Binomial sums via Bailey's cubic transformation

Wenchang Chu

Displaying similar documents to “Binomial sums via Bailey's cubic transformation”

Gauss Sums of the Cubic Character over $G F (2^{m})$ : an Elementary Derivation

Davide Schipani, Michele Elia (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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By an elementary approach, we derive the value of the Gauss sum of a cubic character over a finite field $_{2^{s}}$ without using Davenport-Hasse’s theorem (namely, if s is odd the Gauss sum is -1, and if s is even its value is $- {(- 2)}^{s / 2}$ ).

Generalizations of Milne’s $U (n + 1)$ $q$ -Chu-Vandermonde summation

Jian-Ping Fang (2016)

Czechoslovak Mathematical Journal

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We derive two identities for multiple basic hyper-geometric series associated with the unitary $U (n + 1)$ group. In order to get the two identities, we first present two known $q$ -exponential operator identities which were established in our earlier paper. From the two identities and combining them with the two $U (n + 1)$ $q$ -Chu-Vandermonde summations established by Milne, we arrive at our results. Using the identities obtained in this paper, we give two interesting identities involving binomial...

The reduced ideals of a special order in a pure cubic number field

Abdelmalek Azizi, Jamal Benamara, Moulay Chrif Ismaili, Mohammed Talbi (2020)

Archivum Mathematicum

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Let $K = ℚ (θ)$ be a pure cubic field, with $θ^{3} = D$ , where $D$ is a cube-free integer. We will determine the reduced ideals of the order $𝒪 = ℤ [θ]$ of $K$ which coincides with the maximal order of $K$ in the case where $D$ is square-free and $\neg \equiv \pm 1 (mod 9)$ .

Contribution to construction of global cubic $C^{1}$ or $C^{2}$ -spline on equidistant knotset

Kobza, Jiří

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Approach of $q$ -Derivative Operators to Terminating $q$ -Series Formulae

Xiaoyuan Wang, Wenchang Chu (2018)

Communications in Mathematics

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The $q$ -derivative operator approach is illustrated by reviewing several typical summation formulae of terminating basic hypergeometric series.

Number of solutions of cubic Thue inequalities with positive discriminant

N. Saradha, Divyum Sharma (2015)

Acta Arithmetica

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Let F(X,Y) be an irreducible binary cubic form with integer coefficients and positive discriminant D. Let k be a positive integer satisfying $k < ({(3 D)}^{1 / 4}) / 2 π$ . We give improved upper bounds for the number of primitive solutions of the Thue inequality $| F (X, Y) | \leq k$ .

On the $k$ -polygonal numbers and the mean value of Dedekind sums

Jing Guo, Xiaoxue Li (2016)

Czechoslovak Mathematical Journal

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For any positive integer $k \geq 3$ , it is easy to prove that the $k$ -polygonal numbers are $a_{n} (k) = (2 n + n (n - 1) (k - 2)) / 2$ . The main purpose of this paper is, using the properties of Gauss sums and Dedekind sums, the mean square value theorem of Dirichlet $L$ -functions and the analytic methods, to study the computational problem of one kind mean value of Dedekind sums $S (a_{n} (k) {\bar{a}}_{m} (k), p)$ for $k$ -polygonal numbers with $1 \leq m, n \leq p - 1$ , and give an interesting computational formula for it.

A new proof of the $q$ -Dixon identity

Victor J. W. Guo (2018)

Czechoslovak Mathematical Journal

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We give a new and elementary proof of Jackson’s terminating $q$ -analogue of Dixon’s identity by using recurrences and induction.

A note on signs of Kloosterman sums

Kaisa Matomäki (2011)

Bulletin de la Société Mathématique de France

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We prove that the sign of Kloosterman sums $Kl (1, 1; n)$ changes infinitely often as $n$ runs through the square-free numbers with at most $15$ prime factors. This improves on a previous result by Sivak-Fischler who obtained 18 instead of 15. Our improvement comes from introducing an elementary inequality which gives lower and upper bounds for the dot product of two sequences whose individual distributions are known.

On a cubic Hecke algebra associated with the quantum group $U_{q} (2)$

Janusz Wysoczański (2010)

Banach Center Publications

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We define an operator α on ℂ³ ⊗ ℂ³ associated with the quantum group $U_{q} (2)$ , which satisfies the Yang-Baxter equation and a cubic equation (α² - 1)(α + q²) = 0. This operator can be extended to a family of operators $h_{j} : = I_{j} \otimes α \otimes I_{n - 2 - j}$ on ${(ℂ ³)}^{\otimes n}$ with 0 ≤ j ≤ n - 2. These operators generate the cubic Hecke algebra $ℋ_{q, n} (2)$ associated with the quantum group $U_{q} (2)$ . The purpose of this note is to present the construction.

Proof of a conjectured three-valued family of Weil sums of binomials

Daniel J. Katz, Philippe Langevin (2015)

Acta Arithmetica

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We consider Weil sums of binomials of the form $W_{F, d} (a) = \sum_{x \in F} ψ (x^{d} - a x)$ , where F is a finite field, ψ: F → ℂ is the canonical additive character, $g c d (d, | F^{\times} |) = 1$ , and $a \in F^{\times}$ . If we fix F and d, and examine the values of $W_{F, d} (a)$ as a runs through $F^{\times}$ , we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo $| F^{\times} |$ ). Choices of F and d for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if F is a field of order 3ⁿ with n...