Factors of Sums and Alternating Sums Involving Binomial Coefficients and Powers of Integers

Abstract

Abstract. We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers n 1 , . . . , n m , n m+1 = n 1 , and any nonnegative integer r, there holdsand conjecture that for any nonnegative integer r and positive integer s such that r + s is odd,where ε = ±1.

Factors of Sums and Alternating Sums of Products of $q$-binomial Coefficients and Powers of $q$-integers

“…The third aim of this paper is to give the following congruences involving q-ballot numbers. Note that the q = 1 case confirms a conjecture of Guo and Zeng [10,Conjecture 1.3].…”

“…Note that the q = 1 case of Theorem 1.4 confirms another conjecture of Guo and Zeng [10,Conjecture 6.10]. It should also be mentioned that Theorem 1.4 in the case where m = n gives the s 2 case of Theorem 1.3 (see (5.2)).…”

“…Note that there are exactly similar consequences of Theorem 1.2. We shall also confirm some conjectures in [10,Section 6]. For convenience, we let ε k = q j(k 2 +k)−(2r+1)k or ε k = (−1) k q ( k 2 )+j(k 2 +k)−2rk throughout this section.…”

“…Note that sums involving the ballot numbers A n,k := A n,k (1) have been considered by Miana and Romero [13, Theorem 10] and Guo and Zeng [10].…”

Proofs of two conjectures on Catalan triangle numbers

“…Conjecture 4.1]: Let n and a be positive integers with n > a, and let r be a non-negative integer.Note that Conjecture 1.4 is true for some special cases (see Guo and Zeng [10, Theorem 1.4], Guo and Zeng[11, Theorem 1.4], and Guo and Wang[7, Theorem 1.3]).…”