k x n − k = x n F − n , − n − n (cid:12)(cid:12)(cid:12)(cid:12) − sx ! , n ≥ satisfying the following recursion relation L n ( x, s | q ) = F n +1 ( x, s ) + sF n − ( x, s ) (64) and admitting the following generating function [3] f L ( x, s ; t ) = ∞ X n =0 F n ( x, s ) t n = 1 + st − xt − st , | t | < , (65) which can be easily derived from the tree-term recursion relation (58) and (36). For s = 1 , we recover the normalized Lucas polynomials l n ( x ) = L n ( x, investigatedby Bicknell [18].(ii) For x = s = 1 , the ( p, q ) − deformed Lucas polynomials become the ( p, q ) − Lucasnumbers: L n ( p, q ) := ⌊ n/ ⌋ X k =0 ( pq ) ( k ) [ n ] p,q [ n − k ] p,q (cid:20) n − kk (cid:21) p,q , (66) generalizing the q − Lucas numbers [7] L n ( q ) := ⌊ n/ ⌋ X k =0 q ( k ) [ n ] q [ n − k ] q (cid:20) n − kk (cid:21) q . (67) When ( p, q ) → (1 , one obtains the well known Lucas number: L n = 12 n − n X k =0 (cid:18) n k (cid:19) k . (68) The ( p, q ) − generating function associated to the ( p, q ) − Lucas number is given by f L ( t | p, q ) = 1 + pt − pt ϕ ( p, q ) , p, tpq ) , ( p, (cid:12)(cid:12)(cid:12)(cid:12) ( p, q ); − qt ! , | t | < generalizing the q − Lucas number f L ( t | p, q ) = 1 + t − t ϕ qtq (cid:12)(cid:12)(cid:12)(cid:12) q ; − qt ! , | t | < p, q ) − deformed Fibonacci and Lucas polynomials: characterization and Fourier integral transforms while the classical generating function f L ( t ) associated with the undeformed Lucasnumbers are given by f L ( t ) = ∞ X n =0 L n t n = 1 + t − t − t , | t | < . (71) See [22] for more details on the Lucas numbers { L n } n . Proposition 2.20 D ( p,q ) L n ( n, s | p, q ) = [ n ] p,q F n ( x, s | p, q ) , [ n ] p,q = n − X k =0 p n − − k q k . (72)The proof stems from the observation that the Fibonacci polynomials F n ( x, s ) (resp.the Lucas polynomials L n ( x, s )) are essentially the Chebyshev polynomials of thesecond kind U n ( x ) (resp. of the first kind T n ( x )) (see [3] for more details) with d x T n ( x ) = nU n − ( x ) [17].
3. Fourier transforms of F n ( x, s | p, q ) and L n ( x, s | p, q )In this section, we derive explicit formulas for the classical Fourier integral transformsof the ( p, q ) − deformed Fibonacci F n ( x, s | p, q ) and Lucas L n ( x, s | p, q ) polynomials. ( p, q ) − Fibonacci polynomials
Rewrite the ( p, q ) − deformed Fibonacci polynomials (2) in the following form: F n +1 ( x, s | p, q ) = ⌊ n/ ⌋ X k =0 c ( F ) n, k ( p, q ) s k x n − k , (73)where the associated ( p, q ) − deformed coefficients are given by c ( F ) n,k ( p, q ) := ( pq ) k ( k +1) / (cid:20) n − kk (cid:21) p,q . (74)Using the relations (cid:20) n − kk (cid:21) p − ,q − = ( pq ) k (2 k − n ) (cid:20) n − kk (cid:21) p,q (75)we can express the ( p − , q − ) − deformed coefficients from (74) as c ( F ) n,k ( p − , q − ) = ( pq ) − k ( n +1 − k ) c ( F ) n,k ( p, q ) , (76)allowing to define the ( p − , q − ) − Fibonacci polynomials in the form: F n +1 ( x, s | p − , q − ) := ⌊ n/ ⌋ X k =0 c ( F ) n,k ( p − , q − ) s k x n − k = x n ϕ ( p − n , q − n ) , ( p − n , q − n ) , ( p − n , − q − n ) , ( p − n , − q − n )( p − n , q − n ) , , (cid:12)(cid:12)(cid:12)(cid:12) ( p, q ); − sx . (77) p, q ) − deformed Fibonacci and Lucas polynomials: characterization and Fourier integral transforms p →
1, we immediately obtain the q − − Fibonacci polynomials asfollows: F n +1 ( x, s | q − ) : = ⌊ n/ ⌋ X k =0 c ( F ) n,k ( q − ) s k x n − k = x n ϕ q − n , q − n , − q − n , − q − n q − n , , (cid:12)(cid:12)(cid:12)(cid:12) q ; − sx . (78)The associated q − − Fibonacci number is given by F n +1 ( q − ) : = ⌊ n/ ⌋ X k =0 c ( F ) n,k ( q − ) = ϕ q − n , q − n , − q − n , − q − n q − n , , (cid:12)(cid:12)(cid:12) q ; − . (79) Theorem 3.1
The Fourier transform of the function e − x / F n +1 ( ae iκx , s | p, q ) is givenby √ π Z R F n +1 ( ae iκx , s | p, q ) e ixy − x / dx = ( pq ) n F n +1 ( ae − κy , pqs | p − , q − ) e − y / (80) leading to the formula F n +1 ( a, s | p, q ) = 12 π Z R Z R F n +1 ( ae iκx , s | p, q ) e ixy − x / dxdy, (81) where a is an arbitrary constant factor and q = p − e − κ . Proof.
Using (73) and (76), we obtain:1 √ π Z R F n +1 ( ae iκx , s | p, q ) e ixy − x / dx = ⌊ n/ ⌋ X k =0 c ( F ) n,k ( p, q ) s k a n − k √ π Z R e ixy + i ( n − k ) κx − x / dx = ⌊ n/ ⌋ X k =0 c ( F ) n,k ( p, q ) s k a n − k e − [ κ ( n − k )+ y ] = ( pq ) n / F n +1 ( ae − κy , pqs | p − , q − ) e − y / , where the Gauss integral transform R R e ixy − x / dx = √ πe − y / is used. The proof isachieved by integrating (80) with respect to y . (cid:3) In the limit when the parameter p →
1, the Fourier transform (80) is reduced tothe well-known results investigated by Atakishiyev et al [3],1 √ π Z R F n +1 ( ae iκx , s | q ) e ixy − x / dx = q n F n +1 ( ae − κy , qs | q − ) e − y / , (82)where a is an arbitrary constant factor and q = e − κ . p, q ) − deformed Fibonacci and Lucas polynomials: characterization and Fourier integral transforms ( p, q ) − Lucas integral transform
Rewrite here also the ( p, q ) − deformed Lucas polynomials (47) as L n ( x, s | p, q ) = ⌊ n/ ⌋ X k =0 c ( L ) n,k ( p, q ) s k x n − k , (83)where the coefficients c ( L ) n,k ( p, q ) are given by c ( L ) n,k ( p, q ) := ( pq ) k ( k − / [ n ] p,q [ n − k ] p,q (cid:20) n − kk (cid:21) p,q . (84)By using (75), one can show that the ( p − , q − ) − coefficients (84) can be expressed as: c ( L ) n,k ( p − , q − ) = ( pq ) k ( k − n ) c ( L ) n,k ( p, q ) (85)permiting to define the ( p − , q − ) − deformed Lucas polynomials as follows: L n ( x, s | p − , q − ) := ⌊ n/ ⌋ X k =0 c ( L ) n,k ( p − , q − ) s k x n − k = x n ϕ ( p − n , q − n ) , ( p − n , q − n ) , ( p − n , − q − n ) , ( p − n , − q − n )( p − n , q − n ) , , (cid:12)(cid:12)(cid:12)(cid:12) ( p, q ); − spqx . (86)The limit when p → q − − Lucas polynomials [3]: L n ( x, s | q − ) = ⌊ n/ ⌋ X k =0 c ( L ) n,k ( q − ) s k x n − k = x n ϕ q − n , q − n , − q − n , − q − n q − n , , (cid:12)(cid:12)(cid:12)(cid:12) q ; − sqx . (87)Their associated q − − Lucas numbers are found when x = s = 1 as: L n ( q − ) = ϕ q − n , q − n , − q − n , − q − n q − n , , (cid:12)(cid:12)(cid:12)(cid:12) q ; − q . (88)Finally, we have the following: Theorem 3.2
The Fourier transform of the function e − x / L n ( be iκx , s | p, q ) is given by √ π Z R L n ( be iκx , s | p, q ) e ixy − x / dx = ( pq ) n L n ( be − κ y , ( pq ) − s | p − , q − ) e − y / (89) providing the formula L n ( b, s | p, q ) = 12 π Z R Z R L n ( be iκx , s | p, q ) e ixy − x / dxdy, (90) where b is an arbitrary constant factor and q = p − e − κ . Proof.
The proof is immediate from the Theorem 3.1. (cid:3)
In the limit when the parameter p →
1, the Fourier transform (89) is reduced tothe well-known formula derived by Atakishiyev et al [3]1 √ π Z R L n ( be iκx , s | q ) e ixy − x / dx = q n L n ( be − κ y , q − s | q − ) e − y / , (91)where a is an arbitrary constant factor and q = e − κ . p, q ) − deformed Fibonacci and Lucas polynomials: characterization and Fourier integral transforms
4. Conclusion
In the present work, a full characterization of ( p, q ) − deformed Fibonacci and Lucaspolynomials has been achieved. These polynomials obey non-conventional three-termrecursion relations as previously shown for their q − analogs. Besides, the formulae for thecomputation of the associated Fourier integral transforms have been deduced. Previousknown results have been recovered as particular cases and properly discussed. Acknowledgements
This work is partially supported by the Abdus Salam International Centre forTheoretical Physics (ICTP, Trieste, Italy) through the Office of External Activities(OEA) -Prj-15. The ICMPA is in partnership with the Daniel Iagolnitzer Foundation(DIF), France.
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