Related papers: Discriminant Formulas and Applications
We consider here a particular quadratic equation linking two elements of a C-Algebra. By analysing powers of the unknowns, it appears a double sequence of polynomials related to classical Bernoulli polynomials. We get the generating…
In this paper we consider carlitz q-Bernoulli numbers and q-stirling numbers of the first and the second kind. From these numbers we derive many interesting formulae associated with q-Bernoulli numbers.
A class of determinants is introduced. Different kind of mathematical objects, such as Fibonacci, Lucas, Tchebychev, Hermite, Laguerre, Legendre polynomials, sums and covergents are represented as determinants from this class. A closed…
The problem of exponentiating derivations of quasi *-algebras is considered in view of applying it to the determination of the time evolution of a physical system. The particular case where observables constitute a proper CQ*-algebra is…
We conjecture an exact formula for the Kontsevich integral of the unknot, and also conjecture a formula (also conjectured independently by Deligne) for the relation between the two natural products on the space of Chinese characters. The…
A construction of differential constraints compatible with partial differential equations is considered. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the classical…
In this paper, we apply high level versions of Jacobi's derivative formula to number theory such as quarternary quadratic forms and convolution sums of some arithmetical functions.
The notion of different kind of algebraic Casorati curvatures are introduced. Some results expressing basic Casorati inequalities for algebraic Casorati curvatures are presented. Equality cases are also discussed. As a simple application,…
We extend the authors' previous work on Wiener-Wintner double recurrence theorem to the case of polynomials.
Based on Jensen formulae and the second kind of Chebyshev polynomials, another proof is presented for an extension of a curious binomial identity due to Z. W. Sun and K. J. Wu.
We prove a binomial formula for Macdonald polynomials and consider applications of it.
Some applications of a result, which is proved recently, is considered. We first prove three determinantal identities concerning the binomial coefficient and Stirling numbers of the first and the second kind. We also easily obtain the…
The discriminant of a multivariate polynomial with indeterminate coefficients is not necessarily a hypersurface, and characterizing its codimension was an open problem for quite a while. We resolve this problem for the discriminants of…
This work presents closed formulas for determinant, permanent, inverse, and Drazin inverse of circulant matrices with two non-zero coefficients.
We prove a strengthened form of a conjecture of Sun on a determinant attached to a binary quadratic form. Let $n>3$ and let $c,d\in\Z$. If $n$ is composite, then \[ \det\big[(i^2+cij+dj^2)^{n-2}\big]_{0\leq i,j\leq n-1}\equiv 0\pmod {n^2}…
We present identities for permutations with fixed points. The formulas are based on successive derivations or integrations of the determinant of a particular matrix.
Sulanke and Xin developed a continued fraction method that applies to evaluate Hankel determinants corresponding to quadratic generating functions. We use their method to give short proofs of Cigler's Hankel determinant conjectures, which…
In [Tha15], we looked at two (`multiplicative' and `Carlitz-Drinfeld additive') analogs each, for the well-known basic congruences of Fermat and Wilson, in the case of polynomials over finite fields. When we look at them modulo higher…
We give simple proofs for the Hankel determinants of q-exponential polynomials.
A novel combinatorial formula is developed for for tensor product multiplicities in representation theory. We introduce a difference formula linking these multiplicities to restricted occupancy coefficients via a shifted operator. This…