We decribe various aspects of the Al-Salam-Chihara q-Laguerre polynomials. These include combinatorial descriptions of the polynomials, the moments, the orthogonality relation and a combinatorial interpretation of the linearization coefficients. It is remarkable that the corresponding moment sequence appears also in the recent work of Postnikov and Williams on enumeration of totally positive Gr...

Using the well-known fact that the Fourier transform is unitary, we obtain a class of orthogonal polynomials on the unit circle from the Fourier transform of the Laguerre polynomials (with suitable weights attached). Some related extremal problems which arise naturally in this setting are investigated.

We present a direct proof of a known result that the Hardy operator Hf(x) = 1 x R x 0 f(t) dt in the space L = L2(0,∞) can be written as H = I − U , where U is a shift operator (Uen = en+1, n ∈ Z) for some orthonormal basis {en}. The basis {en} is constructed by using classical Laguerre polynomials. We also explain connections with the Euler differential equation of the first order y′− 1 x y = ...

We consider the compressive wave for the modified Korteweg–de Vries equation with background constants c > 0 for x→ −∞ and 0 for x→ +∞. We study the asymptotics of solutions in the transition zone 4ct− εt < x < 4ct − βt ln t for ε > 0, σ ∈ (0, 1), β > 0. In this region we have a bulk of nonvanishing oscillations, the number of which grows as εt ln t . Also we show how to obtain Khruslov–Kotlyar...

We suggest the modification of the standard approach to TMDs. The modification consists in the consideration of the small bT operator product expansion in the different operator basis. Instead of power expansion we suggest to use the Laguerre polynomial expansion. Within such a scheme the first term of OPE saturates TMDs in the wider range of bT in comparison to the power expansion that decreas...

We consider several generalizations of rook polynomials. In particular we develop analogs of the theory of rook polynomials that are related to general Laguerre and Charlier polynomials in the same way that ordinary rook polynomials are related to simple Laguerre polynomials.

We consider a sequence of polynomials that are orthogonal with respect to a complex analytic weight function which depends on the index n of the polynomial. For such polynomials we obtain an asymptotic expansion in 1/n. As an example, we present the asymptotic expansion for Laguerre polynomials with a weight that depends on the index of the polynomial.

We describe the behavior as n → ∞ of the Laplace transforms of P, where P a fixed complex polynomial. As a consequence we obtain a new elementary proof of an result of Gillis-Ismail-Offer [2] in the combinatorial theory of derangements. 1 Statement of the main results The generalized derangement problem in combinatorics can be formulated as follows. Suppose X is a finite set and ∼ is an equival...

The cumulants and factorial moments of photon distribution for squeezed and correlated light are calculated in terms of Chebyshev, Legendre and Laguerre polynomials. The phenomenon of strong oscillations of the ratio of the cumulant to factorial moment is found. Running title: Oscillations of cumulants in squeezed states.