A Newton Interpolation Approach to Generalized Stirling Numbers

We employ the generalized factorials to define a Stirling-type pair {s(n,k;α,β,r),S(n,k;α,β,r)} which unifies various Stirling-type numbers investigated by previous authors. We make use of the Newton interpolation and divided differences to obtain some basic properties of the generalized Stirling numbers including the recurrence relation, explicit expression, and generating function. The generalizations of the well-known Dobinski's formula are further investigated

Triangular Recurrences, Generalized Eulerian Numbers, and Related Number Triangles

Many combinatorial and other number triangles are solutions of recurrences of the Graham-Knuth-Patashnik (GKP) type. Such triangles and their defining recurrences are investigated analytically. They are acted on by a transformation group generated by two involutions: a left-right reflection and an upper binomial transformation, acting row-wise. The group also acts on the bivariate exponential generating function (EGF) of the triangle. By the method of characteristics, the EGF of any GKP triangle has an implicit representation in terms of the Gauss hypergeometric function. There are several parametric cases when this EGF can be obtained in closed form. One is when the triangle elements are the generalized Stirling numbers of Hsu and Shiue. Another is when they are generalized Eulerian numbers of a newly defined kind. These numbers are related to the Hsu-Shiue ones by an upper binomial transformation, and can be viewed as coefficients of connection between polynomial bases, in a manner that generalizes the classical Worpitzky identity. Many identities involving these generalized Eulerian numbers and related generalized Narayana numbers are derived, including closed-form evaluations in combinatorially significant cases.Comment: 62 pages, final version, accepted by Advances in Applied Mathematic

A Dual Quark Model with Spin

A dual quark model is developed from the usual Veneziano model by explicitly including the Dirac spin of the quarks. Resonances appear without the parity doubling and new ghosts present in previous models with spin. This is accomplished by eliminating the contributions of the negative parity components (MacDowell-twins) of the spin quarks 1/2 through the introduction of fixed J-plane cuts. The resonances belong to an SU6 symmetric spectrum identical, on the leading trajectory, with that of the usual static symmetrical quark model. All resonances couple via SU6w x O2Lz symmetric vertices and the model factorizes with essentially the same degeneracy as the usual Veneziano model. As a consequence of requiring these two features the model acquires further new structure which is studied in detail in terms of the asymptotic behavior of the model. This new structure leads to unavoidable "background" contributions to the imaginary parts of the amplitudes not present in previous dual models. This situation is examined and interpreted in the language of Finite Energy Sum Rules. In order to test the basic features of the model explicit calculations are made for the case of pion-nucleon scattering in the Regge limit. To make the numerical work easier a somewhat simplified version of the model is used. Although the results of the calculations are suggestive of reasonable J-plane structure for the various amplitudes, i.e., the location of Regge pole-fixed cut interference is reasonable from the standpoint of the data, the overall kinematic behavior of the amplitudes is definitely not compatible with what is measured. However, it is noted that this kinematic behavior depends strongly on those details of the model which were simplified in the present study. If such models are to be unambiguously and successfully tested against data, future studies must treat these details more completely and realistically, including both unitarity and symmetry breaking effects.</p

Two-parameter noncommutative Gaussian processes

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 225-237).The reality of billion-user networks and multi-terabyte data sets brings forth the need for accurate and computationally tractable descriptions of large random structures, such as random matrices or random graphs. The modern mathematical theory of free probability is increasingly giving rise to analysis tools specifically adapted to such large-dimensional regimes and, more generally, non-commutative probability is emerging as an area of interdisciplinary interest. This thesis develops a new non-commutative probabilistic framework that is both a natural generalization of several existing frameworks (viz. free probability, q-deformed probability) and a setting in which to describe a broader class of random matrix limits. From the practical perspective, this new setting is particularly interesting in its ability to characterize the behavior of large random objects that asymptotically retain a certain degree of commutative structure and therefore fall outside the scope of free probability. The type of commutative structure considered is modeled on the two-parameter families of generalized harmonic oscillators found in physics and the presently introduced framework may be viewed as a two-parameter deformation of classical probability. Specifically, we introduce (1) a generalized Non-commutative Central Limit Theorem giving rise to a two-parameter deformation of the classical Gaussian statistics and (2) a two-parameter continuum of non-commutative probability spaces in which to realize these statistics. The framework that emerges has a remarkably rich combinatorial structure and bears upon a number of well-known mathematical objects, such as a quantum deformation of the Airy function, that had not previously played a prominent role in a probabilistic setting. Finally, the present framework paves the way to new types of asymptotic results, by providing more general asymptotic theorems and revealing new layers of structure in previously known results, notably in the "correlated process version" of Wigner's Semicircle Law.by Natasha Blitvić.Ph.D