Abstract. We establish new connection formulae between Fibonacci polynomials and Chebyshev polynomials of the first and second kinds. These formulae are expressed in terms of certain values of hypergeometric functions of the type 2 F 1 . Consequently, we obtain some new expressions for the celebrated Fibonacci numbers and their derivatives sequences. Moreover, we evaluate some definite integrals involving products of Fibonacci and Chebyshev polynomials.
Let E be an elliptic curve described by either an Edwards model or a twisted Edwards model over F p , namely, E is defined by one of the following equationsWe express the number of rational points of E over F p using the Gaussian hypergeometric where ǫ and φ are the trivial and quadratic characters over F p respectively. This enables us to evaluate |E(F p )| for some elliptic curves E, and prove the existence of isogenies between E and Legendre elliptic curves over F p .
A Huff curve over a field K is an elliptic curve defined by the equation ax(y 2 − 1) = by(x 2 − 1) where a, b ∈ K are such that a 2 = b 2 . In a similar fashion, a general Huff curve over K is described by the equation x(ay 2 − 1) = y(bx 2 − 1) where a, b ∈ K are such that ab(a − b) = 0. In this note we express the number of rational points on these curves over a finite field F q of odd characteristic in terms of Gaussian hypergeometric serieswhere φ and ǫ are the quadratic and trivial characters over F q , respectively. Consequently, we exhibit the number of rational points on the elliptic curves y 2 = x(x + a)(x + b) over F q in terms of 2 F 1 (λ). This generalizes earlier known formulas for Legendre, Clausen and Edwards curves. Furthermore, using these expressions we display several transformations of 2 F 1 . Finally, we present the exact value of 2 F 1 (λ) for different λ's over a prime field F p extending previous results of Greene and Ono.
Let F ∈ Z[x, y] and m ≥ 2 be an integer. A set A ⊂ Z is called an (F, m)-Diophantine set if F (a, b) is a perfect m-power for any a, b ∈ A where a = b. If F is a bivariate polynomial for which there exist infinite (F, m)-Diophantine sets, then there is a complete qualitative characterization of all such polynomials F . Otherwise, various finiteness results are known. We prove that given a finite set of distinct integers S of size n, there are infinitely many bivariate polynomials F such that S is an (F, 2)-Diophantine set. In addition, we show that the degree of F can be as small as 4⌊n/3⌋.
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