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Let $p$ be a prime with $p>3$, and let $a,b$ be two rational $p-$integers. In this paper we present general congruences for $sum_{k=0}^{p-1}binom akbinom{-1-a}kfrac p{k+b}pmod {p^2}$. For $n=0,1,2,ldots$ let $D_n$ and $b_n$ be Domb and Almkvist-Zudilin numbers, respectively. We also establish congruences for $$sum_{n=0}^{p-1}frac{D_n}{16^n},quad sum_{n=0}^{p-1}frac{D_n}{4^n}, quad sum_{n=0}^{p-1}frac{b_n}{(-3)^n},quad sum_{n=0}^{p-1}frac{b_n}{(-27)^n}pmod {p^2}$$ in terms of certain binary quadratic forms.
The sequence $A(n)_{n geq 0}$ of Apery numbers can be interpolated to $mathbb{C}$ by an entire function. We give a formula for the Taylor coefficients of this function, centered at the origin, as a $mathbb{Z}$-linear combination of multiple zeta valu
In this paper, we investigate the existence of Sierpi{n}ski numbers and Riesel numbers as binomial coefficients. We show that for any odd positive integer $r$, there exist infinitely many Sierpi{n}ski numbers and Riesel numbers of the form $binom{k}{
The Apery numbers $A_n$ and the Franel numbers $f_n$ are defined by $$A_n=sum_{k=0}^{n}{binom{n+k}{2k}}^2{binom{2k}{k}}^2 {rm and } f_n=sum_{k=0}^{n}{binom{n}{k}}^3(n=0, 1, cdots,).$$ In this paper, we prove three supercongruences for Apery
We present several sequences involving harmonic numbers and the central binomial coefficients. The calculational technique is consists of a special summation method that allows, based on proper two-valued integer functions, to calculate different fam
In contrast to all other known Ramanujan-type congruences, we discover that Ramanujan-type congruences for Hurwitz class numbers can be supported on non-holomorphic generating series. We establish a divisibility result for such non-holomorphic congru