Exponential sums related to binomial coefficient parity
نویسندگان
چکیده
منابع مشابه
Stepanov’s Method Applied to Binomial Exponential Sums
For a prime p and binomial axk+bxl with 1 ≤ l < k < 1 32 (p−1) 2 3 , we use Stepanov’s method to obtain the bound ∣∣∣∣∣ p−1 ∑ x=1 ep(ax k + bx) ∣∣∣∣∣ max { 1, l∆− 1 3 } 1 4 k 1 4 p 3 4 , where ∆ = k−l (k,l,p−1) .
متن کاملExplicit Bounds on Monomial and Binomial Exponential Sums
Let p be a prime and ep(·) = e2πi·/p. First, we make explicit the monomial sum bounds of Heath-Brown and Konyagin: ̨̨̨Pp−1 x=1 ep(ax d) ̨̨̨ ≤ min{λ d5/8p5/8, λ d3/8p3/4}, where λ = 2/ 4 √ 3 = 1.51967 . . .. Second, letting d = (k, l, p− 1), we obtain the explicit binomial sum bound ̨̨̨Pp−1 x=1 ep(ax k + bxl) ̨̨̨ ≤ (k − l, p− 1) + 2.292 d13/46p89/92, for any nonconstant binomial axk + bxl on Zp, by sharpeni...
متن کاملSome New Binomial Sums Related to the Catalan Triangle
In this paper, we derive many new identities on the classical Catalan triangle C = (Cn,k)n>k>0, where Cn,k = k+1 n+1 ( 2n−k n ) are the well-known ballot numbers. The first three types are based on the determinant and the fourth is relied on the permanent of a square matrix. It not only produces many known and new identities involving Catalan numbers, but also provides a new viewpoint on combin...
متن کاملOn Sums Related to Central Binomial and Trinomial Coefficients
A generalized central trinomial coefficient Tn(b, c) is the coefficient of x in the expansion of (x+bx+c) with b, c ∈ Z. In this paper we investigate congruences and series for sums of terms related to central binomial coefficients and generalized central trinomial coefficients. The paper contains many conjectures on congruences related to representations of primes by certain binary quadratic f...
متن کاملCombinatorial interpretations of binomial coefficient analogues related to Lucas sequences
Let s and t be variables. Define polynomials {n} in s, t by {0} = 0, {1} = 1, and {n} = s {n− 1}+ t {n− 2} for n ≥ 2. If s, t are integers then the corresponding sequence of integers is called a Lucas sequence. Define an analogue of the binomial coefficients by {n k } = {n}! {k}! {n− k}! where {n}! = {1} {2} · · · {n}. It is easy to see that { n k } is a polynomial in s and t. The purpose of th...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 1980
ISSN: 0002-9939
DOI: 10.1090/s0002-9939-1980-0581019-4