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By using the Rodriguez-Villegas–Mortenson supercongruences, we prove four supercongruences on sums involving binomial coefficients, which were originally conjectured by Sun. We also confirm a related conjecture of Guo on integer-valued polynomials. [ABSTRACT FROM AUTHOR]

Let F be a finite extension of ℚp. We show that every Schwartz function on F, with values in ℚp, is the p-adic uniform limit of a sequence of Schwartz functions, whose Fourier transforms tend uniformly to 0. The proof uses the notion of q-binomial coefficients and some classical identities between these polynomials. [ABSTRACT FROM AUTHOR]

We prove a weighted version of the sum of binomial coefficients in arithmetic progression. As an application, we deduce a random walk theorem which generalizes a result of Aldous and Diaconis. [ABSTRACT FROM AUTHOR]

We use the notion of subprime factorization to establish recurrence relations for the number of binomial coefficients in a given row of Pascal's triangle that are divisible by and not divisible by , where is a prime. Using these relations to compute this number can provide significant savings in the number of computational steps. [ABSTRACT FROM AUTHOR]

We establish two binomial coefficient-generalized harmonic sum identities using the partial fraction decomposition method. These identities are a key ingredient in the proofs of numerous supercongruences. In particular, in other works of the author, they are used to establish modulo pk ( k > 1) congruences between truncated generalized hypergeometric series, and a function which extends Greene's hypergeometric function over finite fields to the p-adic setting. A specialization of one of these congruences is used to prove an outstanding conjecture of Rodriguez-Villegas which relates a truncated generalized hypergeometric series to the p-th Fourier coefficient of a particular modular form. [ABSTRACT FROM AUTHOR]

Let s and t be variables. Define polynomials { n} in s, t by {0} = 0, {1} = 1, and { n} = s { n - 1} + t { n - 2} for n ≥ 2. If s, t are integers then the corresponding sequence of integers is called a Lucas sequence. Define an analogue of the binomial coefficients by where { n}! = {1} {2} ⋯ { n}. It is easy to see that is a polynomial in s and t. The purpose of this note is to give two combinatorial interpretations for this polynomial in terms of statistics on integer partitions inside a k × ( n - k) rectangle. When s = t = 1 we obtain combinatorial interpretations of the fibonomial coefficients which are simpler than any that have previously appeared in the literature. [ABSTRACT FROM AUTHOR]

In this short paper we establish identities involving sums of products of binomial coefficients and coefficients that satisfy the general second-order linear recurrence. We obtain generalizations of identities of Carlitz, Prodinger and Haukkanen. [ABSTRACT FROM AUTHOR]

Abstract: We consider the problem of deciding whether a given rational function has a power series expansion with all its coefficients positive. Introducing an elementary transformation that preserves such positivity we are able to provide an elementary proof for the positivity of Szegö''s function which has been at the historical root of this subject starting with Szegö. We then demonstrate how to apply the transformation to prove a 4-dimensional generalization of the above function, and close with discussing the set of parameters such that has positive coefficients. [Copyright &y& Elsevier]

We describe exercises in which students use PascGaloisJE to formulate conjectures about certain binomial identities which hold when the binomial coefficients are interpreted as elements in the cyclic group [image omitted] of integers modulo a prime integer p. In addition to having an appealing visual component, these exercises are open-ended and expose students to the investigative nature of mathematical discovery. [ABSTRACT FROM AUTHOR]

Abstract: It is shown that all the assumptions for symmetric monoidal categories follow from a unifying principle involving natural isomorphisms of the type , called medial commutativity. Medial commutativity in the presence of the unit object enables us to define associativity and commutativity natural isomorphisms. In particular, Mac Lane’s pentagonal and hexagonal coherence conditions for associativity and commutativity are derived from the preservation up to a natural isomorphism of medial commutativity by the biendofunctor . This preservation boils down to an isomorphic representation of the Yang–Baxter equation of symmetric and braid groups. The assumptions of monoidal categories, and in particular Mac Lane’s pentagonal coherence condition, are explained in the absence of commutativity, and also of the unit object, by a similar preservation of associativity by the biendofunctor . In the final section one finds coherence conditions for medial commutativity in the absence of the unit object. These conditions are obtained by taking the direct product of the symmetric groups for . [Copyright &y& Elsevier]

Abstract: The -operator for -modules plays an important role in the study of Iwasawa theory via Fontaine''s big rings. In this note, we prove several sharp estimates for the -operator in the cyclotomic case. These estimates immediately imply a number of sharp p-adic combinatorial congruences, one of which extends the classical congruences of Fleck (1913) and Weisman [Some congruences for binomial coefficients, Michigan Math. J. 24 (1977) 141–151]. [Copyright &y& Elsevier]

The article offers an annotation to the mathematical congruence (mdnd) ≡ (mn) (modq). Mathematical problems that involve binomial coefficients modulo a prime power have been the focus of many great mathematicians of the 19th century. These mathematicians have derived different theorems that could be easily proved. One of these theorems is the congruence (mdnd) ≡ (mn) (modq). The author presented a partial converse theorem based on this established equation and observed that the same products of binomial coefficients are established on the right of Lucas's theorem.