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Register a new userSuper congruences concerning binomial coefficients and Apery-like numbers
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Zhi-Hong Sun
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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.
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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 values. We then show that for integers $n$ whose base-$p$ digits belong to a certain set, $A(n)$ satisfies a Lucas congruence modulo $p^2$.
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 $binom{k}{r}$. Let $S(x)$ be the number of positive integers $r$ satisfying $1leq rleq x$ for which $binom{k}{r}$ is a Sierpi{n}ski number for infinitely many $k$. We further show that the value $S(x)/x$ gets arbitrarily close to 1 as $x$ tends to infinity. Generalizations to base $a$-Sierpi{n}ski numbers and base $a$-Riesel numbers are also considered. In particular, we prove that there exist infinitely many positive integers $r$ such that $binom{k}{r}$ is simultaneously a base $a$-Sierpi{n}ski and base $a$-Riesel number for infinitely many $k$.
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 numbers or Franel numbers conjectured by Z.-W. Sun. Let $pgeq 5$ be a prime and let $nin mathbb{Z}^{+}$. We show that begin{align} otag frac{1}{n}bigg(sum_{k=0}^{pn-1}(2k+1)A_k-psum_{k=0}^{n-1}(2k+1)A_kbigg)equiv0pmod{p^{4+3 u_p(n)}} end{align} and begin{align} otag frac{1}{n^3}bigg(sum_{k=0}^{pn-1}(2k+1)^3A_k-p^3sum_{k=0}^{n-1}(2k+1)^3A_kbigg)equiv0pmod{p^{6+3 u_p(n)}}, end{align} where $ u_p(n)$ denotes the $p$-adic order of $n$. Also, for any prime $p$ we have begin{align} otag frac{1}{n^3}bigg(sum_{k=0}^{pn-1}(3k+2)(-1)^kf_k-p^2sum_{k=0}^{n-1}(3k+2)(-1)^kf_kbigg)equiv0pmod{p^{3}}. end{align}
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 families of power series which involve odd harmonic numbers and central binomial coefficients. Furthermore it is shown that based on these series a new type of nonlinear Euler sums that involve odd harmonic numbers can be calculated in terms of zeta functions.
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 congruences of Hurwitz class numbers. The two keys tools in our proof are the holomorphic projection of products of theta series with a Hurwitz class number generating series and a theorem by Serre, which allows us to rule out certain congruences.
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