Related papers: Divisibility results concerning truncated hypergeo…

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…

For any $m,n\in\mathbb{N}=\{0,1,2\ldots\}$, the truncated hypergeometric series ${}_{m+1}F_m$ is defined by $$ {}_{m+1}F_m\bigg[\begin{matrix}x_0&x_1&\ldots&x_m\\…

For $n\in\mathbb{N}=\{0,1,2,\ldots\}$ and $b,c\in\mathbb{Z}$, the $n$th generalized central trinomial coefficient $T_n(b,c)$ is the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. In particular, $T_n=T_n(1,1)$ is the central…

For various positive integers $k$, the sums of $k$th powers of the first $n$ positive integers, $S_k(n+1)=1^k+2^k+...+n^k$, have got to be some of the most popular sums in all of mathematics. In this note we prove that for each $k\ge 2 $$…

For any positive integers $m$ and $\alpha$, we prove that $$\sum_{k=0}^{n-1}\epsilon^k(2k+1)A_k^{(\alpha)}(x)^m\equiv0\pmod{n}, $$ where $\epsilon\in\{1,-1\}$ and $$…

Motivated by recent work of George Andrews and Mircea Merca on the expansion of the quotient of the truncation of Euler's pentagonal number series by the complete series, we provide similar expansion results for averages involving…

Many combinatorial sequences (for example, the Catalan and Motzkin numbers) may be expressed as the constant term of $P(x)^k Q(x)$, for some Laurent polynomials $P(x)$ and $Q(x)$ in the variable $x$ with integer coefficients. Denoting such…

We show that if p is an odd prime then $$\sum_{k=0}^{p-1}E_kE_{p-1-k}=1 (mod p)$$ and $$\sum_{k=0}^{p-3}E_kE_{p-3-k}=(-1)^{(p-1)/2}2E_{p-3} (mod p),$$ where E_0,E_1,E_2,... are Euler numbers. Moreover, we prove that for any positive integer…

Let $W$ be a smooth test function with compact support in $(0,\infty)$. Conditional on the Generalized Riemann Hypothesis for Hecke $L$-functions over $\mathbb{Q}(\omega)$, we prove that $$\sum_{p \equiv 1 \pmod{3}} \frac{1}{2 \sqrt{p}}…

Kemnitz Conjecture [9] states that if we take a sequence of elements in $Z_{p}^{2}$ of length $4p-3$, $p$ is a prime number, then it has a subsequence of length $p$, whose sum is $0$ modulo $p$. It is known that in $Z_{p}^{3}$ to get a…

The Ap\'ery polynomials are given by $$A_n(x)=\sum_{k=0}^n\binom nk^2\binom{n+k}k^2x^k\ \ (n=0,1,2,\ldots).$$ (Those $A_n=A_n(1)$ are Ap\'ery numbers.) Let $p$ be an odd prime. We show that…

For primes $p\geq 5 $, $K(KU_p)$ -- the algebraic $K$-theory spectrum of $(KU)^{\wedge}_p$, Morava $K$-theory $K(1)$, and Smith-Toda complex $V(1)$, Ausoni and Rognes conjectured (alongside related conjectures) that $L_{K(1)}S^0…

Let $p$ be prime. We prove that, for $n$ odd, the $p$-torsion part of $\pi_q(S^{n})$ has cardinality at most $p^{2^{\frac{1}{p-1}(q-n+3-2p)}}$, and hence has rank at most $2^{\frac{1}{p-1}(q-n+3-2p)}$. For $p=2$ these results also hold for…

Let $p$ be a prime. In this paper, we present a detailed $p$-adic analysis to factorials and double factorials and their congruences. We give good bounds for the $p$-adic sizes of the coefficients of the divided universal Bernoulli number…

In 2017, He [Proc. Amer. Math. Soc. 145 (2017), 501--508] established two spuercongruences on truncated hypergeometric series and further proposed two related conjectures. Subsequently, Liu [Results Math. 72 (2017), 2057--2066] extended…

Let $p>5$ be a prime. We prove congruences modulo $p^{3-d}$ for sums of the general form $\sum_{k=0}^{(p-3)/2}\binom{2k}{k}t^k/(2k+1)^{d+1}$ and $\sum_{k=1}^{(p-1)/2}\binom{2k}{k}t^k/k^d$ with $d=0,1$. We also consider the special case…

Motivated by recent works of Sun and Tauraso, we prove some variations on the Green-Krammer identity involving central q-binomial coefficients, such as $$ \sum_{k=0}^{n-1}(-1)^kq^{-{k+1\choose 2}}{2k\brack k}_q \equiv (\frac{n}{5})…

The main aim of this paper is to derive some new summation theorems for terminating and truncated Clausen's hypergeometric series with unit argument, when one numerator parameter and one denominator parameter are negative integers. Further,…

We prove several power series identities involving the refined generating function of interval orders, as well as the refined generating function of the self-dual interval orders. These identities may be expressed as $\sum_{n\ge…

We establish the first moment bound $$ \sum_{\varphi} L(\varphi \otimes \varphi \otimes \Psi, \tfrac{1}{2}) \ll_{\varepsilon} p^{5/4+\varepsilon} $$ for triple product $L$-functions, where $\Psi$ is a fixed Hecke-Maass form on…