Related papers: Some supercongruences arising from symbolic summat…
We naturally obtain some combinatorial identities finding the difference analogs of hyperbolic and trigonometric functions of order $n.$ In particular, we obtain the identities connected with the proved in the paper the addition formulas…
In [7], Higashitani, Kummer, and Micha{\l}ek pose a conjecture about the symmetric edge polytopes of complete multipartite graphs and confirm it for a number of families in the bipartite case. We confirm that conjecture for a number of new…
Hypergeometric functions over finite fields were introduced by Greene in the 1980s as a finite field analogue of classical hypergeometric series. These functions, and their generalizations, naturally lend themselves to, and have been widely…
We provide elliptic extensions of elementary identities such as the sum of the first $n$ odd or even numbers, the geometric sum and the sum of the first $n$ cubes. Many such identities, and their $q$-analogues, are indefinite sums, and can…
Utilizing spectral residues of parameterized, recursively defined sequences, we develop a general method for generating identities of composition sums. Specific results are obtained by focusing on coefficient sequences of solutions of first…
We investigate a remarkable class of exponential sums which are derived from the symmetric groups and which display a diverse array of visually appealing features. Our interest in these expressions stems not only from their astounding…
We derive a general recurrence relation for squares of Fibonacci-like numbers. Various properties are developed, including double binomial summation identites.
We perform certain alternating binomial summations with parameters that occur in the analysis of algorithms. A combination of integral and special function and special number representations is used. The results are sufficiently general to…
There are many instances known when the Fourier coefficients of modular forms are congruent to partial sums of hypergeometric series. In our previous work arXiv:1803.01830, such partial sums are related to the radial asymptotics of infinite…
Using a probabilistic approach, we derive some interesting combinatorial identities involving gamma and beta functions. These results generalize certain well-known combinatorial identities involving binomial coefficients and special…
Some class of sums which naturally include the sums of powers of integers is considered. A number of conjectures concerning a representation of these sums is made.
We show that double cosets of the infinite symmetric group with respect to some special subgroups admit natural structures of semigroups. We interpret elements of such semigroups in combinatorial terms (chips, colored graphs,…
We derive two general transformations for certain basic hypergeometric series from the recurrence formulae for the partial numerators and denominators of two $q$-continued fractions previously investigated by the authors. By then…
By using various expansions of the parametric digamma function and the method of residue computations, we study three variants of the linear Euler sums, related Hoffman's double $t$-values and Kaneko-Tsumura's double $T$-values, and…
In this paper, we prove supercongruence relations for truncated $N$-tuple sums of basic hypergeometric series. As an application, we give double, triple, and quadruple sum analogs of some Ramanujan-type supercongruences.
Binomial coefficients and harmonic numbers are important in many branches of number theory. With the help of the operator method and several summation and transformation formulas for hypergeometric series, we prove eight conjectural series…
Chu and Zhang, in 2014, introduced hypergeometric transforms derived through the application of an Abel-type summation lemma to Dougall's ${}_{5}H_{5}$-series. These transforms were applied by Chu and Zhang to obtain accelerated rates of…
In this paper we give an elementary proof of the local sum conjecture in two dimensions. In a remarkable paper [CMN, arXiv:1810.11340], this conjecture has been established in all dimensions using sophisticated, powerful techniques from a…
We study the hyperbolicity of singular quotients of bounded symmetric domains. We give effective criteria for such quotients to satisfy Green-Griffiths-Lang's conjectures in both analytic and algebraic settings. As an application, we show…
We give summation formulae for the bilateral basic hypergeometric series ${}_1\psi_1( a; b; q, z )$ through Ramanujan's summation formula, which are generalizations of nontrivial identities found in the physics of three-dimensional Abelian…