Classes of hypercomplex polynomials of discrete variable based on the quasi-monomiality principle

With the aim of derive a quasi-monomiality formulation in the context of discrete hypercomplex variables, one will amalgamate through a Clifford-algebraic structure of signature $(0,n)$ the umbral calculus framework with Lie-algebraic symmetries. The exponential generating function ({\bf EGF}) carrying the {\it continuum} Dirac operator D=\sum_{j=1}^n\e_j\partial_{x_j} together with the Lie-algebraic representation of raising and lowering operators acting on the lattice h\BZ^n is used to derive the corresponding hypercomplex polynomials of discrete variable as Appell sets with membership on the space Clifford-vector-valued polynomials. Some particular examples concerning this construction such as the hypercomplex versions of falling factorials and the Poisson-Charlier polynomials are introduced. Certain applications from the view of interpolation theory and integral transforms are also discussed.Comment: 24 pages. 1 figure. v2: a major revision, including numerous improvements throughout the paper was don

Monomiality principle, Sheffer-type polynomials and the normal ordering problem

We solve the boson normal ordering problem for $(q(a^\dag)a+v(a^\dag))^n$ with arbitrary functions $q(x)$ and $v(x)$ and integer $n$, where $a$ and $a^\dag$ are boson annihilation and creation operators, satisfying $[a,a^\dag]=1$. This consequently provides the solution for the exponential $e^{\lambda(q(a^\dag)a+v(a^\dag))}$ generalizing the shift operator. In the course of these considerations we define and explore the monomiality principle and find its representations. We exploit the properties of Sheffer-type polynomials which constitute the inherent structure of this problem. In the end we give some examples illustrating the utility of the method and point out the relation to combinatorial structures.Comment: Presented at the 8'th International School of Theoretical Physics "Symmetry and Structural Properties of Condensed Matter " (SSPCM 2005), Myczkowce, Poland. 13 pages, 31 reference

On the combinatorics of partition functions in AdS3/LCFT2

Three-dimensional Topologically Massive Gravity at its critical point has been conjectured to be holographically dual to a Logarithmic CFT. However, many details of this correspondence are still lacking. In this work, we study the 1-loop partition function of Critical Cosmological Topologically Massive Gravity, previously derived by Gaberdiel, Grumiller and Vassilevich, and show that it can be usefully rewritten as a Bell polynomial expansion. We also show that there is a relationship between this Bell polynomial expansion and the Plethystic Exponential. Our reformulation allows us to match the TMG partition function to states on the CFT side, including the multi-particle states of t (the logarithmic partner of the CFT stress tensor) which had previously been elusive. We also discuss the appearance of a ladder action between the different multi-particle sectors in the partition function, which induces an interesting sl(2) structure on the n-particle components of the partition function.Comment: 26 pages. Typos fixed, references and clarifications adde

Adjoint Appell-Euler and First Kind Appell-Bernoulli Polynomials

The adjunction property, recently introduced for Sheffer polynomial sets, is considered in the case of Appell polynomials. The particular case of adjoint Appell-Euler and Appell-Bernoulli polynomials of the first kind is analyzed

Monomiality Principle and Eigenfunctions of Differential Operators

We apply the so-calledmonomiality principlein order to construct eigenfunctions for a wide set of ordinary differential operators, relevant to special functions and polynomials, including Bessel functions and generalized Gould-Hopper polynomials

Quasi-monomiality and operational identities for Laguerre–Konhauser-type matrix polynomials and their applications

It is shown that an appropriate combination of methods, relevant to matrix polynomials and to operational calculus can be a very useful tool to establish and treat a new class of matrix Laguerre–Konhauser polynomials. We explore the formal properties of the operational identities to derive a number of properties of the new class of Laguerre–Konhauser matrix polynomials and discuss the links with classical polynomials