New expansions for x n ± y n in terms of quadratic forms
aa r X i v : . [ m a t h . G M ] F e b NEW EXPANSIONS FOR x n ± y n IN TERMS OF QUADRATIC FORMS
MOUSTAFA IBRAHIMDEPARTMENT OF MATHEMATICS, COLLEGE OF SCIENCE,UNIVERSITY OF BAHRAIN, KINGDOM OF BAHRAIN
Abstract.
We prove new theorems for the polynomial expansions of x n ± y n in terms of thebinary quadratic forms αx + βxy + αy and ax + bxy + ay . The paper gives new arithmeticdifferential approach to compute the coefficients. Also, the paper gives generalization towell-known polynomial identity in the history of number theory. The paper highlights theemergence of a new class of polynomials that unify many well-known sequences including theChebyshev polynomials of the first and second kind, Dickson polynomials of the first andsecond kind, Lucas and Fibonacci numbers, Mersenne numbers, Pell polynomials, Pell-Lucaspolynomials, and Fermat numbers. Also, this paper highlights the emergence of the notions oftrajectories and orbits of certain integers that passes through many well-known polynomialsand sequences. The Lucas-Fibonacci trajectory, the Lucas-Pell trajectory, the Fibonacci-Pelltrajectory, the Fibonacci-Lucas trajectory, the Chebyshev-Dickson trajectory of the first kind,the Chebyshev-Dickson trajectory of the second kind, and others are new trajectories includedin this paper. Also, the Lucas orbit, Fibonacci orbit, Mersenne orbit, Lucas-Fibonacci orbit,Fermat orbit, and others are new orbits included in this paper. Summary Of The Main Results Of The Paper
The Main Theorems of The Paper.
In this paper, we give new arithmetic differentialapproach to compute the coefficients of the polynomial expansions of x n ± y n in terms of thebinary quadratic forms αx + βxy + αy and ax + bxy + ay , βa − αb = 0. Regarding thepolynomials x n + y n , we prove the following new polynomial expansions for any given variables α, β, a, b , and for any natural number n ,( βa − αb ) ⌊ n ⌋ x n + y n ( x + y ) δ ( n ) = ⌊ n ⌋ X r =0 ( − r r ! ( αx + βxy + αy ) ⌊ n ⌋− r ( ax + bxy + ay ) r (cid:16) α ∂∂a + β ∂∂b (cid:17) r Ψ( a, b, n ) (1.1)where δ ( n ) = 1 for n odd and δ ( n ) = 0 for n even and ⌊ n ⌋ is the highest integer less than orequal n . Regarding the polynomials x n − y n , we prove and study the following new polynomialsexpansions ( βa − αb ) ⌊ n − ⌋ x n − y n ( x − y )( x + y ) δ ( n − = ⌊ n − ⌋ X r =0 ( − r r ! ( αx + βxy + αy ) ⌊ n − ⌋− r ( ax + bxy + ay ) r (cid:16) α ∂∂a + β ∂∂b (cid:17) r Φ( a, b, n ) (1.2) Computing The Polynomials Ψ r ( a, b, n ) and Φ r ( a, b, n ) . The polynomials Ψ r ( a, b, n ) andΦ r ( a, b, n ) enjoy fascinating arithmetical and differential properties and unify many well-knownFEBRUARY 9, 2020 1olynomials. This paper presents three different approaches to compute the polynomialsΨ r ( a, b, n ) and Φ r ( a, b, n ). We prove in this paper the following explicit formulasΨ( a, b, n ) = (2 a − b ) ⌊ n ⌋ n ( r b + 2 ab − a ! n + − r b + 2 ab − a ! n ) , Φ( a, b, n ) = (2 a − b ) ⌊ n − ⌋ n q b +2 ab − a ( r b + 2 ab − a ! n − − r b + 2 ab − a ! n ) . Example For (1.1) . To show simple example for (1.1), we take n = 4. Simple computationsshow that ( − (cid:16) α ∂∂a + β ∂∂b (cid:17) Ψ( a, b,
4) = Ψ( a, b,
4) = − a + b , ( − (cid:16) α ∂∂a + β ∂∂b (cid:17) Ψ( a, b,
4) = 4 aα − bβ, ( − (cid:16) α ∂∂a + β ∂∂b (cid:17) Ψ( a, b,
4) = − α + β . (1.3)Therefore we get the following polynomial identity which is a special case for the main resultsof the current paper( βa − αb ) ( x + y ) = ( − a + b )( αx + βxy + αy ) + (4 aα − bβ )( αx + βxy + αy )( ax + bxy + ay )+ ( − α + β )( ax + bxy + ay ) . (1.4)This particular example gives natural generalization to one of the well-known polynomialidentities in the history of number theory as we see in section (8). The Emergence of New Class of Polynomials.
For any natural number n , and for any r = 0 , , , . . . , ⌊ n ⌋ , as we see in the current paper, the following polynomials( − r r ! (cid:16) α ∂∂a + β ∂∂b (cid:17) r Ψ r ( a, b, n ) , ( − r r ! (cid:16) α ∂∂a + β ∂∂b (cid:17) r Φ r ( a, b, n )enjoy unexpected new arithmetical and differential properties. Actually, it gives new class ofpolynomials. Therefore, we need to define the following new polynomialsΨ (cid:18) a b nα β r (cid:19) = ( − r r ! (cid:16) α ∂∂a + β ∂∂b (cid:17) r Ψ r ( a, b, n ) , Φ (cid:18) a b nα β r (cid:19) = ( − r r ! (cid:16) α ∂∂a + β ∂∂b (cid:17) r Φ r ( a, b, n ) . Also, for any a, b, α, β, η, ξ, n , βa − αb = 0 , we prove the following new and unexpected identities ⌊ n ⌋ X r =0 Ψ (cid:18) a b nα β r (cid:19) ξ ⌊ n ⌋− r η r = Ψ( aξ − αη, bξ − βη, n ) , ⌊ n − ⌋ X r =0 Φ (cid:18) a b nα β r (cid:19) ξ ⌊ n − ⌋− r η r = Φ( aξ − αη, bξ − βη, n ) . (1.5)2 VOLUME, NUMBEREW EXPANSIONS FOR X N ± Y N IN TERMS OF QUADRATIC FORMS
Unification of Well-Known Sequences.
The polynomials Ψ r ( a, b, n ) and Φ r ( a, b, n ) enjoyfascinating arithmetical and differential properties and unify many well-known polynomialsand sequences including the Chebyshev polynomials of the first and second kind, Dicksonpolynomials of the first kind and second kind, Lucas numbers, Fibonacci numbers, Fermatnumbers, Pell-Lucas polynomials, Pell numbers and others, see [13], [14]. Links With Fibonacci and Lucas Sequences.
We should notice that Ψ and Φ polynomialsare greatly linked with Lucas and Fibonacci sequences through variety of formulas includingthe following formulas, for any variables α, β ,Ψ( − , − , n ) = L ( n ) , Ψ(1 , − , n ) = ( F ( n ) for n odd L ( n ) for n even , Φ( − , − , n ) = F ( n ) , Φ(1 , − , n ) = ( F ( n ) for n even L ( n ) for n odd , where L ( n ) are the Lucas numbers defined by the recurrence relation L (0) = 2 , L (1) = 1 and L ( n + 1) = L ( n ) + L ( n −
1) and F ( n ) are the Fibonacci numbers defined by the recurrencerelation F (0) = 0 , F (1) = 1 and F ( n + 1) = F ( n ) + F ( n − Links With Dickson Polynomials.
The Dickson polynomial of the first kind of degree n with parameter α , D n ( x, α ), and the Dickson polynomial of the second kind of degree n withparameter α , E n ( x, α ), are defined by the following formulas, where δ ( n ) = 1 for n odd and δ ( n ) = 0 for n even, D n ( x, α ) = ⌊ n ⌋ X i =0 nn − i (cid:18) n − ii (cid:19) ( − α ) i x n − i , E n ( x, α ) = ⌊ n ⌋ X i =0 (cid:18) n − ii (cid:19) ( − α ) i x n − i . We show in this paper that the following relations hold D n ( x, α ) = x δ ( n ) Ψ( α, α − x , n ) , E n ( x, α ) = x δ ( n ) Φ( α, α − x , n + 1) . Links With Chebyshev Polynomials.
The Chebyshev polynomial of the first kind ofdegree n , T n ( x ), and the Chebyshev polynomial of the second kind of degree n , U n ( x ), aredefined by the following formulas T n ( x ) = ⌊ n ⌋ X i =0 ( − i nn − i (cid:18) n − ii (cid:19) (2) n − i − x n − i , U n ( x ) = ⌊ n ⌋ X i =0 (cid:18) n − ii (cid:19) ( − i (2 x ) n − i . We show in this paper that the following relations hold T n ( x ) = x δ ( n ) δ ( n +1) Ψ(1 , − x , n ) , U n ( x ) = (2 x ) δ ( n ) Φ(1 , − x , n + 1) . Links With Mersenne and Fermat Numbers.
The selections (2 , −
5) and ( − , −
5) for( a, b ) will connect the Ψ − Polynomial with Mersenne numbers. Actually, for each naturalnumber n we get the following desirable formulasΨ( − , − , n ) = 2 n + ( − n , Φ( − , − , n ) = 2 n − ( − n , Ψ(2 , − , n ) = 2 n + 13 δ ( n ) , Φ(2 , − , n ) = 2 n − δ ( n − . (1.6)FEBRUARY 9, 2020 3nd hence for p odd we get Ψ( − , − , p ) = Φ(2 , − , p ) = 2 p − M p , where M p are calledMersenne numbers. Mersenne primes M p , for some prime p , are also noteworthy due to theirconnection with primality tests, and perfect numbers. For some useful properties for theMersenne numbers, the readers should consult with [3]. In number theory, a perfect numberis a positive integer that is equal to the sum of its proper positive divisors, that is, the sumof its positive divisors excluding the number itself. It is unknown whether there is any oddperfect number. Recently, [11] showed that odd perfect numbers are greater than 10 . Also,we should observe that Ψ( − , − , n ) = Ψ(2 , − , n ) = 2 n + 1 = F n , where F n is the Fermatnumber. Links With Pell Numbers and Polynomials.
The Pell numbers P n are defined by therecurrence relation P = 0, P = 1, P n = 2 P n − + P n − . The following formula shows thatthe Φ polynomial is also natural generalization for Pell numbers P n = 2 δ ( n − Φ( − , − , n ).Actually, the Pell numbers P n = P n (1), where P n ( x ) is the Pell polynomial defined by therecurrence relation P ( x ) = 0 , P ( x ) = 1 and P n +1 ( x ) = 2 xP n ( x ) + P n − ( x ). Similarly, we candeduce the following desirable relation P n ( x ) = (2 x ) δ ( n − Φ( − , − − x , n ). The Pell-Lucasnumbers Q n are defined by the recurrence relation Q = 2, Q = 2, Q n = 2 Q n − + Q n − . Thefollowing formula shows that Ψ polynomial is a generalization for Pell-Lucas numbers Q n =2 δ ( n ) Ψ( − , − , n ). Also, the Pell-Lucas numbers Q n = Q n (1), where Q n ( x ) is the Pell-Lucaspolynomial defined by the recurrence relation Q ( x ) = 2 , Q ( x ) = 2 x, Q n +1 ( x ) = 2 xQ n ( x ) + Q n − ( x ). Similarly, we can deduce the following relation Q n ( x ) = (2 x ) δ ( n ) Ψ( − , − − x , n ). Trajectories and Orbits Connect Well-Known Polynomials and Sequences.
Thenotions of “trajectories”, specific sequence of polynomials, and “orbits”, which are specialcases of trajectories, are naturally arise up in this paper. The Ψ and Φ polynomials are notonly fundamental for the future developments of the study of the polynomial expansions of x n + y n and x n − y n in terms of binary quadratic forms but also to discover new trajectoriesand orbits “connecting” well-known polynomials and numbers and relate them with unsolvedDiophantine problems. For example, the following particular trajectory has arithmetic interest. Trajectory From Sums of Powers to Another and Open Questions.
For any naturalnumber n , we show in this paper that we can establish a trajectory from the polynomialsΨ r ( n ) that “connects” the polynomials x n + y n ( x + y ) δ ( n ) with the polynomials z n + t n ( z + t ) δ ( n ) . x n + y n ( x + y ) δ ( n ) z n + t n ( z + t ) δ ( n ) We consider the following trajectory of polynomials x n + y n ( x + y ) δ ( n ) = Ψ ( n ) , Ψ ( n ) , . . . , Ψ ⌊ n ⌋ ( n ) = z n + t n ( z + t ) δ ( n ) where Ψ r ( n ) = Ψ (cid:18) + xy − x − y n − zt + z + t r (cid:19) , which are thecoefficients of the following polynomial expansion( zx − ty ) ⌊ n ⌋ ( zy − tx ) ⌊ n ⌋ u n + v n ( u + v ) δ ( n ) = ⌊ n ⌋ X r =0 Ψ r ( n )( zu − tv ) ⌊ n ⌋− r ( zv − tu ) ⌊ n ⌋− r (( ux − vy ) r ( uy − vx )) r X N ± Y N IN TERMS OF QUADRATIC FORMSFor this particular trajectory, we show that for any numbers x, y, z, t, xz = yt, xt = yz , anynatural number n , the following identity hold ⌊ n ⌋ X r =0 Ψ (cid:18) + xy − x − y n − zt + z + t r (cid:19) = Ψ( xy + zt, − x − y − z − t , n ) . This particular trajectory should be surrounded by lots of mathematical questions, from num-ber theory to dynamical systems. One of these, when the endpoints coincident with eachother, it gives an orbit. If such orbit exists, we get a solution to many unsolved Diophantineequations. We should ask, if such orbit exist, what are the general arithmetic characteristicsof such orbit? For example, what is the characteristics of this orbit, if exist, for n = 5. And itis natural to ask what is the formula for n such that this orbit exist. Can such orbits exist forinfinite values for n ? Alternatively, this is equivalent to ask whether there is any nontrivialinteger solutions ( x, y, z, t ) and any natural number n for following Diophantine equation x n + y n ( x + y ) δ ( n ) = z n + t n ( z + t ) δ ( n ) . Fermat Orbit And Open Questions.
We present in this paper what we call “Fermat orbit”which is the sequence η ( r ) of integers2 n + 1 = η (0) , η (1) , η (2) , . . . , η (2 n − ) = 2 n + 1 , which start with Fermat numbers 2 n + 1 and ended with Fermat numbers 2 n + 1 which arethe coefficients of the following polynomial expansion20 n − ( x n + y n ) = n − X r =0 η ( r )( − x + 5 xy − y ) n − − r ( − x − xy − y ) r . We show in this paper that η ( r ) = Ψ (cid:18) − − n − r (cid:19) . F n = 2 n + 1 A Fermat prime is a Fermat number F n = 2 n + 1 that is prime.Studying Fermat orbit should help understand the arithmetic ofFermat numbers which still are extremely ambiguous. There are only five known Fermat primes [12]. The five known Fermat primes are F = 3 , F =5 , F = 17 , F = 257 , F = 65537. Only seven Fermat numbers have been completely factored; F , F , F , F , F , F , F . We should ask what is the arithmetic relations bewteen the primefactors of η (1) , η (2) , . . . , η (2 n − ) −
1, and 2 n + 1. Does an efficient algorithms exist based onthe terms of Fermat orbit to factor 2 n + 1? This orbit should have links with the arithmeticof Fermat numbers.FEBRUARY 9, 2020 5. DEFINITION OF Ψ( a, b, n ) and Φ( a, b, n ) POLYNOMIALS
In [8], I first introduced the definitions for Ψ and Φ polynomials as following
Definition 2.1.
We define δ ( n ) = 1 for n odd and δ ( n ) = 0 for n even. For any given variables a, b and for any natural number n , we define the sequenceΨ( a, b, n ) = Ψ( n ) , Φ( a, b, n ) = Φ( n ) , by the following recurrence relationsΨ(0) = 2 , Ψ(1) = 1 , Ψ( n + 1) = (2 a − b ) δ ( n ) Ψ( n ) − a Ψ( n − , (2.1)Φ(0) = 0 , Φ(1) = 1 , Φ( n + 1) = (2 a − b ) δ ( n +1) Φ( n ) − a Φ( n −
1) (2.2)3.
POLYNOMIAL EXPANSIONS IN TERMS OF BINARY QUADRATIC FORMS
In [8], page 447, using the formal derivation, we proved the following identities x n + y n = ⌊ n ⌋ X i =0 ( − i nn − i (cid:18) n − ii (cid:19) ( xy ) i ( x + y ) n − i (3.1a) x n − y n x − y = ⌊ n − ⌋ X i =0 ( − i (cid:18) n − i − i (cid:19) ( xy ) i ( x + y ) n − i − (3.1b)where ⌊ m ⌋ denotes the largest integer ≤ m . Already Kummer and others used the identities(3.1a), (3.1b), and for more details you can consult with [1], [2], [5] [6], [7], [9], and [10]. Theorem 3.1.
For any given variables α, β, a, b , and for any natural number n , there existpolynomials in a, b, α, β with integer coefficients, that we call Ψ (cid:18) a b nα β r (cid:19) , Φ (cid:18) a b nα β r (cid:19) that depend only on α, β, a, b, n, and r , and satisfy the following polynomial identities ( βa − αb ) ⌊ n ⌋ x n + y n ( x + y ) δ ( n ) = ⌊ n ⌋ X r =0 Ψ (cid:18) a b nα β r (cid:19) ( αx + βxy + αy ) ⌊ n ⌋− r ( ax + bxy + ay ) r , (3.2)( βa − αb ) ⌊ n − ⌋ x n − y n ( x − y )( x + y ) δ ( n − = ⌊ n − ⌋ X r =0 Φ (cid:18) a b nα β r (cid:19) ( αx + βxy + αy ) ⌊ n − ⌋− r ( ax + bxy + ay ) r . (3.3) Proof.
From (3.1a), (3.1b), we obtain x n + y n ( x + y ) δ ( n ) = ⌊ n ⌋ X i =0 ( − i nn − i (cid:18) n − ii (cid:19) ( xy ) i ( x + 2 xy + y ) ⌊ n ⌋ − i (3.4)and x n − y n ( x − y )( x + y ) δ ( n − = ⌊ n − ⌋ X i =0 ( − i (cid:18) n − i − i (cid:19) ( xy ) i ( x + 2 xy + y ) ⌊ n − ⌋ − i (3.5)Now, it is clear that for any variables a, b, α, β , the following identities are true( βa − αb )( x + 2 xy + y ) = (2 a − b )( αx + βxy + αy ) + ( β − α )( ax + bxy + ay ) , (3.6)6 VOLUME, NUMBEREW EXPANSIONS FOR X N ± Y N IN TERMS OF QUADRATIC FORMS( βa − αb ) xy = a ( αx + βxy + αy ) + ( − α )( ax + bxy + ay ) (3.7)Now multiplying (3.4) by ( βa − αb ) ⌊ n ⌋ and (3.5) by ( βa − αb ) ⌊ n − ⌋ , with using (3.6), (3.7),we obtain the proof. (cid:3) ALGEBRAIC INDEPENDENCE OF BINARY QUADRATIC FORMS
For the following special case, applying Jacobian criterion for the special case f , f , [15],which says that polynomials f and f are algebraically independent if the determinant of theJacobian J is non zero where J = ∂f ∂x ∂f ∂y∂f ∂x ∂f ∂y ! . So working out with the Jacobian for the polynomials f = αx + βxy + αy and f = ax + bxy + ay , we get J = ∂f ∂x ∂f ∂y∂f ∂x ∂f ∂y ! = (cid:18) αx + βy βx + 2 αy ax + by bx + 2 ay (cid:19) . So | J | = 2( βa − αb )( y − x ). Therefore we get the following Lemma 4.1.
For any given numbers a, b, α, β , βa − αb = 0 , x = y , the binary quadratic forms αx + βxy + αy and ax + bxy + ay are algebraically independent. COMPUTING Ψ AND Φ POLYNOMIALS FOR r = 0 Theorem 5.1.
For any numbers a, b, α, β, n , a − b = 0 , βa − αb = 0 , we have Ψ (cid:18) a b nα β (cid:19) = Ψ( a, b, n ) = (2 a − b ) ⌊ n ⌋ n ( r b + 2 ab − a ! n + − r b + 2 ab − a ! n ) , (5.1a)Φ (cid:18) a b nα β (cid:19) = Φ( a, b, n ) = (2 a − b ) ⌊ n − ⌋ n q b +2 ab − a ( r b + 2 ab − a ! n − − r b + 2 ab − a ! n ) . (5.1b) Proof.
First, we prove that Ψ (cid:18) a b nα β (cid:19) = Ψ( a, b, n ). Let a, b, α, β, n be given numberssuch that 2 a − b = 0, βa − αb = 0. Define H ( n ) := Ψ (cid:18) a b nα β (cid:19) From (3.2), the following congruence relations are true for any nonnegative integer m ,( βa − αb ) ⌊ m ⌋ x m + y m ( x + y ) δ ( m ) ≡ H ( m )( αx + βxy + αy ) ⌊ m ⌋ (mod ( ax + bxy + ay )) (5.2)We should notice that H (0) = 2 , H (1) = 1. Now, multiply (5.2) by (2 a − b ) ⌊ m ⌋ and notingfrom (3.6) that(2 a − b )( αx + βxy + αy ) ≡ ( βa − αb )( x + 2 xy + y ) (mod ax + bxy + ay ) , FEBRUARY 9, 2020 7e obtain(2 a − b ) ⌊ m ⌋ ( βa − αb ) ⌊ m ⌋ x m + y m ( x + y ) δ ( m ) ≡ ( βa − αb ) ⌊ m ⌋ H ( m )( x + 2 xy + y ) ⌊ m ⌋ (mod ( ax + bxy + ay )) (5.3)As (5.3) true for any x, y , 2 a − b = 0, βa − αb = 0, we can remove ( βa − αb ) ⌊ m ⌋ from bothsides to obtain the following congruence relation(2 a − b ) ⌊ m ⌋ x m + y m ( x + y ) δ ( m ) ≡ H ( m )( x + 2 xy + y ) ⌊ m ⌋ (mod ( ax + bxy + ay )) (5.4)Multiply both sides by ( x + y ) δ ( m ) , and noting that δ ( m ) + 2 ⌊ m ⌋ = m , we obtain(2 a − b ) ⌊ m ⌋ ( x m + y m ) ≡ H ( m )( x + y ) m (mod ( ax + bxy + ay )) (5.5)Therefore, for any nonnegative integer n , we obtain the following three congruence relations(2 a − b ) ⌊ n +12 ⌋ ( x n +1 + y n +1 ) ≡ H ( n + 1)( x + y ) n +1 (mod ( ax + bxy + ay )) (5.6a)(2 a − b ) ⌊ n ⌋ ( x n + y n ) ≡ H ( n )( x + y ) n (mod ( ax + bxy + ay )) (5.6b)(2 a − b ) ⌊ n − ⌋ ( x n − + y n − ) ≡ H ( n − x + y ) n − (mod ( ax + bxy + ay )) (5.6c)It is clear that (2 a − b ) xy ≡ a ( x + y ) (mod ( ax + bxy + ay )) (5.7)Now consider the following polynomial identity(2 a − b ) ⌊ n +12 ⌋ ( x n +1 + y n +1 )= (2 a − b ) δ ( n ) ( x + y )(2 a − b ) ⌊ n ⌋ ( x n + y n ) − (2 a − b ) xy (2 a − b ) ⌊ n − ⌋ ( x n − + y n − ) (5.8)From (5.6a) (5.6b), (5.6c), (5.7), and (5.8) we obtain H ( n +1)( x + y ) n +1 ≡ (2 a − b ) δ ( n ) H ( n )( x + y ) n +1 − aH ( n − x + y ) n +1 (mod ( ax + bxy + ay ))(5.9)As 2 a − b = 0, we can choose x and y such that ax + bxy + ay and x + y are relatively primes,and hence we can remove ( x + y ) n +1 from (5.9) to get H ( n + 1) ≡ (2 a − b ) δ ( n ) H ( n ) − aH ( n −
1) (mod ( ax + bxy + ay )) (5.10)Therefore the congruence (5.10) must turn into identity and hence we obtain H (0) = 2 , H (1) = 1 , H ( n + 1) = (2 a − b ) δ ( n ) H ( n ) − aH ( n −
1) (5.11)From (5.11) and (2.1) we get H ( n ) = Ψ( a, b, n ). Now put x = x = 12 (cid:16) r b + 2 ab − a (cid:17) ,y = y = 12 (cid:16) − r b + 2 ab − a (cid:17) . (5.12)Clearly ax + bx y + ay = 0, and hence the modulus of the congruence (5.6b) vanish forevery natural number n which means that(2 a − b ) ⌊ n ⌋ ( x n + y n ) = H ( n )( x + y ) n . (5.13)Then, from (5.13), and as x + y = 1, H ( n ) = Ψ( a, b, n ), we getΨ( a, b, n ) = (2 a − b ) ⌊ n ⌋ ( x n + y n ) . X N ± Y N IN TERMS OF QUADRATIC FORMSTherefore, we obtain the full proof of (5.1a). By the similar argument we can prove (5.1b). (cid:3)
Simple calculations, together with the polynomial identities (3.1a), (3.1b), and (5.1a),(5.1b), we get the following desirable polynomial expansions
Theorem 5.2.
For any numbers a, b and any natural number n , the following formulas aretrue Ψ( a, b, n ) = ⌊ n ⌋ X i =0 nn − i (cid:18) n − ii (cid:19) ( − a ) i (2 a − b ) ⌊ n ⌋ − i , (5.14a)Φ( a, b, n ) = ⌊ n − ⌋ X i =0 (cid:18) n − i − i (cid:19) ( − a ) i (2 a − b ) ⌊ n − ⌋ − i (5.14b)6. COMPUTING Ψ AND Φ POLYNOMIALS
Also, we should notice that there is some symmetry in (3.2), (3.3) and( βa − αb ) ⌊ m ⌋ = ( − ⌊ m ⌋ ( bα − aβ ) ⌊ m ⌋ . Therefore we obtain the following desirable formulas
Theorem 6.1.
For any numbers a, b, α, β, n , we have Ψ (cid:18) a b nα β ⌊ n ⌋ (cid:19) = ( − ⌊ n ⌋ Ψ( α, β, n ) , (6.1a)Φ (cid:18) a b nα β ⌊ n − ⌋ (cid:19) = ( − ⌊ n − ⌋ Φ( α, β, n ) . (6.1b)7. NOTATIONS
Notation 7.1.
Throughout this paper, and for specific given variables a, b, α, β , we writeΨ r ( n ) = Ψ (cid:18) a b nα β r (cid:19) , Φ r ( n ) = Φ (cid:18) a b nα β r (cid:19) Notation 7.2.
Suppose that P ( x i ) = Q ( x i ) is any polynomial identity in the variables x , x , . . . , x n and suppose a , a , . . . , a n are any given parameters. If we differentiate P ( x i ) = Q ( x i ) with respect to the differential operator defined by a ∂∂x + a ∂∂x + · · · + a n ∂∂x n , then wewould say that we apply the differential map g that sends x −→ a , x −→ a , . . . , x n −→ a n to P ( x i ) = Q ( x i ). Alternatively, we say that we differentiate P ( x i ) = Q ( x i ) with respectto the differential map g that sends x −→ a , x −→ a , . . . , x n −→ a n if we apply theoperator a ∂∂x + a ∂∂x + · · · + a n ∂∂x n to P ( x i ) = Q ( x i ). Notation 7.3.
Suppose that f ( x i ) is any polynomial in the variables x , x , . . . , x n and sup-pose a , a , . . . , a n are any given parameters. We define r ] f ( x i ) := (cid:16) a ∂∂x + a ∂∂x + · · · + a n ∂∂x n (cid:17) r f ( x i )and ] f ( x i ) := f ( x i ). Moreover, we say that f ( x i ) −→ g ( x i ) with respect to the differentialmap g that sends x −→ a , x −→ a , . . . , x n −→ a n if g ( x i ) = a ∂∂x f ( x i ) + a ∂∂x f ( x i ) + · · · + a n ∂∂x n f ( x i ) . FEBRUARY 9, 2020 9 otation 7.4.
If we list parameters, letters, x , x , . . . , x n , we should look at every parameterin this paper as a variable. And parameter x k is considered constant with respect to a particularmap g only if x k −→ Example 7.5.
By differentiating ax + bxy + ay with respect to the differential map a −→ α, b −→ β, α −→ , β −→ , x −→ , y −→ , we obtain (cid:16) α ∂∂a + β ∂∂b + 0 ∂∂α +0 ∂∂β + 0 ∂∂x + 0 ∂∂y (cid:17) ( ax + bxy + ay )= (cid:16) α ∂∂a + β ∂∂b (cid:17) ( ax + bxy + ay )= α ∂∂a ( ax + bxy + ay ) + β ∂∂b ( ax + bxy + ay )= αx + βxy + αy Therefore we say that ax + bxy + ay −→ αx + βxy + αy with respect to the differential map a −→ α, b −→ β, α −→ , β −→ , x −→ , y −→
0. Also, we can show that by differentiating( βa − αb ) ⌊ n ⌋ with respect to the differential map a −→ α, b −→ β, α −→ , β −→ , x −→ , y −→ , we get (cid:16) α ∂∂a + β ∂∂b + 0 ∂∂α + 0 ∂∂β +0 ∂∂x + 0 ∂∂y (cid:17) ( βa − αb ) ⌊ n ⌋ = (cid:16) α ∂∂a + β ∂∂b (cid:17) ( βa − αb ) ⌊ n ⌋ = ⌊ n ⌋ ( βa − αb ) ⌊ n ⌋− (cid:16) α ∂∂a + β ∂∂b (cid:17) ( βa − αb )= ⌊ n ⌋ ( βa − αb ) ⌊ n ⌋− (cid:16) α ∂∂a ( βa − αb ) + β ∂∂b ( βa − αb ) (cid:17) = ⌊ n ⌋ ( βa − αb ) ⌊ n ⌋− (cid:16) αβ ∂∂a a − βα ∂∂b b (cid:17) = 0Therefore we say that ( βa − αb ) ⌊ n ⌋ −→ a −→ α, b −→ β, α −→ , β −→ , x −→ , y −→ . Example 7.6.
With respect to the following particular differential map a −→ − a − b, b −→ − b − a, α −→ − α − β, β −→ − β − α, x −→ y, y −→ x, we get (cid:16) ( − a − b ) ∂∂a +( − b − a ) ∂∂b + ( − α − β ) ∂∂α + ( − β − α ) ∂∂β + y ∂∂x + x ∂∂y (cid:17) ( ax + bxy + ay )= (cid:16) ( − a − b ) ∂∂a + ( − b − a ) ∂∂b + y ∂∂x + x ∂∂y (cid:17) ( ax + bxy + ay )= ( − a − b )( x + y ) + ( − b − a )( xy ) + y (2 ax + by ) + x ( bx + 2 ay )= − ( ax + bxy + ay )Therefore, for this particular differential map, we get ax + bxy + ay −→ − ( ax + bxy + ay )10 VOLUME, NUMBEREW EXPANSIONS FOR X N ± Y N IN TERMS OF QUADRATIC FORMSAlso, with respect to this particular differential map, we can easily check that αx + βxy + αy −→ − αx − βxy − αy ( βa − αb ) −→ − βa − αb )( x + y ) δ ( n ) −→ δ ( n )( x + y ) δ ( n ) x n + y n −→ n ( x n − y + y n − x ) n ( x n − y + y n − x ) −→ n ( x n + y n ) + n ( n − x n − y + y n − x )This particular differential map is useful for studying the expansions of the polynomials of theform x n y m + x m y n DIFFERENTIAL APPROACHES FOR COMPUTING Ψ AND Φ POLYNOMIALS
For βa − αb = 0, defineΨ r ( n ) := Ψ (cid:18) a b nα β r (cid:19) and Φ r ( n ) := Φ (cid:18) a b nα β r (cid:19) To compute Ψ r ( n ) , Φ r ( n ), consider the differential map g that sends a −→ α, b −→ β, α −→ , β −→ , x −→ , y −→ . We differentiate (3.2) with respect to this particular differential map and noting from Example(7.5) the following ( βa − αb ) ⌊ n ⌋ −→ , x n + y n ( x + y ) δ ( n ) −→ . (8.1)Therefore ( α ∂∂a + β ∂∂b )( βa − αb ) ⌊ n ⌋ x n + y n ( x + y ) δ ( n ) = 0 . (8.2)Now acting ( α ∂∂a + β ∂∂b ) on (3.2), we obtain0 = (cid:16) α ∂∂a + β ∂∂b (cid:17) ⌊ n ⌋ X r =0 Ψ r ( n )( αx + βxy + αy ) ⌊ n ⌋− r ( ax + bxy + ay ) r = ⌊ n ⌋ X r =0 ( αx + βxy + αy ) ⌊ n ⌋− r ( ax + bxy + ay ) r (cid:16) α ∂∂a + β ∂∂b (cid:17) Ψ r ( n )+ ⌊ n ⌋ X r =0 Ψ r ( n )( αx + βxy + αy ) ⌊ n ⌋− r (cid:16) α ∂∂a + β ∂∂b (cid:17) ( ax + bxy + ay ) r + ⌊ n ⌋ X r =0 Ψ r ( n )( ax + bxy + ay ) r (cid:16) α ∂∂a + β ∂∂b (cid:17) ( αx + βxy + αy ) ⌊ n ⌋− r (8.3)Consequently, from (8.3), we obtain the following desirable polynomial identities0 = ⌊ n ⌋ X r =0 (cid:16) ^ Ψ r ( n ) + ( r + 1)Ψ r +1 ( n ) (cid:17) ( αx + βxy + αy ) ⌊ n ⌋− r ( ax + bxy + ay ) r . (8.4)FEBRUARY 9, 2020 11here, according to our notations, ^ Ψ r ( n ) := ( α ∂∂a + β ∂∂b )Ψ r ( n ). Similarly, we differentiate (3.3)with same particular differential map. We should have no difficulty to obtain the followingidentity0 = ⌊ n − ⌋ X r =0 (cid:16) ^ Φ r ( n ) + ( r + 1)Φ r +1 ( n ) (cid:17) ( αx + βxy + αy ) ⌊ n − ⌋− r ( ax + bxy + ay ) r (8.5)where, according to our notations, ^ Φ r ( n ) := ( α ∂∂a + β ∂∂b )Φ r ( n ). As βa − αb = 0, from Lemma(4.1), the polynomials ( αx + βxy + αy ) and ( ax + bxy + ay ) are algebraic independentwhich means that all of the coefficients of (8.4), and (8.5) must vanish. This means that withrespect to the differential map a −→ α, b −→ β, α −→ , β −→
0, we obtain ^ Ψ r ( n ) + ( r + 1)Ψ r +1 ( n ) = 0 , ∀ ⌊ n ⌋− r =0 r ^ Φ r ( n ) + ( r + 1)Φ r +1 ( n ) = 0 , ∀ ⌊ n − ⌋− r =0 r (8.6)Similarly, with respect to the differential map α −→ a, β −→ b, a −→ , b −→
0, we get ^ Ψ r ( n ) + ( ⌊ n ⌋ − r + 1)Ψ r − ( n ) = 0 , ∀ ⌊ n ⌋ r =1 r ^ Φ r ( n ) + ( ⌊ n − ⌋ − r + 1)Φ r − ( n ) = 0 , ∀ ⌊ n − ⌋ r =1 r (8.7)This immediately gives the following desirable theorems Theorem 8.1.
For any natural number n , for any parameters a, b, α, β, βa − αb = 0 , let Ψ r ( n ) := Ψ (cid:18) a b nα β r (cid:19) and Φ r ( n ) := Φ (cid:18) a b nα β r (cid:19) . Consider the differential map g that sends a −→ α, b −→ β, α −→ , β −→ . Then with respect to this differential map we get Ψ r ( n ) = − r ^ Ψ r − ( n ) ∀ ⌊ n ⌋ r =1 r , Φ r ( n ) = − r ^ Φ r − ( n ) ∀ ⌊ n − ⌋ r =1 r (8.8) Theorem 8.2.
For any natural number n , for any parameters a, b, α, β, βa − αb = 0 , let Ψ r ( n ) := Ψ (cid:18) a b nα β r (cid:19) and Φ r ( n ) := Φ (cid:18) a b nα β r (cid:19) . Consider the differential map g that sends α −→ a, β −→ b, a −→ , b −→ . Then with respect to this differential map we get Ψ r ( n ) = − ⌊ n ⌋ − r ^ Ψ r +1 ( n ) ∀ ⌊ n ⌋− r =0 r , Φ r ( n ) = − ⌊ n ⌋ − r ^ Φ r +1 ( n ) ∀ ⌊ n − ⌋− r =0 r (8.9)12 VOLUME, NUMBEREW EXPANSIONS FOR X N ± Y N IN TERMS OF QUADRATIC FORMS
FIRST OBSERVATION ABOUT THEOREM (8.1) . Theorem (8.1) has lots of appli-cations. We should realize that Theorem (8.1) is also true for any differentiable map g aslong as e a = α, e b = β, e α = 0 , e β = 0 , βa − αb = 0. Therefore, we should easily get thefollowing desirable generalization Corollary 8.3.
For any natural number n , for any parameters a, b, α, β, θ, βa − αb = 0 , let Ψ r ( n ) := Ψ (cid:18) a − αθ b − βθ nα β r (cid:19) and Φ r ( n ) := Φ (cid:18) a − αθ b − βθ nα β r (cid:19) . Consider the differential map g that sends a −→ , b −→ , α −→ , β −→ , θ −→ − .Then with respect to this particular differential map we get Ψ r ( n ) = − r ^ Ψ r − ( n ) , ∀ ⌊ n ⌋ r =1 r Φ r ( n ) = − r ^ Φ r − ( n ) , ∀ ⌊ n − ⌋ r =1 r. (8.10) SECOND OBSERVATION ABOUT THEOREM (8.1) . In terms of the classical nota-tions of the partial differentiation we get
Theorem 8.4.
The polynomials Ψ (cid:18) a b nα β r (cid:19) and Φ (cid:18) a b nα β r (cid:19) satisfy Ψ (cid:18) a b nα β r (cid:19) = ( − r r ! (cid:16) α ∂∂a + β ∂∂b (cid:17) r Ψ( a, b, n ) , Ψ (cid:18) a b nα β r (cid:19) = ( − r ( ⌊ n ⌋ − r )! (cid:16) a ∂∂α + b ∂∂β (cid:17) ⌊ n ⌋− r Ψ( α, β, n ) , Φ (cid:18) a b nα β r (cid:19) = ( − r r ! (cid:16) α ∂∂a + β ∂∂b (cid:17) r Φ( a, b, n ) , Φ (cid:18) a b nα β r (cid:19) = ( − r ( ⌊ n − ⌋ − r )! (cid:16) a ∂∂α + b ∂∂β (cid:17) ⌊ n − ⌋− r Φ( α, β, n ) . (8.11)An immediate consequence is the following result Theorem 8.5.
For any natural numbers n , we have ⌊ n ⌋ )! (cid:16) α ∂∂a + β ∂∂b (cid:17) ⌊ n ⌋ Ψ( a, b, n ) = Ψ( α, β, n )1( ⌊ n − ⌋ )! (cid:16) α ∂∂a + β ∂∂b (cid:17) ⌊ n − ⌋ Φ( a, b, n ) = Φ( α, β, n ) (8.12) ILLUSTRATIVE EXAMPLE TO APPLY THE METHODS OF THEOREM (8.1) . It is desirable to clarify how we apply the methods of Theorem(8.1) to compute the polynomialcoefficients Ψ (cid:18) a b nα β r (cid:19) or Φ (cid:18) a b nα β r (cid:19) , for any specific value for n . Therefore, in thenext section, we choose to computeΨ (cid:18) a b α β r (cid:19) for r = 0 , , (cid:22) (cid:23) . FEBRUARY 9, 2020 13
INKS WITH WELL-KNOWN IDENTITY IN THE HISTORY OF NUMBERTHEORY.
For example, take n = 4. ThenΨ (cid:18) a b α β (cid:19) = Ψ( a, b,
4) = − a + b . Hence Ψ (cid:18) a b α β (cid:19) = − (cid:16) α ∂∂a + β ∂∂b (cid:17) ( − a + b ) = 4 aα − bβ, Ψ (cid:18) a b α β (cid:19) = − (cid:16) α ∂∂a + β ∂∂b (cid:17) (4 aα − bβ ) = − α + β . Then from (3.2) we immediately obtain the following polynomial identity( βa − αb ) ( x + y ) = ( − a + b )( αx + βxy + αy ) + (4 aα − bβ )( αx + βxy + αy )( ax + bxy + ay )+ ( − α + β )( ax + bxy + ay ) (8.13)Generally, it is desirable to select values for the parameters a, b, α, β to make the middleterm, Ψ (cid:18) a b α β (cid:19) , equal zero. Therefore, put (cid:18) a bα β (cid:19) = (cid:18) (cid:19) . Then the middlecoefficient 4 aα − bβ of (8.13) vanish, and we obtain the following special case for a well-knownidentity in the history of number theory that is used extensively in the study of equal sums oflike powers and in discovering new formulas for Fibonacci numbers x + y + ( x + y ) = 2( x + xy + y ) (8.14)In volume 2, on page 650, [4] attributes this special case to C. B. Haldeman (1905), althoughProth (1878) used it in passing (see page 657 of [4]).9. COMMON FACTORS
Also, it is useful to note the following relations. Replace each α and β by λα and λβ respectively in (3.2), we get the following polynomial identity for any λ ( λβa − λαb ) ⌊ n ⌋ x n + y n ( x + y ) δ ( n ) = ⌊ n ⌋ X r =0 Ψ (cid:18) a b nλα λβ r (cid:19) ( λαx + λβxy + λαy ) ⌊ n ⌋− r ( ax + bxy + ay ) r Then ( βa − αb ) ⌊ n ⌋ x n + y n ( x + y ) δ ( n ) = ⌊ n ⌋ X r =0 λ − r Ψ (cid:18) a b nλα λβ r (cid:19) ( αx + βxy + αy ) ⌊ n ⌋− r ( ax + bxy + ay ) r (9.1)Comparing (9.1) with (3.2), we obtain λ − r Ψ (cid:18) a b nλα λβ r (cid:19) = Ψ (cid:18) a b nα β r (cid:19) Similarly, we can prove the following useful relations.14 VOLUME, NUMBEREW EXPANSIONS FOR X N ± Y N IN TERMS OF QUADRATIC FORMS
Theorem 9.1.
For any numbers a, b, α, β, βa − αb = 0 , λ, r, n , we get Ψ (cid:18) a b nλα λβ r (cid:19) = λ r Ψ (cid:18) a b nα β r (cid:19) , (9.2a)Φ (cid:18) a b nλα λβ r (cid:19) = λ r Φ (cid:18) a b nα β r (cid:19) , (9.2b)Ψ (cid:18) λa λb nα β r (cid:19) = λ ⌊ n ⌋− r Ψ (cid:18) a b nα β r (cid:19) , (9.2c)Φ (cid:18) λa λb nα β r (cid:19) = λ ⌊ n − ⌋− r Φ (cid:18) a b nα β r (cid:19) , (9.2d)Ψ (cid:18) a b nα β r (cid:19) = ( − ⌊ n ⌋ Ψ (cid:18) α β na b ⌊ n ⌋ − r (cid:19) , (9.2e)Φ (cid:18) a b nα β r (cid:19) = ( − ⌊ n − ⌋ Φ (cid:18) α β na b ⌊ n − ⌋ − r (cid:19) , (9.2f) λ ⌊ n ⌋ Ψ( a, b, n ) = Ψ( λa, λb, n ) , (9.2g) λ ⌊ n − ⌋ Φ( a, b, n ) = Φ( λa, λb, n ) . (9.2h)10. FORMULAS FOR SUMS INCLUDING Ψ AND Φ POLYNOMIALS
We now prove the following theorem.
Theorem 10.1.
For any a, b, α, β, θ, n , βa − αb = 0 , the following identities are true ⌊ n ⌋ X r =0 Ψ (cid:18) a b nα β r (cid:19) θ r = Ψ( a − αθ, b − βθ, n ) , (10.1a) ⌊ n − ⌋ X r =0 Φ (cid:18) a b nα β r (cid:19) θ r = Φ( a − αθ, b − βθ, n ) (10.1b) Proof.
Define q := αx + βxy + αy and q := ax + bxy + ay andΛ θ := θq − q = ( αθ − a ) x + ( βθ − b ) xy + ( αθ − a ) y . From (3.2), we know that( βa − αb ) ⌊ n ⌋ x n + y n ( x + y ) δ ( n ) = ⌊ n ⌋ X r =0 Ψ (cid:18) a b nα β r (cid:19) ( q ) ⌊ n ⌋− r ( q ) r , As q ≡ θq (mod Λ θ ), we get( βa − αb ) ⌊ n ⌋ x n + y n ( x + y ) δ ( n ) ≡ q ⌊ n ⌋ ⌊ n ⌋ X r =0 Ψ (cid:18) a b nα β r (cid:19) θ r (mod Λ θ ) (10.2)Replace each of a, b by αθ − a, βθ − b respectively, in (3.2), we obtain( β [ αθ − a ] − α [ βθ − b ]) ⌊ n ⌋ x n + y n ( x + y ) δ ( n ) ≡ Ψ( αθ − a, βθ − b, n ) q ⌊ n ⌋ (mod Λ θ ) (10.3)As β [ αθ − a ] − α [ βθ − b ] = − ( βa − αb ), and noting from (9.2g) that( − ⌊ n ⌋ Ψ( αθ − a, βθ − b, n ) = Ψ( a − αθ, b − βθ, n ) , FEBRUARY 9, 2020 15e obtain the following congruence( βa − αb ) ⌊ n ⌋ x n + y n ( x + y ) δ ( n ) ≡ Ψ( a − αθ, b − βθ, n ) q ⌊ n ⌋ (mod Λ θ ) (10.4)Now, subtracting (10.2) and(10.4), we obtain0 ≡ (cid:16) ⌊ n ⌋ X r =0 Ψ (cid:18) a b nα β r (cid:19) θ r − Ψ( a − αθ, b − βθ, n ) (cid:17) q ⌊ n ⌋ (mod Λ θ ) (10.5)As the congruence (10.5) is true for any x, y , and as ( βθ − b ) α − ( αθ − a ) β = βa − αb = 0,then from Lemma (4.1) the binary quadratic forms Λ θ and q are algebraic independent. Thisimmediately leads to 0 = ⌊ n ⌋ X r =0 Ψ (cid:18) a b nα β r (cid:19) θ r − Ψ( a − αθ, b − βθ, n ) . (10.6)Hence we obtain the proof of (10.1a). Similarly, we can prove (10.1b). (cid:3) The following desirable generalization is important
Theorem 10.2.
For any a, b, α, β, η, ξ, n , βa − αb = 0 , the following identities are true ⌊ n ⌋ X r =0 Ψ (cid:18) a b nα β r (cid:19) ξ ⌊ n ⌋− r η r = Ψ( aξ − αη, bξ − βη, n ) , ⌊ n − ⌋ X r =0 Φ (cid:18) a b nα β r (cid:19) ξ ⌊ n − ⌋− r η r = Φ( aξ − αη, bξ − βη, n ) . (10.7) Proof.
Without loss of generality, let ξ = 0. We obtain the proof by replacing each θ in (10.1a)and (10.1b) by ηξ , and multiplying each side of (10.1a) by ξ ⌊ n ⌋ , and (10.1b) by ξ ⌊ n − ⌋ , andnoting that ξ ⌊ n ⌋ Ψ( a − α ηξ , b − β ηξ , n ) = Ψ( aξ − αη, bξ − βη, n )and ξ ⌊ n − ⌋ Φ( a − α ηξ , b − β ηξ , n ) = Φ( aξ − αη, bξ − βη, n ) (cid:3) Replacing θ by ± Theorem 10.3.
For any a, b, α, β, n , βa − αb = 0 the following identities are true ⌊ n ⌋ X r =0 Ψ (cid:18) a b nα β r (cid:19) = Ψ( a − α, b − β, n ) , ⌊ n − ⌋ X r =0 Φ (cid:18) a b nα β r (cid:19) = Φ( a − α, b − β, n ) (10.8)16 VOLUME, NUMBEREW EXPANSIONS FOR X N ± Y N IN TERMS OF QUADRATIC FORMS
Theorem 10.4.
For any a, b, α, β, n , βa − αb = 0 the following identities are true ⌊ n ⌋ X r =0 Ψ (cid:18) a b nα β r (cid:19) ( − r = Ψ( a + α, b + β, n ) , ⌊ n − ⌋ X r =0 Φ (cid:18) a b nα β r (cid:19) ( − r = Φ( a + α, b + β, n ) (10.9)10.1. GENERALIZATIONS FOR THEOREM (10.1) . Now we can generalize Theorem(10.1) by applying the following specific differential map, for k times, on (10.1a) and (10.1b),that sends a −→ , b −→ , α −→ , β −→ , θ −→ − a − αθ −→ α , b − βθ −→ β. Now, from (5.1a), (5.1b) of Theorem (5.1) and from Corollary (8.3), we get (cid:16) − ∂∂θ (cid:17) k Ψ( a − αθ, b − βθ, n )= (cid:16) ∂∂a + 0 ∂∂b + 0 ∂∂α + 0 ∂∂β − ∂∂θ (cid:17) k Ψ (cid:18) a − αθ b − βθ nα β (cid:19) = ( − k ( k !)Ψ (cid:18) a − αθ b − βθ nα β k (cid:19) , (cid:16) − ∂∂θ (cid:17) k Φ( a − αθ, b − βθ, n )= (cid:16) ∂∂a + 0 ∂∂b + 0 ∂∂α + 0 ∂∂β − ∂∂θ (cid:17) k Φ (cid:18) a − αθ b − βθ nα β (cid:19) = ( − k ( k !)Φ (cid:18) a − αθ b − βθ nα β k (cid:19) . Hence we obtain the following desirable generalization for Theorem (10.1).
Theorem 10.5.
For any n, k, a, b, α, β, θ, n , βa − αb = 0 , the following identities are true ⌊ n ⌋ X r = k (cid:18) rk (cid:19) Ψ (cid:18) a b nα β r (cid:19) θ r − k = Ψ (cid:18) a − αθ b − βθ nα β k (cid:19) , (10.10a) ⌊ n − ⌋ X r = k (cid:18) rk (cid:19) Φ (cid:18) a b nα β r (cid:19) θ r − k = Φ (cid:18) a − αθ b − βθ nα β k (cid:19) (10.10b)Again, without loss of generality, let ξ = 0. By replacing each θ in (10.10a) and (10.10b)by ηξ , and multiplying each side of (10.10a) by ξ ⌊ n ⌋− k and (10.10b) by ξ ⌊ n − ⌋− k , and notingfrom the properties of Ψ and Φ polynomials, given in (9.2c), (9.2d) of Theorem (9.1), that ξ ⌊ n ⌋− k Ψ (cid:18) a − α ηξ b − β ηξ nα β k (cid:19) = Ψ (cid:18) aξ − αη bξ − βη nα β k (cid:19) ,ξ ⌊ n − ⌋− k Φ (cid:18) a − α ηξ b − β ηξ nα β k (cid:19) = Φ (cid:18) aξ − αη bξ − βη nα β k (cid:19) . Therefore, we obtain the following generalization for Theorem (10.5).FEBRUARY 9, 2020 17 heorem 10.6.
For any n, k, a, b, α, β, θ, n , βa − αb = 0 , the following identities are true ⌊ n ⌋ X r = k (cid:18) rk (cid:19) Ψ (cid:18) a b nα β r (cid:19) ξ ⌊ n ⌋− r η r − k = Ψ (cid:18) aξ − αη bξ − βη nα β k (cid:19) , (10.11) ⌊ n − ⌋ X r = k (cid:18) rk (cid:19) Φ (cid:18) a b nα β r (cid:19) ξ ⌊ n − ⌋− r η r − k = Φ (cid:18) aξ − αη bξ − βη nα β k (cid:19) (10.12)11. SPECIFIC FORMULAS FOR Ψ AND Φ POLYNOMIALS
Now, put ξ = αx + βxy + αy and η = ax + bxy + ay in Theorem (10.2), we obtain ⌊ n ⌋ X r =0 Ψ (cid:18) a b nα β r (cid:19) ( αx + βxy + αy ) ⌊ n ⌋− r ( ax + bxy + ay ) r = ( βa − αb ) ⌊ n ⌋ Ψ( xy, − x − y , n ) , (11.1) ⌊ n − ⌋ X r =0 Φ (cid:18) a b nα β r (cid:19) ( αx + βxy + αy ) ⌊ n − ⌋− r ( ax + bxy + ay ) r = ( βa − αb ) ⌊ n − ⌋ Φ( xy, − x − y , n ) . (11.2)Now, from (3.2), and (3.3), we get ⌊ n ⌋ X r =0 Ψ (cid:18) a b nα β r (cid:19) ( αx + βxy + αy ) ⌊ n ⌋− r ( ax + bxy + ay ) r = ( βa − αb ) ⌊ n ⌋ x n + y n ( x + y ) δ ( n ) , (11.3)and ⌊ n − ⌋ X r =0 Φ (cid:18) a b nα β r (cid:19) ( αx + βxy + αy ) ⌊ n − ⌋− r ( ax + bxy + ay ) r = ( βa − αb ) ⌊ n − ⌋ x n − y n ( x − y )( x + y ) δ ( n − (11.4)From (11.1), (11.3), we get formula (11.5) for Ψ polynomial, and from (11.2), and (11.4), weget formula (11.6) for Φ polynomial. Theorem 11.1.
For any natural number n , the Ψ and Φ polynomials satisfy the followingpolynomial identities Ψ( xy, − x − y , n ) = x n + y n ( x + y ) δ ( n ) , (11.5)Φ( xy, − x − y , n ) = x n − y n ( x − y )( x + y ) δ ( n − (11.6)18 VOLUME, NUMBEREW EXPANSIONS FOR X N ± Y N IN TERMS OF QUADRATIC FORMS12.
FORMULA TO COMPUTE Φ r ( n ) DIRECTLY FROM Ψ r ( n )First, let Ψ r ( n ) := Ψ (cid:18) a b nα β r (cid:19) , Φ r ( n ) := Φ (cid:18) a b nα β r (cid:19) . By differentiating (3.2) with respect to the differential map a −→ , b −→ , α −→ , β −→ , x −→ , y −→ − , and noting that with this differentiation we have( βa − αb ) ⌊ n ⌋ −→ , Ψ (cid:18) a b nα β r (cid:19) −→ ,x n + y n −→ n ( x n − − y n − ) , ( x + y ) δ ( n ) −→ ,ax + bxy + ay −→ (2 a − b )( x − y ) , αx + βxy + αy −→ (2 α − β )( x − y ) , we get the following polynomial identity( βa − αb ) ⌊ n ⌋ n ( x n − − y n − )( x + y ) δ ( n ) = ⌊ n ⌋− X r =0 (2 α − β )Ψ r ( n )( ⌊ n ⌋ − r )( αx + βxy + αy ) ⌊ n ⌋− r − ( ax + bxy + ay ) r ( x − y )+ ⌊ n ⌋ X r =1 (2 a − b )Ψ r ( n )( r )( αx + βxy + αy ) ⌊ n ⌋− r ( ax + bxy + ay ) r − ( x − y ) . (12.1)Raising each n by one, and noting that ( βa − αb ) ⌊ n +12 ⌋ = ( βa − αb ) ⌊ n − ⌋ ( βa − αb ) , ⌊ n +12 ⌋ − ⌊ n − ⌋ , δ ( n + 1) = δ ( n − n + 1)( x − y ), we get the following formula( βa − αb ) ⌊ n − ⌋ ( x n − y n )( x − y )( x + y ) δ ( n ) = ⌊ n − ⌋ X r =0 (2 α − β )( ⌊ n +12 ⌋ − r )Ψ r ( n + 1)( βa − αb )( n + 1) ( αx + βxy + αy ) ⌊ n − ⌋− r ( ax + bxy + ay ) r + ⌊ n − ⌋ X r =0 (2 a − b )( r + 1)Ψ r +1 ( n + 1)( βa − αb )( n + 1) ( αx + βxy + αy ) ⌊ n − ⌋− r ( ax + bxy + ay ) r . (12.2)Now, comparing (12.2) with (3.3), we immediately get the following desirable formula. Theorem 12.1.
For any natural number n , βa − αb = 0 , the following formula is true Φ r ( n ) = (2 α − β )( ⌊ n +12 ⌋ − r )Ψ r ( n + 1) + (2 a − b )( r + 1)Ψ r +1 ( n + 1)( βa − αb )( n + 1) (12.3)FEBRUARY 9, 2020 193. THE Ψ AND Φ TRAJECTORIES AND ORBITS OF ORDER n From Theorems (5.1), (6.1), (9.1), we know that for any numbers a, b, α, β , the followingrelations are true for any natural number n Ψ ( n ) = Ψ (cid:18) a b nα β (cid:19) = Ψ( a, b, n ) , Ψ ⌊ n ⌋ ( n ) = Ψ (cid:18) a b nα β ⌊ n ⌋ (cid:19) = ( − ⌊ n ⌋ Ψ( α, β, n ) = Ψ( − α, − β, n ) , Φ ( n ) = Φ (cid:18) a b nα β (cid:19) = Φ( a, b, n ) , Φ ⌊ n − ⌋ ( n ) = Φ (cid:18) a b nα β ⌊ n − ⌋ (cid:19) = ( − ⌊ n − ⌋ Φ( α, β, n ) = Φ( − α, − β, n ) . Therefore, Theorems (3.1), (5.1), (8.5) should motivate us define the trajectories which areconstructed from the sequence of polynomials Ψ r ( n ) and Φ r ( n ) which connects Ψ( a, b, n ) withΨ( − α, − β, n ) and Φ( a, b, n ) with Φ( − α, − β, n ) respectively. Definition 13.1.
Let a, b, α, β be given real numbers, βa − αb = 0, andΨ r ( n ) := Ψ (cid:18) a b nα β r (cid:19) , Φ r ( n ) := Φ (cid:18) a b nα β r (cid:19) . We call the sequence of polynomialsΨ ( n ) , Ψ ( n ) , Ψ ( n ) , . . . , Ψ ⌊ n ⌋ ( n ) (13.1)the Ψ trajectory of order n from ( a, b ) to ( α, β ). Alternatively, we call that the Ψ trajectory“connect” Ψ( a, b, n ) with Ψ( − α, − β, n ). When Ψ( − α, − β, n ) = Ψ( a, b, n ) for given naturalnumber n then we call the Ψ trajectory of order n from ( a, b ) to ( α, β ) by the Ψ orbit of order n from ( a, b ) to ( α, β ). Similarly, we call the sequence of polynomialsΦ ( n ) , Φ ( n ) , Φ ( n ) , . . . , Φ ⌊ n − ⌋ ( n ) (13.2)the Φ trajectory of order n from ( a, b ) to ( α, β ). Alternatively, we call that the Φ trajectory“connect” Φ( a, b, n ) with Φ( − α, − β, n ). When Φ( − α, − β, n ) = Φ( a, b, n ) for given naturalnumber n then we call the Φ trajectory of order n from ( a, b ) to ( α, β ) by the Φ orbit of order n from ( a, b ) to ( α, β ). MAJOR QUESTIONS ABOUT THE Ψ AND Φ TRAJECTORIES AND ORBITS.
Major focus of future research on the Ψ and Φ trajectories and orbits are the following questions(1) Does an efficient algorithms exist to identify whether a given Ψ and Φ trajectory canbe turned to an orbit ?(2) What are the common arithmetical characteristics of the terms of the Ψ and Φ orbit?(3) What are the common properties of the prime factors of the terms of Ψ and Φ orbit?(4) Does the arithmetic of the integer terms of the Fermat orbit help finding an efficientalgorithms to factorize Fermat numbers ?20 VOLUME, NUMBEREW EXPANSIONS FOR X N ± Y N IN TERMS OF QUADRATIC FORMSΨ( a, b, n ) Ψ( − α, − β, n ) The notions of Ψ and Φ trajectories arose up naturally in this paper. Many well-known sequencesare now not isolated; meaning we can connect them by a certain trajectory. When the endpointΨ( − α, − β, n ) coincidences with starting point Ψ( a, b, n ), then the trajectory gives the Ψ orbit. Also,when the endpoint Φ( − α, − β, n ) coincidences with starting point Φ( a, b, n ), then the trajectory gives theΦ orbit. These trajectories and orbits might be a new area of research in the future. Φ( a, b, n )Φ( − α, − β, n )From the relations (5.1a), (5.1b) of Theorem (5.1), we should have no difficulties to provethe following relations of the following theorems. Theorem 13.2.
For any real numbers a, b , a = ± b , and any natural number n , the followingidentities are true Ψ( a, − b, n ) = Ψ( − a, − b, n ) for n even Ψ( a, − b, n ) = Φ( − a, − b, n ) for n odd Φ( a, − b, n ) = Φ( − a, − b, n ) for n even Φ( a, − b, n ) = Ψ( − a, − b, n ) for n odd Theorem 13.3.
For any real numbers a, b , a = ± b , and any natural number n , the followingidentity is true Φ( a, b, n ) = Φ( a, b, n )Ψ( a, b, n ) EXAMPLES FOR TRAJECTORIES CONNECT WELL-KNOWN SEQUENCES.
We already proved in the current paper that the Ψ and Φ polynomials unify many well-knownpolynomials and sequences. Now the notions of the Ψ and Φ trajectories should be useful.13.1.
The CHEBYSHEV-LUCAS TRAJECTORY.
The Ψ trajectory of order n from(1 , − x ) to (1 ,
3) starts with δ ( n +1) x δ ( n ) T n ( x ) and ends with L ( n ). So this way we get new pathconnects Chebyshev polynomials of the first kind with Lucas sequences. The vision behindconsidering such trajectory is that this path should help to understand some characteristics ofthe starting party, the Chebyshev Polynomials, by knowing some characteristics of the finalparty, the Lucas numbers, and vice versa.FEBRUARY 9, 2020 21 ( n ) δ ( n +1) x δ ( n ) T n ( x ) Studying this particular trajectory should help understandthe arithmetic of Chebyshev polynomials of the first kind andLucas numbers. For any natural number n , we call this paththe Chebyshev-Lucas trajectory which is the sequence2 δ ( n +1) x δ ( n ) T n ( x ) = Ψ ( n ) , Ψ ( n ) , Ψ ( n ) , . . . , Ψ ⌊ n ⌋ ( n ) = L ( n )where Ψ r ( n ) = Ψ (cid:18) − x n r (cid:19) . Moreover, the termsΨ r ( n ) satisfy the following polynomial identity(1 + 4 x ) ⌊ n ⌋ z n + t n ( z + t ) δ ( n ) = ⌊ n ⌋ X r =0 Ψ r ( n )( z + 3 zt + t ) ⌊ n ⌋− r ( z + (2 − x ) zt + t ) r . THE LUCAS-FIBONACCI TRAJECTORY.
Let n be any odd natural number.The Ψ trajectory of order n from ( − , −
3) to ( − , +3) starts with L ( n ) and ends with F ( n ).So this way we get path connects Lucas numbers with Fibonacci numbers. L ( n ) F ( n ) Studying this particular trajectory should enjoy certain prop-erties that comes from the arithmetic of Lucas numbers andFibonacci. For any odd natural number n , we call this paththe Lucas-Fibonacci trajectory which is the sequence L ( n ) = Ψ ( n ) , Ψ ( n ) , Ψ ( n ) , . . . , Ψ ⌊ n ⌋ ( n ) = F ( n )where Ψ r ( n ) = Ψ (cid:18) − − n − r (cid:19) . Moreover, the termsΨ r ( n ) satisfy the following polynomial identity( − ⌊ n ⌋ z n + t n ( z + t ) δ ( n ) = ⌊ n ⌋ X r =0 Ψ r ( n )( − z + 3 zt − t ) ⌊ n ⌋− r ( − z − zt − t ) r . THE LUCAS ORBIT.
Let n be any even natural number. The Ψ trajectory of order n from ( − , −
3) to ( − , +3) starts with L ( n ) and ends with L ( n ). So this way we get orbit.We call this orbit the Lucas orbit at level n . L ( n ) This particular orbit should enjoy unexpected properties thatcomes from the arithmetic of Lucas numbers. For any evennatural number n , the Lucas orbit at level n is the sequence L ( n ) = Ψ ( n ) , Ψ ( n ) , Ψ ( n ) , . . . , Ψ ⌊ n ⌋ ( n ) = L ( n )where Ψ r ( n ) = Ψ (cid:18) − − n − r (cid:19) . Moreover, the termsΨ r ( n ) satisfy the following polynomial identity( − ⌊ n ⌋ z n + t n ( z + t ) δ ( n ) = ⌊ n ⌋ X r =0 Ψ r ( n )( − z + 3 zt − t ) ⌊ n ⌋− r ( − z − zt − t ) r .
22 VOLUME, NUMBEREW EXPANSIONS FOR X N ± Y N IN TERMS OF QUADRATIC FORMS13.4.
THE LUCAS-PELL TRAJECTORY.
Let n be any natural number. The Ψ tra-jectory of order n from ( − , −
3) to (1 ,
6) starts with L ( n ) and ends with Q n δ ( n ) , where Q n arethe Pell-Lucas numbers. L ( n ) Q n δ ( n ) So, we get trajectory passing through the Lucas numbers andLucas-Pell numbers. We call this trajectory the Lucas-Pelltrajectory. This trajectory should enjoy some properties thatcomes from the arithmetic of Lucas and Lucas-Pell numbers.For any natural number n , the Lucas-Pell trajectory at level n is the sequence L ( n ) = Ψ ( n ) , Ψ ( n ) , Ψ ( n ) , . . . , Ψ ⌊ n ⌋ ( n ) = Q n δ ( n ) where Ψ r ( n ) = Ψ (cid:18) − − n r (cid:19) . Moreover, the termsΨ r ( n ) satisfy the following polynomial identity( − ⌊ n ⌋ z n + t n ( z + t ) δ ( n ) = ⌊ n ⌋ X r =0 Ψ r ( n )( z + 6 zt + t ) ⌊ n ⌋− r ( − z − zt − t ) r . THE FIBONACCI-PELL TRAJECTORY.
Let n be any natural number. The Φtrajectory of order n from ( − , −
3) to (1 ,
6) starts with F ( n ) and ends with P n δ ( n − , where P n are the Pell numbers. F ( n ) P n δ ( n − So, we get trajectory passing through the Lucas numbers andLucas-Pell numbers. We call this trajectory the Lucas-Pelltrajectory. This trajectory should enjoy some properties thatcomes from the arithmetic of Lucas and Lucas-Pell numbers.For any natural number n , the Lucas-Pell trajectory at level n is the sequence F ( n ) = Φ ( n ) , Φ ( n ) , Φ ( n ) , . . . , Φ ⌊ n − ⌋ ( n ) = P n δ ( n − where Φ r ( n ) = Φ (cid:18) − − n r (cid:19) . Moreover, the termsΦ r ( n ) satisfy the following polynomial identity( − ⌊ n − ⌋ z n − t n ( z − t )( z + t ) δ ( n − = ⌊ n − ⌋ X r =0 Φ r ( n )( z + 6 zt + t ) ⌊ n − ⌋− r ( − z − zt − t ) r . THE FIBONACCI ORBIT.
Let n be any even natural number. The Φ trajectoryof order n from ( − , −
3) to ( − , +3) starts with F n and ends with F n , where F n are theFibonacci numbers. So, we get orbit passing through the Fibonacci numbers. We call thisorbit the Fibonacci orbit at level n .FEBRUARY 9, 2020 23 ( n ) Therefore, for any even natural number n , the Fibonacci orbitat level n is the sequence F n = Φ ( n ) , Φ ( n ) , Φ ( n ) , . . . , Φ ⌊ n ⌋ ( n ) = F n where Φ r ( n ) = Φ (cid:18) − − n − r (cid:19) . Moreover, the termsΦ r ( n ) satisfy the following polynomial identity( − ⌊ n − ⌋ z n − t n z − t = ⌊ n − ⌋ X r =0 Φ r ( n )( − z + 3 zt − t ) ⌊ n − ⌋− r ( − z − zt − t ) r . THE FIBONACCI-LUCAS TRAJECTORY.
Let n be any odd natural number.The Φ trajectory of order n from ( − , −
3) to ( − , +3) starts with F ( n ) and ends with L ( n ).So this way we get another new path connects Fibonacci numbers with Lucas numbers. L ( n ) F ( n ) Therefore, for any odd natural number n , we call this path theLucas-Fibonacci trajectory which is the sequence F ( n ) = Φ ( n ) , Φ ( n ) , Φ ( n ) , . . . , Φ ⌊ n ⌋ ( n ) = L ( n )where Φ r ( n ) = Φ (cid:18) − − n − r (cid:19) . Moreover, the termsΦ r ( n ) satisfy the following polynomial identity( − ⌊ n − ⌋ z n − t n z − t = ⌊ n − ⌋ X r =0 Φ r ( n )( − z + 3 zt − t ) ⌊ n − ⌋− r ( − z − zt − t ) r . OBSERVATION.
For any odd natural number n , we should observe that the sequence F ( n ) = Φ ( n ) , Φ ( n ) , . . . , Φ ⌊ n ⌋ ( n ) = L ( n ) = Ψ ( n ) , Ψ ( n ) , . . . , Ψ ⌊ n ⌋ ( n ) = F ( n )is orbit passing through both of Fibonacci and Lucas numbers, where Φ r ( n ) = Φ (cid:18) − − n − r (cid:19) and Ψ r ( n ) = Ψ (cid:18) − − n − r (cid:19) . We call such orbit the “Fibonacci-Lucas orbit”. We shouldask for explicit closed formula for the terms of this orbit. The arithmetical characteristics forthis orbit should be intimately related with the Fibonacci and Lucas numbers.24 VOLUME, NUMBEREW EXPANSIONS FOR X N ± Y N IN TERMS OF QUADRATIC FORMS F ( n ) L ( n ) The Fibonacci-Lucas trajectory of order n from( − , −
3) to ( − , +3) and the Lucas-Fibonaccitrajectory of order n from ( − , −
3) to ( − , +3)can be combined to construct a new orbit that gothrough both of the Fibonacci and Lucas numbers. THE MERSENNE ORBIT.
Let n be any even natural number. The Φ trajectory oforder n from (2 , −
5) to (2 , +5) starts with M n and ends with M n , where M n are the Mersennenumbers defined by M n = 2 n −
1. So, we get orbit. We call this orbit the Mersenne orbit atlevel n . M n For any even natural number n , the Mersenne orbit at level n is the sequence M n ( n ) , Φ ( n ) , Φ ( n ) , . . . , Φ ⌊ n ⌋ ( n ) = M n r ( n ) = Φ (cid:18) − n r (cid:19) . Moreover, the terms Φ r ( n )satisfy the following polynomial identity(20) ⌊ n − ⌋ z n − t n z − t = ⌊ n − ⌋ X r =0 Φ r ( n )(2 z + 5 zt + 2 t ) ⌊ n − ⌋− r (2 z − zt + 2 t ) r . THE MERSENNE TRAJECTORY.
Let n be any odd natural number. The Φ tra-jectory of order n from (2 , −
5) to (2 , +5) starts with M n and ends with n +13 , where M n arethe Mersenne numbers defined by M n = 2 n −
1. We call this specific trajectory the Mersennetrajectory at level n . M n n +13 For any odd natural number n , the Mersenne trajectory atlevel n is the sequence M n = Φ ( n ) , Φ ( n ) , Φ ( n ) , . . . , Φ ⌊ n ⌋ ( n ) = 2 n + 13where Φ r ( n ) = Φ (cid:18) − n r (cid:19) . Moreover, the terms Φ r ( n )satisfy the following polynomial identity(20) ⌊ n − ⌋ z n − t n z − t = ⌊ n − ⌋ X r =0 Φ r ( n )(2 z + 5 zt + 2 t ) ⌊ n − ⌋− r (2 z − zt + 2 t ) r . FEBRUARY 9, 2020 253.10.
THE CHEBYSHEV-DICKSON TRAJECTORY OF THE FIRST KIND.
We can establish trajectory that connects the Chebyshev polynomials of first kind with theDickson polynomials of the first kind with parameter α . For example, the Ψ trajectory of order n from (1 , − x ) to ( − α, − α + x ) starts with δ ( n +1) x δ ( n )1 T n ( x ) and ends with x − δ ( n )2 D n ( x , α ).So this way we get new path connects the Chebyshev polynomials of the first kind with theDickson polynomials of the first kind with parameter α . x − δ ( n )2 D n ( x , α ) δ ( n +1) x δ ( n )1 T n ( x ) Studying this particular trajectory should help understandthe arithmetic of Chebyshev polynomials of the first kind andDickson polynomials of the first kind with parameter α . Wecall this path the Chebyshev-Dickson trajectory of the firstkind which is the sequence2 δ ( n +1) x δ ( n )1 T n ( x ) = Ψ ( n ) , Ψ ( n ) , . . . , Ψ ⌊ n ⌋ ( n ) = x − δ ( n )2 D n ( x , α )where Ψ r ( n ) = Ψ (cid:18) − x n − α − α + x r (cid:19) . Moreover, theterms Ψ r ( n ) satisfy the following polynomial identity( x − αx ) ⌊ n ⌋ z n + t n ( z + t ) δ ( n ) = ⌊ n ⌋ X r =0 Ψ r ( n ) (cid:16) − αz − (2 α − x ) zt − αt (cid:17) ⌊ n ⌋− r ( z +(2 − x ) zt + t ) r . THE CHEBYSHEV-DICKSON TRAJECTORY OF THE SECOND KIND.
We can establish trajectory that connects the Chebyshev polynomials of second kind with theDickson polynomials of the second kind with parameter α . For example, the Φ trajectory oforder n from (1 , − x ) to ( − α, − α + x ) starts with (2 x ) − δ ( n − U n − ( x ) and ends with x − δ ( n − E n − ( x , α ). So this way we get new path connects Chebyshev polynomials of thesecond kind with Dickson polynomials of the second kind with parameter α . x − δ ( n − E n − ( x , α )(2 x ) − δ ( n − U n − ( x ) This trajectory should help understand the arithmetic of theChebyshev polynomials of the second kind and the Dicksonpolynomials of the second kind with parameter α . We callthis path the Chebyshev-Dickson trajectory of the second kindwhich is the sequence U n − ( x )(2 x ) δ ( n − = Φ ( n ) , Φ ( n ) , . . . , Φ ⌊ n ⌋ ( n ) = E n − ( x , α ) x δ ( n − where Φ r ( n ) = Φ (cid:18) − x n − α − α + x r (cid:19) . Moreover, theterms Φ r ( n ) satisfy the following polynomial identity( x − αx ) ⌊ n − ⌋ z n − t n ( z − t )( z + t ) δ ( n − = ⌊ n − ⌋ X r =0 Φ r ( n ) (cid:16) − αz − (2 α − x ) zt − αt (cid:17) ⌊ n − ⌋− r ( z +(2 − x ) zt + t ) r . Therefore, as we see, with the Ψ and Φ trajectories, many well-known sequences are notnow isolated; meaning we can connect different elements from the list that contains Chebyshevpolynomials of the first and second kind, Dickson polynomials of the first and second kind,26 VOLUME, NUMBEREW EXPANSIONS FOR X N ± Y N IN TERMS OF QUADRATIC FORMSLucas and Fibonacci numbers, Mersenne numbers, Pell polynomials, Pell-Lucas polynomials,Fermat numbers, and 2 n ± Trajectory From Difference of Powers to Another and Open Questions.
Also, forany natural number n , we show in this paper that we can also establish a trajectory thatconnects the polynomials x n − y n ( x − y )( x + y ) δ ( n − with the polynomials z n − t n ( z − t )( z + t ) δ ( n − . x n − y n ( x − y )( x + y ) δ ( n − z n − t n ( z − t )( z + t ) δ ( n − There are many open questions to the trajectory of polynomials x n − y n ( x − y )( x + y ) δ ( n − = Φ ( n ) , Φ ( n ) , . . . , Φ ⌊ n − ⌋ ( n ) = z n − t n ( z − t )( z + t ) δ ( n − where Φ r ( n ) = Φ (cid:18) + xy − x − y n − zt + z + t r (cid:19) , which are the coefficientsof the following polynomial expansion( zx − ty ) ⌊ n − ⌋ ( zy − tx ) ⌊ n − ⌋ u n − v n ( u − v )( u + v ) δ ( n − = ⌊ n − ⌋ X r =0 Φ r ( n )( zu − tv ) ⌊ n − ⌋− r ( zv − tu ) ⌊ n − ⌋− r (( ux − vy ) r ( uy − vx )) r For this particular trajectory, we observe that for any numbers x, y, z, t, xz = yt, xt = yz ,and for any natural number n , the following identity hold ⌊ n − ⌋ X r =0 Φ (cid:18) + xy − x − y n − zt + z + t r (cid:19) = Φ( xy + zt, − x − y − z − t , n ) . This particular trajectory should also be surrounded by lots of mathematical questions. Oneof these, when the endpoints coincident with each other, again it gives an orbit. Also, if suchorbit exists, we get a solution to many unsolved Diophantine equations. If such orbit exists,what are the general characteristics of such orbit? For example, what is the characteristicsof this orbit, if exist, for n = 5. And it is natural to ask whether there exist such orbit forany natural number n . Alternatively, this is equivalent to ask whether there is any nontrivialinteger solutions for ( x, y, z, t ) and any natural number n for following Diophantine equation x n − y n ( x − y )( x + y ) δ ( n − = z n − t n ( z − t )( z + t ) δ ( n − . Acknowledgments.
I would like to thank University of Bahrain for their kind support.Special great thanks to Bruce Reznick for his kind motivation, great advice, and for usefulconversations. I am also appreciated for Alexandru Buium, Bruce Berndt, and Ahmed Mater.
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Department of Mathematics, College of Science, University of Bahrain, Kingdom of Bahrain
E-mail address : [email protected]@uob.edu.bh