Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSome variations of a divergent Ramanujan-type $q$-supercongruence
137
0
0.0
(
0
)
Authors
Victor J. W. Guo
Ask ChatGPT about the research
No Arabic abstract
Using the $q$-Wilf--Zeilberger method and a $q$-analogue of a divergent Ramanujan-type supercongruence, we give several $q$-supercongruences modulo the fourth power of a cyclotomic polynomial. One of them is a $q$-analogue of a supercongruence recently proved by Wang: for any prime $p>3$, $$ sum_{k=0}^{p-1} (3k-1)frac{(frac{1}{2})_k (-frac{1}{2})_k^2 }{k!^3}4^kequiv p-2p^3 pmod{p^4}, $$ where $(a)_k=a(a+1)cdots (a+k-1)$ is the Pochhammer symbol.
rate research
Read More
This paper is concerned with a class of partition functions $a(n)$ introduced by Radu and defined in terms of eta-quotients. By utilizing the transformation laws of Newman, Schoeneberg and Robins, and Radus algorithms, we present an algorithm to find Ramanujan-type identities for $a(mn+t)$. While this algorithm is not guaranteed to succeed, it applies to many cases. For example, we deduce a witness identity for $p(11n+6)$ with integer coefficients. Our algorithm also leads to Ramanujan-type identities for the overpartition functions $overline{p}(5n+2)$ and $overline{p}(5n+3)$ and Andrews--Paules broken $2$-diamond partition functions $triangle_{2}(25n+14)$ and $triangle_{2}(25n+24)$. It can also be extended to derive Ramanujan-type identities on a more general class of partition functions. For example, it yields the Ramanujan-type identities on Andrews singular overpartition functions $overline{Q}_{3,1}(9n+3)$ and $ overline{Q}_{3,1}(9n+6)$ due to Shen, the $2$-dissection formulas of Ramanujan and the $8$-dissection formulas due to Hirschhorn.
We prove two supercongruences for specific truncated hypergeometric series. These include an uniparametric extension of a supercongruence that was recently established by Long and Ramakrishna. Our proofs involve special instances of various hypergeometric identities including Whipples transformation and the Karlsson--Minton summation.
Ramanujan graphs are graphs whose spectrum is bounded optimally. Such graphs have found numerous applications in combinatorics and computer science. In recent years, a high dimensional theory has emerged. In this paper these developments are surveyed. After explaining their connection to the Ramanujan conjecture we will present some old and new results with an emphasis on random walks on these discrete objects and on the Euclidean spheres. The latter lead to golden gates which are of importance in quantum computation.
In this paper, we confirm the following conjecture of Guo and Schlosser: for any odd integer $n>1$ and $M=(n+1)/2$ or $n-1$, $$ sum_{k=0}^{M}[4k-1]_{q^2}[4k-1]^2frac{(q^{-2};q^4)_k^4}{(q^4;q^4)_k^4}q^{4k}equiv (2q+2q^{-1}-1)[n]_{q^2}^4pmod{[n]_{q^2}^4Phi_n(q^2)}, $$ where $[n]=[n]_q=(1-q^n)/(1-q),(a;q)_0=1,(a;q)_k=(1-a)(1-aq)cdots(1-aq^{k-1})$ for $kgeq 1$ and $Phi_n(q)$ denotes the $n$-th cyclotomic polynomial.
Let $E_n$ be the $n$-th Euler number and $(a)_n=a(a+1)cdots (a+n-1)$ the rising factorial. Let $p>3$ be a prime. In 2012, Sun proved the that $$ sum^{(p-1)/2}_{k=0}(-1)^k(4k+1)frac{(frac{1}{2})_k^3}{k!^3} equiv p(-1)^{(p-1)/2}+p^3E_{p-3} pmod{p^4}, $$ which is a refinement of a famous supercongruence of Van Hamme. In 2016, Chen, Xie, and He established the following result: $$ sum_{k=0}^{p-1}(-1)^k (3k+1)frac{(frac{1}{2})_k^3}{k!^3} 2^{3k} equiv p(-1)^{(p-1)/2}+p^3E_{p-3} pmod{p^4}, $$ which was originally conjectured by Sun. In this paper we give $q$-analogues of the above two supercongruences by employing the $q$-WZ method. As a conclusion, we provide a $q$-analogue of the following supercongruence of Sun: $$ sum_{k=0}^{(p-1)/2}frac{(frac{1}{2})_k^2}{k!^2} equiv (-1)^{(p-1)/2}+p^2 E_{p-3} pmod{p^3}. $$
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
