Related papers: A simple approach to some Hankel determinants
We prove characterization theorems for relative entropy (also known as Kullback-Leibler divergence), q-logarithmic entropy (also known as Tsallis entropy), and q-logarithmic relative entropy. All three have been characterized axiomatically…
Motivated by the Hankel determinant evaluation of moment sequences, we study a kind of Pfaffian analogue evaluation. We prove an LU-decomposition analogue for skew-symmetric matrices, called Pfaffian decomposition. We then apply this…
A very elementary introduction to quantum algebras is presented and a few examples of their physical applications are mentioned.
We consider a family of all analytic and univalent functions (i.e., one-to-one) in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$ of the form $f(z)=z+a_2z^2+a_3z^3+\cdots$. In this paper, we obtain the sharp bounds of the second…
We provide a short proof of the 1-dimensional flat chain conjecture.
We study the asymptotics of Hankel determinants constructed using the values $\zeta(an+b)$ of the Riemann zeta function at positive integers in an arithmetic progression. Our principal result is a Diophantine application of the asymptotics.
We provide an elementary derivation of the Bessel analog of the celebrated Riesz composition formula and use the former to effortlessly derive the latter.
First a few reformulations of Frankl's conjecture are given, in terms of reduced families or matrices, or analogously in terms of lattices. These lead naturally to a stronger conjecture with a neat formulation which might be easier to…
Using the notion of quantum integers associated with a complex number $q\neq 0$, we define the quantum Hilbert matrix and various extensions. They are Hankel matrices corresponding to certain little $q$-Jacobi polynomials when $|q|<1$, and…
A model of a q-harmonic oscillator based on q-Charlier polynomials of Al-Salam and Carlitz is discussed. Simple explicit realization of q-creation and q-annihilation operators, q-coherent states and an analog of the Fourier transformation…
Instantaneous derivation of the Thomas precession with only basic vector calculus.
We solve the connection coefficient problem between the Al-Salam-Chihara polynomials and the q-Hermite polynomials, and we use the resulting identity to answer a question from probability theory. We also derive the distribution of some…
A short and elementary proof of the joint convexity of relative entropy is presented, using nothing beyond linear algebra. The key ingredients are an easily verified integral representation and the strategy used to prove the Cauchy-Schwarz…
A $q$-analogue of combinatorics concerning the Cartan matrix for the Iwahori-Hecke algebra of type $A$ is investigated. We give several descriptions for the determinant of the graded Cartan matrix, which imply some combinatorial identities.…
In this paper we characterise the indeterminate case by the eigenvalues of the Hankel matrices being bounded below by a strictly positive constant. An explicit lower bound is given in terms of the orthonormal polynomials and we find…
We consider Kemp's q-analogue of the binomial distribution. Several convergence results involving the classical binomial, the Heine, the discrete normal, and the Poisson distribution are established. Some of them are q-analogues of…
A q-analogue of Erdelyi's formula for the Hankel transform of the product of Laguerre polynomials is derived using the quantum linking groupoid between the quantum SU(2) and E(2) groups. The kernel of the q-Hankel transform is given by the…
Elementary proofs of Sylvester's, Wolstenholme's, Morley's and Lehmer's congruence theorems
In the current note, we present a new, short proof of the famous AM-GM-HM inequality using only induction and basic calculus.
We show that Genocchi and Bernoulli numbers are closely related to Fibonacci polynomials and derive some q-analogues.