Related papers: Supercongruences arising from a ${}_7F_6$ hypergeo…
We obtain a small improvement of Gallagher's larger sieve and we extend it to higher dimensions. We also obtain two interesting upper bounds for the number of solutions to polynomial congruences.
The confluent hypergeometric functions (the Kummer functions) defined by ${}_{1}F_{1}(\alpha;\gamma;z):=\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!(\gamma)_{n}}z^{n}\ (\gamma\neq 0,-1,-2,\cdots)$, which are of many properties and great…
A simple proof of a new summation formula for a terminating r+3Fr+2(1) hypergeometric series, representing an extension of Saalschutz's formula for a 3F2(1) series, is given for the case of r pairs of numeratorial and denominatorial…
A new class of integrals involving the confluent hypergeometric function ${}_1F_{1}(a;c;z)$ and the Riemann $\Xi$-function is considered. It generalizes a class containing some integrals of S. Ramanujan, G.H. Hardy and W.L. Ferrar and gives…
In this article we give an example of a matrix version of the famous congruence for hypergeometric functions found by Dwork in 'p-adic cycles'.
In this paper, we prove two supercongruences conjectured by Z.-W. Sun via the Wilf-Zeilberger method. One of them is, for any prime $p>3$, \begin{align*} \sum_{n=0}^{p-1}\frac{6n+1}{256^n}\binom{2n}n^3&\equiv…
Superconformal symmetry in six-dimensions is analyzed in terms of coordinate transformations on superspace. A superconformal Killing equation is derived and its solutions are identified in terms of supertranslations, dilations, Lorentz…
Euler's transformation formula for the Gauss hypergeometric function 2F1 is extended to hypergeometric functions of higher order. Unusually, the generalized transformation constrains the hypergeometric function parameters algebraically but…
In this note, we confirm two conjectural supercongruences on double sums of binomial coefficients due to El Bachraoui.
In investigating the properties of a certain class of homogeneous polynomials, we discovered an identity satisfied by their coefficients which involves simple 2F1 Gauss hypergeometric functions. This result appears to be new and we supply a…
Wierzba and Wisniewski proved that in dimension 4, every bimeromorphic map of hyperkahler manifolds is represented as a composition of Mukai flops. Hu and Yau conjectured that this result can be generalized to arbitrary dimension. They…
For every parabolic subgroup $P$ of a Lie supergroup $G$ the homogeneous superspace $G/P$ carries a $G$-invariant supergeometry. We address the quesiton whether $\mathfrak{g}=\operatorname{Lie}(G)$ is the maximal symmetry of this…
In this paper, we obtain analytical solutions of Laplace transform based some generalized class of the hyperbolic integrals in terms of hypergeometric functions ${}_3F_2 (\pm1)$, ${}_4F_3 (\pm1)$, ${}_5F_4(\pm1)$, ${}_6F_5(\pm1)$,…
The objective of this short note is to provide two closed-form evaluations for the generalized hypergeometric function $_4F_3$ of the argument $\frac1{16}$. This is achieved by means of separating a generalized hypergeometric function…
The classical hypergeometric summation theorems are exploited to derive several striking identities on harmonic numbers including those discovered recently by Paule and Schneider (2003).
We prove sum representations of Appell-Lauricella functions over a finite field using confluent hypergeometric functions over the finite field. As an application, we also prove transformation formulas, summation formulas and reduction…
We discuss classical and quantum symmetries of extended Hubbard models. The quantum symmetries are shown to be related to the known superconducting SU(2) symmetry of the original Hubbard model at half filling via generalized Lang-Firsov…
It is shown how the gluing theorem due to Corvino and Shoen can be naturally extended to accommodate supertranslations at spatial infinity. Logarithmic supertranslations play an intersting role in the construction.
Several papers have been written studying unexpected hypersurfaces. We say a finite set of points Z admits unexpected hypersurfaces if a general union of fat linear subspaces imposes less that the expected number of conditions on the ideal…
The most detailed constructions of microstate geometries, and particularly of superstrata, are done using $\mathcal{N} = (1,0)$ supergravity coupled to two anti-self-dual tensor multiplets in six dimensions. We show that an important…