We develop a theory of Darboux transformations for CMV matrices, canonical representations of the unitary operators. In perfect analogy with their self-adjoint version – the Darboux transformations of Jacobi matrices – they are equivalent to Laurent polynomial modifications of the underlying measures. We address other questions which emphasize the similarities between Darboux transformations fo...

We introduce two biparametric families of Apostol-Frobenius-Euler polynomials level-$m$. give some algebraic properties, as well other identities which connect these polynomial class with the generalized $\lambda$-Stirling type numbers second kind, Apostol--Bernoulli polynomials, Apostol--Genocchi Apostol--Euler and Jacobi polynomials. Finally, we will show differential properties this new family

In this paper, we consider a natural extension of several results related to Krall-type polynomials introducing a modification of a q-classical linear functional via the addition of one or two mass points. The limit relations between the q-Krall type modification of big q-Jacobi, little q-Jacobi, big q-Laguerre, and other families of the q-Hahn tableau are established.

We use generating functions to express orthogonality relations in the form of q-beta integrals. The integrand of such a q-beta integral is then used as a weight function for a new set of orthogonal or biorthogonal functions. This method is applied to the continuous q-Hermite polynomials, the Al-Salam-Carlitz polynomials, and the polynomials of Szegő and leads naturally to the Al-Salam-Chihara p...

Jacobi polynomials were introduced by Ozeki in analogy with Jacobi forms of lattices They are useful for coset weight enumeration and weight enumeration of children We determine them in most interesting cases in length at most and in some cases in length We use them to construct group divisible designs packing designs covering designs and t r designs in the sense of Calderbank Delsarte A major ...

Chain sequences are positive sequences fang of the form an 1⁄4 gnð1 gn 1Þ for a nonnegative sequence fgng: This concept has been introduced by Wall in connection with continued fractions. These sequences are very useful in determining the support of orthogonality measure for orthogonal polynomials. Equivalently, they can be used for localizing spectra of Jacobi matrices associated with orthogon...