Mellin's formula and some combinatorial identities

Mellin's formula is applied to expandt/e ande/t in powers of log logt, and to obtain certain identities for the Stirling and Bell numbers.     

Some remarks on the Hook formula

Summary Without the Frame-Robinson-Thral hook formula we establish some divisibility properties of degrees of irreducible representations of the symmetric group.

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Riassunto Senza utilizzare una nota formula di Frame-Robinson-Thrall stabiliamo alcune proprietà di divisibilità di gradi di rappresentazioni irriducibili del grupposimmetrico.

     

Catalan matrix and related combinatorial identities

We introduce the notion of the Catalan matrix whose non-zero elements are expressions which contain the Catalan numbers arranged into a lower triangular Toeplitz matrix. Inverse of the Catalan matrix is derived. Correlations between the matrix and the generalized Pascal matrix are considered. Some combinatorial identities involving Catalan numbers, binomial coefficients and the generalized hypergeometric function are derived using these correlations. Moreover, an additional explicit representation of the Catalan number, as well as an explicit representation of the sum of the first m Catalan numbers are given.     

Polynomial identities of related rings

The set of polynomial identities of a ringA is considered, as well as some types of minimal identities. The change which occurs in these identities upon passage to related rings is then studied. This paper was written while both authors were doing their Ph. D. these at the Hebrew University of Jerusalem under the supervision of S. A. Amitsur, to whom they wish to express their warm thanks.     

m-Dissections of some infinite products and related identities

The Ramanujan Journal - For integers t and m with $$mge 5$$ relatively prime to 6 such that $$1 le t n_0$$ , the signs of the $$c_n$$ are periodic with period m. In addition, a formula is given...     

Some formulas related to the Rogers-Ramanujan identities

Summary The paper contains a number of identities related to theRogers-Ramanujan identities. In particular, the formulas (13) and (14) are generalizations of those identities.     

More hypergeometric identities related to Ramanujan-type series

We find new hypergeometric identities which, in a certain aspect, are stronger than others of the same style found by the author in a previous paper. The identities in Sect. 3 are related to some Ramanujan-type series for 1/π. We derive them by using WZ-pairs associated to some interesting formulas by Wenchang Chu. The identities we prove in Sect. 4 are of the same style but related to Ramanujan-like series for 1/π ².     

Ballot matrix as Catalan matrix power and related identities

We use an analytical approach to find the kth power of the Catalan matrix. Precisely, it is proven that the power of the Catalan matrix is a lower triangular Toeplitz matrix which contains the well-known ballot numbers. A result from [H. S. Wilf, Generatingfunctionology, Academic Press, New York, 1990, Free download available from http://www.math.upenn.edu/~wilf/Downld.html.], related to the generating function for Catalan numbers, is extended to the negative integers. Three interesting representations for Catalan numbers by means of the binomial coefficients and the hypergeometric functions are obtained using relations between Catalan matrix powers.     

An intuitionistic formula hierarchy based on high‐school identities

We revisit the notion of intuitionistic equivalence and formal proof representations by adopting the view of formulas as exponential polynomials. After observing that most of the invertible proof rules of intuitionistic (minimal) propositional sequent calculi are formula (i.e., sequent) isomorphisms corresponding to the high‐school identities, we show that one can obtain a more compact variant of a proof system, consisting of non‐invertible proof rules only, and where the invertible proof rules have been replaced by a formula normalization procedure. Moreover, for certain proof systems such as the G4ip sequent calculus of Vorob'ev, Hudelmaier, and Dyckhoff, it is even possible to see all of the non‐invertible proof rules as strict inequalities between exponential polynomials; a careful combinatorial treatment is given in order to establish this fact. Finally, we extend the exponential polynomial analogy to the first‐order quantifiers, showing that it gives rise to an intuitionistic hierarchy of formulas, resembling the classical arithmetical hierarchy, and the first one that classifies formulas while preserving isomorphism.     

Four identities related to third-order mock theta functions

The Ramanujan Journal - Ramanujan presented four identities for third-order mock theta functions in his Lost Notebook. In 2005, with the aid of complex analysis, Yesilyurt first proved these four...     

Further determinant identities related to classical root systems

By introducing polynomials in matrix entries, six determinants are evaluated which may be considered extensions of Vandermonde-like determinants related to the classical root systems.     

Two identities related to dirichlet character of polynomials

Let q be a positive integer, χ denote any Dirichlet character mod q. For any integer m with (m, q) = 1, we define a sum C(χ, k,m; q) analogous to high-dimensional Kloosterman sums as follows: , where a · ā ≡ 1 mod q. The main purpose of this paper is to use elementary methods and properties of Gauss sums to study the computational problem of the absolute value |C(χ, k,m; q)|, and give two interesting identities for it.     

Bivariate identities related to Chebyshev and Dickson polynomials over Fq

Let ${\mathbb{F}}_q$ be a field of q elements, where q is a power of an odd prime p. The polynomial $f(y)\in {\mathbb{F}}_q[y]$ defined by$f(y):={(1+\sqrt{y})}^{(q+1)/2}+{(1-\sqrt{y})}^{(q+1)/2}$ has the property that$f(1-y)=\rho (2)f(y),$ where ρ is the quadratic character on ${\mathbb{F}}_q$. This univariate identity was applied to prove a recent theorem of N. Katz. We formulate and prove a bivariate extension, and give an application to quadratic residuacity.     

A family of identities related to zero-sum and team games

We list and prove a family of binomial identities by calculating in two ways the probabilities of approximate saddlepoints occurring in random m×n matrices. The identities are easily seen to be equivalent to the evaluation of a family of Gauss ₂F₁ polynomials according to a formula of Vandermonde. We also consider some implications concerning the number of approximate pure strategy Nash equilibria we can expect in large matrix zero-sum and team games.