We solve the problem of finding the inverse connection formulae for the generalised Bessel polynomials and their reciprocals, the reverse generalised Bessel polynomials. The connection formulae express monomials in terms of the generalised Bessel polynomials. They enable formulae for the elements of change of basis matrices for both kinds of generalised Bessel polynomials to be derived and proved correct directly.

For the kernel $B_{\kappa ,a}(x,y)$ of the $(\kappa ,a)$-generalized Fourier transform $\mathcal {F}_{\kappa ,a}$, acting in $L^{2}(\mathbb {R}^{d})$ with the weight $|x|^{a-2}v_{\kappa }(x)$, where $v_{\kappa }$ is the Dunkl weight, we study the important question of when $\|B_{\kappa ,a}\|_{\infty }=B_{\kappa ,a}(0,0)=1$. The positive answer was known for $d\ge 2$ and $\frac {2}{a}\in \mathbb {N}$. We investigate the case $d=1$ and $\frac {2}{a}\in \mathbb {N}$. Moreover, we give sufficient conditions on parameters for $\|B_{\kappa ,a}\|_{\infty }>1$ to hold with $d\ge 1$ and any a.

We also study the image of the Schwartz space under the $\mathcal {F}_{\kappa ,a}$ transform. In particular, we obtain that $\mathcal {F}_{\kappa ,a}(\mathcal {S}(\mathbb {R}^d))=\mathcal {S}(\mathbb {R}^d)$ only if $a=2$. Finally, extending the Dunkl transform, we introduce nondeformed transforms generated by $\mathcal {F}_{\kappa ,a}$ and study their main properties.

We consider the spectral analysis of several examples of bilateral birth–death processes and compute explicitly the spectral matrix and the corresponding orthogonal polynomials. We also use the spectral representation to study some probabilistic properties of the processes, such as recurrence, the invariant distribution (if it exists), and the probability current.

In this paper, we present explicit and computable error bounds for the asymptotic expansions of the Hermite polynomials with Plancherel–Rotach scale. Three cases, depending on whether the scaled variable lies in the outer or oscillatory interval, or it is the turning point, are considered separately. We introduce the ‘branch cut’ technique to express the error terms as integrals on the contour taken as the one-sided limit of curves approaching the branch cut. This new technique enables us to derive simple error bounds in terms of elementary functions. We also provide recursive procedures for the computation of the coefficients appearing in the asymptotic expansions.

A joint algebraic interpretation of the biorthogonal Askey polynomials on the unit circle and of the orthogonal Jacobi polynomials is offered. It ties their bispectral properties to an algebra called the meta-Jacobi algebra $m\mathfrak {J}$ .

Motivated by a famous question of Lehmer about the Mahler measure, we study and solve its analytic analogue.

For a finite-dimensional Hopf algebra $\mathsf {A}$ , the McKay matrix $\mathsf {M}_{\mathsf {V}}$ of an $\mathsf {A}$ -module $\mathsf {V}$ encodes the relations for tensoring the simple $\mathsf {A}$ -modules with $\mathsf {V}$ . We prove results about the eigenvalues and the right and left (generalized) eigenvectors of $\mathsf {M}_{\mathsf {V}}$ by relating them to characters. We show how the projective McKay matrix $\mathsf {Q}_{\mathsf {V}}$ obtained by tensoring the projective indecomposable modules of $\mathsf {A}$ with $\mathsf {V}$ is related to the McKay matrix of the dual module of $\mathsf {V}$ . We illustrate these results for the Drinfeld double $\mathsf {D}_n$ of the Taft algebra by deriving expressions for the eigenvalues and eigenvectors of $\mathsf {M}_{\mathsf {V}}$ and $\mathsf {Q}_{\mathsf {V}}$ in terms of several kinds of Chebyshev polynomials. For the matrix $\mathsf {N}_{\mathsf {V}}$ that encodes the fusion rules for tensoring $\mathsf {V}$ with a basis of projective indecomposable $\mathsf {D}_n$ -modules for the image of the Cartan map, we show that the eigenvalues and eigenvectors also have such Chebyshev expressions.

We introduce a generalization ${\rm{\pounds}}_d^{(\alpha)}(X)$ of the finite polylogarithms ${\rm{\pounds}}_d^{(0)}(X) = {{\rm{\pounds}}_d}(X) = \sum\nolimits_{k = 1}^{p - 1} {X^k}/{k^d}$ , in characteristic p, which depends on a parameter α. The special case ${\rm{\pounds}}_1^{(\alpha)}(X)$ was previously investigated by the authors as the inverse, in an appropriate sense, of a parametrized generalization of the truncated exponential which is instrumental in a grading switching technique for nonassociative algebras. Here, we extend such generalization to ${\rm{\pounds}}_d^{(\alpha)}(X)$ in a natural manner and study some properties satisfied by those polynomials. In particular, we find how the polynomials ${\rm{\pounds}}_d^{(\alpha)}(X)$ are related to the powers of ${\rm{\pounds}}_1^{(\alpha)}(X)$ and derive some consequences.

The Jacobi coefficients $c_{j}^{\ell }\left( \alpha ,\,\beta \right)\,\left( 1\,\le \,j\,\le \,\ell ,\,\alpha ,\,\beta \,>\,-1 \right)$ are linked to the Maclaurin spectral expansion of the Schwartz kernel of functions of the Laplacian on a compact rank one symmetric space. It is proved that these coefficients can be computed by transforming the even derivatives of the Jacobi polynomials $P_{k}^{\left( \alpha ,\,\beta \right)}\,\left( k\,\ge \,0,\,\alpha ,\,\beta \,>\,-1 \right)$ into a spectral sum associated with the Jacobi operator. The first few coefficients are explicitly computed, and a direct trace interpretation of the Maclaurin coefficients is presented.

We propose amixed spectral method for heat transfer in unbounded domains, using generalised Hermite functions and Legendre polynomials. Some basic results on the mixed generalised Hermite-Legendre orthogonal approximation are established, which plays important roles in spectral methods for various problems defined on unbounded domains. As an example, the mixed generalised Hermite-Legendre spectral scheme is constructed for anisotropic heat transfer. Its convergence is proven, and some numerical results demonstrate the spectral accuracy of this approach.

Assume that $A$ is a finite-dimensional algebra over some field, and assume that $A$ is weakly symmetric and indecomposable, with radical cube zero and radical square nonzero. We show that such an algebra of wild representation type does not have a nonprojective module $M$ whose ext-algebra is finite dimensional. This gives a complete classification of weakly symmetric indecomposable algebras which have a nonprojective module whose ext-algebra is finite dimensional. This shows in particular that existence of ext-finite nonprojective modules is not equivalent with the failure of the finite generation condition (Fg), which ensures that modules have support varieties.

We show that if $v$ is a regular semi-classical form (linear functional), then the symmetric form $u$ defined by the relation ${{x}^{2}}\sigma u\,=\,-\lambda v$, where $\left( \sigma f \right)\left( x \right)\,=\,f\left( {{x}^{2}} \right)$ and the odd moments of $u$ are 0, is also regular and semi-classical form for every complex $\lambda $ except for a discrete set of numbers depending on $v$. We give explicitly the three-term recurrence relation and the structure relation coefficients of the orthogonal polynomials sequence associated with $u$ and the class of the form $u$ knowing that of $v$. We conclude with an illustrative example.

A 𝔻-semiclassical weight is one which satisfies a particular linear, first-order homogeneous equation in a divided-difference operator 𝔻. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first-order homogeneous matrix equation in the divided-difference operator termed the spectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case of quadratic lattices. The simplest examples of the 𝔻-semiclassical orthogonal polynomial systems are precisely those in the Askey table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the 𝔻-semiclassical class it is entirely natural to define a generalization of the Askey table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from such deformations and their relations with the other elements of the theory. As an example we treat the first nontrivial deformation of the Askey–Wilson orthogonal polynomial system defined by the q-quadratic divided-difference operator, the Askey–Wilson operator, and derive the coupled first-order divided-difference equations characterizing its evolution in the deformation variable. We show that this system is a member of a sequence of classical solutions to the q-Painlevé system.

We give an explicit formula for the resultant of Chebyshev polynomials of the first, second, third, and fourth kinds. We also compute the resultant of modified cyclotomic polynomials.

We relate a one-parametric generating function for the squares of Legendre polynomials to an arithmetic hypergeometric series whose parametrisation by a level 7 modular function was recently given by Cooper. By using this modular parametrisation we resolve a subfamily of identities involving $1/ \pi $ which was experimentally observed by Sun.

In this paper, we give analogues of the local uncertainty inequality for the Dunkl transform on ℝd, and indicate how the local uncertainty inequality implies a global uncertainty inequality.

In this paper, we propose a mixed Fourier-Jacobi spectral method for two dimensional Neumann boundary value problem. This method differs from the classical spectral method. The homogeneous Neumann boundary condition is satisfied exactly. Moreover, a tridiagonal matrix is employed, instead of the full stiffness matrix encountered in the classical variational formulation. For analyzing the numerical error, we establish the mixed Fourier-Jacobi orthogonal approximation. The convergence of proposed scheme is proved. Numerical results demonstrate the efficiency of this approach.

The aim of this paper is to develop the mixed spectral and pseudospectral methods for nonlinear problems outside a disc, using Fourier and generalized Laguerre functions. As an example, we consider a nonlinear strongly damped wave equation. The mixed spectral and pseudospectral schemes are proposed. The convergence is proved. Numerical results demonstrate the efficiency of this approach.