Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userProof of a q-supercongruence conjectured by Guo and Schlosser
86
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
By means of the $q$-Zeilberger algorithm, we prove a basic hypergeometric supercongruence modulo the fifth power of the cyclotomic polynomial $Phi_n(q)$. This result appears to be quite unique, as in the existing literature so far no basic hypergeometric supercongruences modulo a power greater than the fourth of a cyclotomic polynomial have been proved. We also establish a couple of related results, including a parametric supercongruence.
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.
In this note, using the derangement polynomials and their umbral representation, we give another simple proof of an identity conjectured by Lacasse in the study of the PAC-Bayesian machine learning theory.
In light of the well-known fact that the $n$th divided difference of any polynomial of degree $m$ must be zero while $m<n$,the present paper proves the $(alpha,beta)$-inversion formula conjectured by Hsu and Ma [J. Math. Res. $&$ Exposition 25(4) (2005) 624]. As applications of $(alpha,beta)$-inversion, we not only recover some known matrix
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
