The Sokolov case,integrable Kirchhoff elasticae,and genus 2 theta functions via discriminantly separable polynomials

We use the discriminantly separable polynomials of degree 2 in each of three variables to integrate explicitly the Sokolov case of a rigid body in an ideal fluid and integrable Kirchhoff elasticae in terms of genus 2 theta functions. The integration procedure is a natural generalization of the one used by Kowalevski in her celebrated 1889 paper. The algebraic background for the most important changes of variables in this integration procedure is associated to the structure of the two-valued groups on an elliptic curve. Such two-valued groups have been introduced by V.M. Buchstaber.     

New examples of systems of the Kowalevski type

A new examples of integrable dynamical systems are constructed. An integration procedure leading to genus two theta-functions is presented. It is based on a recent notion of discriminantly separable polynomials. They have appeared in a recent reconsideration of the celebrated Kowalevski top, and their role here is analogue to the situation with the classical Kowalevski integration procedure.     

Generalized and mixed type Gegenbauer polynomials

In this paper, a new generalized form of the Gegenbauer polynomials is introduced by using the integral representation method. Further, the Hermite–Gegenbauer and the Laguerre–Gegenbauer polynomials are introduced by using the operational identities associated with the generalized Hermite and Laguerre polynomials of two variables.     

Bell polynomials and binomial type sequences

This paper concerns the study of the Bell polynomials and the binomial type sequences. We mainly establish some relations tied to these important concepts. Furthermore, these obtained results are exploited to deduce some interesting relations concerning the Bell polynomials which enable us to obtain some new identities for the Bell polynomials. Our results are illustrated by some comprehensive examples.     

Convex envelopes of separable functions over regions defined by separable functions of the same type

In this paper we derive the convex envelope of separable functions obtained as a linear combination of strictly convex coercive one-dimensional functions over compact regions defined by linear combinations of the same one-dimensional functions. As a corollary of the main result, we are able to derive the convex envelope of any quadratic function (not necessarily separable) over any ellipsoid, and the convex envelope of some quadratic functions over a convex region defined by two quadratic constraints.     

Embeddings and separable differential operators in spaces of Sobolev-Lions type

We study embedding theorems for anisotropic spaces of Bessel-Lions type H _p,γ ^l (Ω; E ₀, E), where E ₀ and E are Banach spaces. We obtain the most regular spaces E _α for which mixed differentiation operators D ^α from H _p,γ ^l (Ω; E ₀, E) to L _p,γ(Ω; E _α ) are bounded. The spaces E _α are interpolation spaces between E ₀ and E, depending on α = (α ₁, α ₂, …, α _n ) and l = (l ₁, l ₂, …, l _n ). The results obtained are applied to prove the separability of anisotropic differential operator equations with variable coefficients.     

Sequences of polynomials of fractional binomial type

Sequences of polynomials which satisfy a binomial theorem involving fractional binomial coefficients can be characterized as umbral left inverses of singular sequences of binomial type.     

Inequalities of Rafalson type for algebraic polynomials

For a positive Borel measure dμ, we prove that the constantcan be represented by the zeros of orthogonal polynomials corresponding to dμ in case (i) dν(x)=(A+Bx)dμ(x), where A+Bx is nonnegative on the support of dμ and (ii) dν(x)=(A+Bx²)dμ(x), where dμ is symmetric and A+Bx² is nonnegative on the support of dμ. The extremal polynomials attaining the constant are obtained and some concrete examples are given including Markov-type inequality when dμ is a measure for Jacobi polynomials.     

On cubic polynomials III. Systems ofp-adic equations

We study systems of equations $$\mathfrak{F}_1 = \ldots = \mathfrak{F}_r = 0,$$ where \(\mathfrak{F}_1 , \ldots ,\mathfrak{F}_r \) are cubic forms withp-adic coefficients. Such a system has a nontrivialp-adic solution if the number of variables is at least 50,000 ·r ³. Further we will give estimates for the number of solutions of certain systems of cubic congruences.     

d-Orthogonality of Humbert and Jacobi type polynomials

In this paper, we treat three questions related to the d-orthogonality of the Humbert polynomials. The first one consists to determinate the explicit expression of the d-dimensional functional vector for which the d-orthogonality holds. The second one is the investigation of the components of Humbert polynomial sequence. That allows us to introduce, as far as we know, new d-orthogonal polynomials generalizing the classical Jacobi ones. The third one consists to solve a characterization problem related to a generalized hypergeometric representation of the Humbert polynomials.     

Sobolev inner products and orthogonal polynomials of Sobolev type

This survey paper deals with polynomials which are orthogonal with respect to scalar products of the form _R F ^T[A]G withF ^T=[f(x), f(Ⅎ(x),...f ^(y)(x)], [A] A _ji =A _ji =A _ij =d _ji (I _ji ) where d _ji is a measure of supportI _ij and [A] is positive semi-definite. Basic properties are indicated or proved in particular cases.     

Integral equations and potential-theoretic type integrals of orthogonal polynomials

Abel's theorem is used to solve the Fredholm integral equations of the first kind with Gauss's hypergeometric kernel. The case that the kernel takes general form is discussed in detail and the solution is given. Here, the works of some authors are considered as special cases. The discussion focuses on the three dimensional contact problems in the theory of elasticity with general kernel.     

Some identities for Carlitz type q-Daehee polynomials

The Ramanujan Journal - Motivated by Carlitz’s q-Bernoulli polynomials, properties and identities are given for a new type of polynomial, Carlitz type q-Daehee polynomials. Higher order...