Hao Pan, Zhi-Wei Sun
We prove that for any nonnegative integers $n$ and $r$ the binomial sum $$ \sum_{k=-n}^n\binom{2n}{n-k}k^{2r} $$ is divisible by $2^{2n-\min\{α(n),α(r)\}}$, where $α(n)$ denotes the number of 1's in the binary expansion of $n$. This confirms a recent conjecture of Guo and Zeng.
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We prove some 3-adic congruences for binomial sums, which were conjectured by Sun.
We propose higher-order generalizations of Jacobsthal's $p$-adic approximation for binomial coefficients. Our results imply explicit formulae for linear combinations of binomial coefficients $\binom{ip}{p}$ ($i=1,2,\dots$) that are divisible by arbitrarily large powers of prime $p$.
The coefficients c(n,k) defined by (1-k^2x)^(-1/k) = sum c(n,k) x^n reduce to the central binomial coefficients for k=2. Motivated by a question of H. Montgomery and H. Shapiro for the case k=3, we prove that c(n,k) are integers and study their divisibility properties.
We prove that a certain family of sums of products of three binomials has alternating behavior modulo a prime $p$. To accomplish this we rewrite these sums as signed sums of products of three binomials, the better to handle $p$, and we give closed-form expressions for two related sums of signed products of three binomials.
We confirm several conjectures of Guo, Jouhet and Zeng concerning the factors of alternative binomials sums.
In this note, we confirm two conjectural supercongruences on double sums of binomial coefficients due to El Bachraoui.
We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.
Let k>1 be an integer and let p be a prime. We show that if $p^a\le k<2p^a$ or $k=p^aq+1$ (with 2q<p) for some a=1,2,..., then the set {\binom{n}{k}: n=0,1,2,...} is dense in the ring Z_p of p-adic integers, i.e., it contains a complete system of residues modulo any power of p.
We establish $q$-analogs for four congruences involving central binomial coefficients. The $q$-identities necessary for this purpose are shown via the $q$-WZ method.
Due to some results by John P. D Angelo and Dusty Grundmeier about CR- mappings the main results of my 2001 paper about recurrences for some sequences of binomial sums can be simplified.
By using the Rodriguez-Villegas-Mortenson supercongruences, we prove four supercongruences on sums involving binomial coefficients, which were originally conjectured by Sun. We also confirm a related conjecture of Guo on integer-valued polynomials.
We prove the divisibility conjecture on sums of even powers of $q$-binomial coefficients, which was recently proposed by Guo, Schlosser and Zudilin. Our proof relies on two $q$-harmonic series congruences due to Shi and Pan.
In this paper, we prove a divisibility result for the lacunary $q$-binomial sum $$ \sum_{k\equiv r\pmod{c}}(-1)^kq^{\binom{k}{2}}\qbinom{n}{k}{q}\qbinom{(k-r)/c}{l}{q^{c}}. $$
Periodic Weighted Sums of Binomial Coefficients
本文用初等方法建立了二项式系数与斐波那契数、卢卡斯数等之间的新旧关系。
A $q$-analogue of Boyadzhiev-Mneimneh-type binomial sums of finite multi-polylogarithms
该研究给出有限多重多重对数的Boyadzhiev-Mneimneh型二项式和的$q$-模拟公式，其$q\ o1$极限等价于Sakugawa-Seki恒等式，还给出对应柯西二项式定理的该类和的公式。
We establish a $q$-analogue of Sun--Zhao's congruence on harmonic sums. Based on this $q$-congruence and a $q$-series identity, we prove a congruence conjecture on sums of central $q$-binomial coefficients, which was recently proposed by Guo. We also deduce a $q$-analogue of a congruence due to Apagodu and Zeilberger from Guo's $q$-congruence.
With help of $q$-congruence, we prove the divisibility of some binomial sums. For example, for any integers $ρ,n\geq 2$, $$\sum_{k=0}^{n-1}(4k+1) \binom{2k}{k}^ρ\cdot (-4)^{ρ(n-1-k)} \equiv 0\pmod{2^{ρ-2}n\binom{2n}{n}}.$$
Three combinatorial sums involving central binomial coefficients
本文研究三类含中心二项式系数及不同调和数的组合和，通过分部求和建立递归式，并用第二类 Stirling 数与 r-Stirling 数给出闭合形式。
In this paper we initiate the study of products and sums divisible by central binomial coefficients. We show that 2(2n+1)binom(2n,n)| binom(6n,3n)binom(3n,n) for every n=1,2,3,... Also, for any nonnegative integers $k$ and $n$ we have $$\binom {2k}k | \binom{4n+2k+2}{2n+k+1}\binom{2n+k+1}{2k}\binom{2n-k+1}n$$ and $$\binom{2k}k | (2n+1)\binom{2n}nC_{n+k}\binom{n+k+1}{2k},$$ where $C_m$ denotes the Catalan number $\binom{2m}m/(m+1)=\binom{2m}m-\binom{2m}{m+1}$. Applying this result we obtain two sums divisible by central binomial coefficients.
Let f(n)= Sum binomial(n,k)^(-1). First, we show that f:N to Q_p is nowhere continuous in the p-adic topology. If x is a p-adic integer, we say that f(x) is p-definable if lim f(x_j) exists in Q_p, where x_j denotes the jth partial sum for x. We prove that f(-1) is p-definable for all primes p, and if p is odd, then -1 is the only element of Z_p - N for which f(x) is p-definable. For p=2, we show that if k is a positive integer, then f(-k-1) is not 2-definable, but that if the 1's in the binary expansion of x are eventually very sparse, then f(x) is 2-definable. Some of our proofs require that…
In this paper, we confirm several conjectured congruences of Sun concerning the divisibility of binomial sums. For example, with help of a quadratic hypergeometric transformation, we prove that $$ \sum_{k=0}^{p-1}\binom{p-1}k\binom{2k}k^2\frac{P_k}{8^k}\equiv0\pmod{p^2} $$ for any prime $p\equiv 7\mod{8}$, where $P_k$ is the $k$-th Pell number. Further, we also propose three new congruences of the same type.
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 holds {align*} \sum_{k=0}^{n_1}ε^k (2k+1)^{2r+1}\prod_{i=1}^{m} {n_i+n_{i+1}+1\choose n_i-k} \equiv 0 \mod (n_1+n_m+1){n_1+n_m\choose n_1}, {align*} and conjecture that for any nonnegative integer $r$ and positive integer $s$ such that $r+s$ is odd, $$ \sum_{k=0}^{n}ε^k (2k+1)^{r}({2n\choose n-k}-{2n\choose n-k-1})^{s} \equiv 0 \mod{2n\choose n}, $$…
We evaluate various binomial sums involving the powers of Fibonacci and Lucas numbers.
In this paper, we consider two particular binomial sums \begin{align*} \sum_{k=0}^{n-1}(20k^2+8k+1){\binom{2k}{k}}^5 (-4096)^{n-k-1} \end{align*} and \begin{align*} \sum_{k=0}^{n-1}(120k^2+34k+3){\binom{2k}{k}}^4\binom{4k}{2k} 65536^{n-k-1}, \end{align*} which are inspired by two series for $\frac{1}{π^2}$ obtained by Guillera. We consider their divisibility properties and prove that they are divisible by $2n^2 \binom{2n}{n}^2$ for all integer $n\geq 2$. These divisibility properties are stronger than those divisibility results found by He, who proved the above two sums are divisible by $2n \b…
We give some results and conjectures about recurrence relations for certain sequences of binomial sums.
In this paper we study the factors of some alternating sums of products of binomial and q-binomial coefficients. We prove that for all positive integers n_1,...,n_m, n_{m+1}=n_1, and 0\leq j\leq m-1, {n_1+n_{m}\brack n_1}^{-1}\sum_{k=-n_1}^{n_1}(-1)^kq^{jk^2+{k\choose 2}} \prod_{i=1}^m {n_i+n_{i+1}\brack n_i+k}\in \N[q], which generalizes a result of Calkin [Acta Arith. 86 (1998), 17--26]. Moreover, we show that for all positive integers n, r and j, {2n\brack n}^{-1}{2j\brack j} \sum_{k=j}^n(-1)^{n-k}q^{A}\frac{1-q^{2k+1}}{1-q^{n+k+1}} {2n\brack n-k}{k+j\brack k-j}^r\in N[q], where A=(r-1){n\c…
We present some new linear, quadratic, cubic and quartic binomial Fibonacci, Lucas and Fibonacci--Lucas summation identities.
Modifying an idea of E. Brietzke we give simple proofs for the recurrence relations of some sequences of binomial sums which have previously been obtained by other more complicated methods.
By using the Newton interpolation formula, we generalize the recent identities on the Catalan triangle obtained by Miana and Romero as well as those of Chen and Chu. We further study divisibility properties of sums of products of binomial coefficients and an odd power of a natural number. For example, we prove that for all positive integers $n_1, ..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer $r$, the expression $$n_1^{-1}{n_1+n_{m}\choose n_1}^{-1} \sum_{k=1}^{n_1}k^{2r+1}\prod_{i=1}^{m} {n_i+n_{i+1}\choose n_i+k}$$ is either an integer or a half-integer. Moreover, several related conj…
If p is a prime and n a positive integer, let v(n) denote the exponent of p in n, and u(n)=n/p^{v(n)} the unit part of n. If k is a positive integer not divisible by p, we show that the p-adic limit of (-1)^{pke} u((kp^e)!) as e goes to infinity is a well-defined p-adic integer, which we call z_k. In terms of these, we give a formula for the p-adic limit of binom{a p^e +c, b p^e +d) as e goes to infinity, which we call binom(a p^\infty +c, b p^\infty +d). Here a \ge b are positive integers, and c and d are integers.
In this short note, we establish some identities containing sums of binomials with coefficients satisfying third order linear recursive relations. As a result and in particular, we obtain general forms of earlier identities involving binomial coefficients and Fibonacci type sequences.
We consider binomial and inverse binomial sums at infinity and rewrite them in terms of a small set of constants, such as powers of $π$ or $\log(2)$. In order to perform these simplifications, we view the series as specializations of generating series. For these generating series, we derive integral representations in terms of root-valued iterated integrals. Using substitutions, we express the interated integrals as cyclotomic harmonic polylogarithms. Finally, by applying known relations among the cyclotomic harmonic polylogarithms, we derive expressions in terms of several constants.
Let q>1 and m>0 be relatively prime integers. We find an explicit period $ν_m(q)$ such that for any integers n>0 and r we have $[n+ν_m(q),r]_m(a)=[n,r]_m(a) (mod q)$ whenever a is an integer with $\gcd(1-(-a)^m,q)=1$, or a=-1 (mod q), or a=1 (mod q) and 2|m, where $[n,r]_m(a)=\sum_{k=r(mod m)}\binom{n}{k}a^k$. This is a further extension of a congruence of Glaisher.
Spivey presented a new approach to evaluate combinatorial sums by using finite differences. We present some closed forms for sums involving the binomial coefficients, Fibonacci and Lucas numbers in terms of the falling factorial.
In this paper, we prove two conjectures of Z.-W. Sun: $$2n\binom{2n}n\big|\sum_{k=0}^{n-1}(3k+1)\binom{2k}k^3{16}^{n-1-k}\ \mbox{for}\ \mbox{all}\ n=2,3,\cdots,$$ and $$\sum_{k=0}^{(p-1)/2}\frac{3k+1}{16^k}\binom{2k}{k}^3\equiv p+2\left(\frac{-1}{p}\right)p^3E_{p-3}\pmod{p^4},$$ where $p>3$ is a prime and $E_0,E_1,E_2,\cdots$ are Euler numbers.
Sums of powers of binomials, their Apéry limits, and Franel's suspicions
本文显式确定了二项式系数幂和的Apéry极限，并在假设递归可通过创造性望远镜法获得的前提下，证明了关于这些序列递归阶的Franel猜想的最小阶版本。
We prove that, for all positive integers $n_1, \ldots, n_m$, $n_{m+1}=n_1$, and non-negative integers $j$ and $r$ with $j\leqslant m$, the following two expressions \begin{align*} &\frac{1}{[n_1+n_m+1]}{n_1+n_{m}\brack n_1}^{-1}\sum_{k=0}^{n_1} q^{j(k^2+k)-(2r+1)k}[2k+1]^{2r+1}\prod_{i=1}^m {n_i+n_{i+1}+1\brack n_i-k},\\[5pt] &\frac{1}{[n_1+n_m+1]}{n_1+n_{m}\brack n_1}^{-1}\sum_{k=0}^{n_1}(-1)^k q^{{k\choose 2}+j(k^2+k)-2rk}[2k+1]^{2r+1}\prod_{i=1}^m {n_i+n_{i+1}+1\brack n_i-k} \end{align*} are Laurent polynomials in $q$ with integer coefficients, where $[n]=1+q+\cdots+q^{n-1}$ and ${n\brack k…
In this paper we study recurrences concerning the combinatorial sum $[n,r]_m=\sum_{k\equiv r (mod m)}\binom {n}{k}$ and the alternate sum $\sum_{k\equiv r (mod m)}(-1)^{(k-r)/m}\binom{n}{k}$, where m>0, $n\ge 0$ and r are integers. For example, we show that if $n\ge m-1$ then $$\sum_{i=0}^{\lfloor(m-1)/2\rfloor}(-1)^i\binom{m-1-i}i [n-2i,r-i]_m=2^{n-m+1}.$$ We also apply such results to investigate Bernoulli and Euler polynomials. Our approach depends heavily on an identity established by the author [Integers 2(2002)].
Double sums involving binomial coefficients and special numbers
该研究提出一种初等方法处理内层为不完全二项式和的双重求和问题，并将核心恒等式应用于含调和数、Fibonacci数等多种特殊数的双重求和场景。
The main purpose of this paper is to establish some new properties of Horadam numbers in terms of binomial sums. By that, we can obtain these special numbers in a new and direct way. Moreover, some connections between Horadam and generalized Lucas numbers are revealed to get a more strong result.
Recently the first author proved a congruence proposed in 2006 by Adamchuk: $\sum_{k=1}^{\lfloor\frac{2p}{3}\rfloor}\binom{2k}{k}\equiv 0\pmod{p^2}$ for any prime $p=1 \pmod{3}$. In this paper, we provide more examples (with proofs) of congruences of the same kind $$\sum_{k=1}^{\lfloor\frac{ap}{r}\rfloor}\binom{2k}{k}x^k \pmod{p^2}$$ where $p$ is a prime such that $p\equiv 1 \pmod{r}$, $a/r$ is a fraction in $(1/2,1)$ and $x$ is a $p$-adic integer. The key ingredients are the $p$-adic Gamma functions $Γ_p$ and a special class of computer-discovered hypergeometric identities.
Double sums associated with binomial transforms
本文针对内层为不完整二项式和的二重和展开研究，证明其与二项式变换对相关的新结果，推导了多类特殊数、多项式族相关的二重和恒等式及含双曲函数的新二重和。
Carnevale and Voll conjectured that j (--1) j $λ$ 1 j $λ$ 2 j = 0 when $λ$ 1 and $λ$ 2 are two distinct integers. We check the conjecture when either $λ$ 2 or $λ$ 1 -- $λ$ 2 is small. We investigate the asymptotic behaviour of their sum when the ratio r := $λ$ 1 /$λ$ 2 is fixed and $λ$ 2 goes to infinity. We find an explicit range r $\ge$ 5.8362 on which the conjecture is true. We show that the conjecture is almost surely true for any fixed r. For r close to 1, we give several explicit intervals on which the conjecture is also true.
Binomial sums and properties of the Bernoulli transform
本文研究了二项式和Sₙ(q)的性质，探讨了其与伯努利变换的关系，并给出了与阿佩尔多项式相关的恒等式。
In this paper we discuss a class of double sums involving ratios of binomial coefficients. The sums are of the form \[ \sum_{j=0}^{n} \sum_{i=0}^j \frac{\binom{f_1(n)}{i}}{\binom{f_2(n)}{j}}\,c^{i-j}, \] where $f_1, f_2$ are functions of $n$. Such sums appear in the analyses of the Mabinogion urn and the Ehrenfest urn in probability. Using hypergeometric functions, we are able to simplify these sums, and in some cases express them in terms of the harmonic numbers.
Let $p>3$ be a prime, and let $a$ be a rational $p$-adic integer, using WZ method we establish the congruences modulo $p^3$ for $$\sum_{k=0}^{p-1} \binom ak\binom{-1-a}k\binom{2k}k\frac {w(k)}{4^k},$$ where $$w(k)=1,\frac 1{k+1},\frac 1{(k+1)^2},\frac 1{(k+1)^3},\frac 1{2k-1},\frac 1{k+2}, \frac 1{k+3}, k,k^2,k^3,\frac 1{a+k},\frac 1{a+k-1}.$$ As consequences, taking $a=-\frac 12,-\frac 13,-\frac 14,-\frac 16$ we deduce many congruences modulo $p^3$ and so solve some conjectures posed by the author earlier.
Binomial Fibonacci sums from Chebyshev polynomials
本文通过将特定二项式系数求和与切比雪夫多项式建立联系，推导出涉及斐波那契数和卢卡斯数的新型二项式和恒等式，并进一步研究组合和及其与 Chu 和 Guo 2022 年工作的联系。
A Formal Refutation of the Hypergeometric Parametric Extension for Reciprocal Binomial Sums
本文通过逻辑一致性检查、积分推导分析和符号计算，证明Pain提出的参数扩展在递归二项式和中的超几何闭形式表示是错误的。
Sums of products of binomial coefficients mod 2 and run length transforms of sequences
该研究推导了二项式系数乘积之和模2的递推关系，证明其生成序列为已知基本序列的游程长度变换，并给出正整数、斐波那契数等序列游程长度变换的公式。
A generalized central trinomial coefficient $T_n(b,c)$ is the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$ with $b,c\in\mathbb Z$. 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 forms, and 62 proposed new series for $1/π$ motivated by congruences and related dualities.
Universal Matrices for Counting Fibonomial and $C$-nomial Coefficients by their $p$-adic Valuations
针对由序列 $C$ 定义的二项式系数推广形式 $C$-nomial 系数，提出当 $C$ 为强整除序列时，其按 $p$-adic 赋值计数的生成函数可通过通用矩阵乘积公式表示，并将该公式推广至 $C$-多项式系数。
We provide a unified, probabilistic approach using renewal theory to derive some novel limits of sums for the normalized binomial coefficients and for the normalized Eulerian numbers. We also investigate some corresponding results for their associated distributions -- the binomial distributions for the binomial coefficients and the Irwin-Hall distributions (uniform B-splines) for the Eulerian numbers.
Sums of products of binomial coefficients mod 2 and 2-regular sequences
本文证明了模2二项式系数乘积之和对应的线性递推序列游程长度变换是2-正则序列，从而利用计算机程序 Walnut 复现并扩展了 Wu 的结果，免去了冗长的技术证明。
We prove exact asymptotic expansions for the partial sums of the sequences of central binomial coefficients and Catalan numbers, $\sum_{k=0}^n \binom{2k}{k}$ and $\sum_{k=0}^n C_n$. We also obtain closed forms for the polynomials $\sum_{k=0}^{q-1}\binom{2k}{k}x^k$ and $\sum_{k=0}^{q-1}C_kx^k$ over the field of $p$ elements, where $q$ is a power of the prime $p$.
Let p be an odd prime and let a be a positive integer. In this paper we investigate the sum $\sum_{k=0}^{p^a-1}\binom{hp^a-1}{k}\binom{2k}{k}/m^k$ mod p^2, where h,m are p-adic integers with m\not=0 (mod p). For example, we show that if h\not=0 (mod p) and p^a>3 then $$ sum_{k=0}^{p^a-1}\binom{hp^a-1}{k}\binom{2k}{k}(-h/2)^k =(\frac{1-2h}{p^a})(1+h((4h-2)^{p-1}/h^{p-1}-1)) (mod p^2),$$ where (-) denotes the Jacobi symbol. Here is another remarkable congruence: If p>3 then $$\sum_{k=0}^{p^a-1}\binom{p^a-1}{k}\binom{2k}{k}(-1)^k =3^{p-1}(\frac{p^a}3) (mod p^2).$$
In this paper, we pose lots of challenging conjectures on congruences for the sums involving binomial coefficients and Apéry-like numbers modulo $p^3$, where $p$ is an odd prime.
Using the WZ-method we find some of the easiest Ramanujan's formulae and also some new interesting Ramanujan-like sums.
We present a survey on Weil sums in which an additive character of a finite field $F$ is applied to a binomial whose individual terms (monomials) become permutations of $F$ when regarded as functions. Then we indicate how these Weil sums are used in applications, especially how they characterize the nonlinearity of power permutations and the correlation of linear recursive sequences over finite fields. In these applications, one is interested in the spectrum of Weil sum values that are obtained as the coefficients in the binomial are varied. We review the basic properties of such spectra, and …
We show that, for all positive integers $n_1, \ldots, n_m$, $n_{m+1}=n_1$, and any non-negative integers $j$ and $r$ with $j\leqslant m$, the expression $$ \frac{1}{[n_1]}{n_1+n_{m}\brack n_1}^{-1} \sum_{k=1}^{n_1}[2k][k]^{2r}q^{jk^2-(r+1)k}\prod_{i=1}^{m} {n_i+n_{i+1}\brack n_i+k} $$ is a Laurent polynomial in $q$ with integer cofficients, where $[n]=1+q+\cdots+q^{n-1}$ and ${n\brack k}=\prod_{i=1}^k(1-q^{n-i+1})/(1-q^i)$. This gives a $q$-analogue of a divisibility result on the Catalan triangle obtained by the first author and Zeng, and also confirms a conjecture of the first author and Zen…
Recently several authors have proved results on Ehrhart series of free sums of rational polytopes. In this note we treat these results from an algebraic viewpoint. Instead of attacking combinatorial statements directly, we derive them from structural results on affine monoids and their algebras that allow conclusions for Hilbert and Ehrhart series. We characterize when a binomial regular sequence generates a prime ideal or even normality is preserved for the residue class ring.
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