Decomposition of partial orders

A decomposition theory for partial orders which arises from the split decomposition of submodular functions is introduced. As a consequence of this theory, any partial order has a unique decomposition consisting of indecomposable partial orders and certain highly decomposable partial orders. The highly decomposable partial orders are completely characterized. As a special case of partial orders, we consider lattices and distributive lattices. It occurs, that the highly decomposable distributive lattices are precisely the Boolean lattices.     

On the convergence properties of basic series representing special monogenic functions

In this paper, it is shown that certain classes of special monogenic functions cannot be represented by the basic series in the whole space. New definitions for the order of basis of special monogenic polynomials are given together with theorems on representation of classes of special monogenic functions in certain balls and at a point. Received: 8 January 2002     

Partial theta functions and mock modular forms as q-hypergeometric series

Ramanujan studied the analytic properties of many q-hypergeometric series. Of those, mock theta functions have been particularly intriguing, and by work of Zwegers, we now know how these curious q-series fit into the theory of automorphic forms. The analytic theory of partial theta functions however, which have q-expansions resembling modular theta functions, is not well understood. Here we consider families of q-hypergeometric series which converge in two disjoint domains. In one domain, we show that these series are often equal to one another, and define mock theta functions, including the classical mock theta functions of Ramanujan, as well as certain combinatorial generating functions, as special cases. In the other domain, we prove that these series are typically not equal to one another, but instead are related by partial theta functions.     

Multivariate Meixner classes of invariant distributions

We define multivariate Meixner classes of invariant distributions of random matrices as those whose generating functions for the associated orthogonal polynomials are of certain special integral or summation forms, generalizing the univariate Meixner classes of distributions which were first characterized by Meixner [21]. Characterization theorems and properties of these multivariate Meixner classes are established. The zonal polynomials, the extended invariant polynomials with matrix arguments, and their related results in multivariate distribution theory are utilized in the discussion.     

Convergent Expansions for Solutions of Linear Ordinary Differential Equations Having a Simple Pole, with an Application to Associated Legendre Functions

Second-order linear ordinary differential equations with a large parameter u are examined. Asymptotic expansions involving modified Bessel functions are applicable for the case where the coefficient function of the large parameter has a simple pole. In this paper, we examine such equations in the complex plane, and convert the asymptotic expansions into uniformly convergent series, where u appears in an inverse factorial, rather than an inverse power. Under certain mild conditions, the region of convergence containing the simple pole is unbounded. The theory is applied to obtain exact connection formulas for general solutions of the equation, and also, in a special case, to obtain convergent expansions for associated Legendre functions of complex argument and large degree.     

Feynman path integrals and asymptotic expansions for transition probability densities of some Lévy driven financial markets

In this paper we implement the method of Feynman path integral for the analysis of option pricing for certain Lévy process driven financial markets. For such markets, we find closed form solutions of transition probability density functions of option pricing in terms of various special functions. Asymptotic analysis of transition probability density functions is provided. We also find expressions for transition probability density functions in terms of various special functions for certain Lévy process driven market where the interest rate is stochastic.     

A certain class of q-series transformations

A large number of summation and transformation formulas for a certain class of double hypergeometric series are observed here to follow fairly readily from a single known result which, in turn, is a very special case of one of six general expansion formulas given in the literature. Generalizations and unifications of these expansion formulas involving series with essentially arbitrary terms are presented. It is also shown how the various series transformations considered in this paper admit themselves of $\text{q}$-extensions which are capable of unifying numerous scattered results in the theory of basic double hypergeometric functions.     

Notes on Ruscheweyh's Multiplier Conjecture

In this note we show that Ruscheweyh's multiplier conjecture is true in several special cases. The results obtained here are closely related to the partial sums of certain analytic functions defined by means of univalent functions. We also answer a question of Ponnusamy concerning thenth partial sums of certain univalent functions.     

Special functions and characterizations of probability distributions by zero regression properties

Characterizations of the binomial, negative binomial, gamma, Poisson, and normal distributions are obtained by the property of zero regression of certain polynomial statistics of arbitrary degree, on the mean. In each case, the equations which express zero regression are derived from the recurrence relations of a set of special functions. The differential recurrence relations of these special functions are used in the proofs of the characterization theorems.     

Continuous ranking of zeros of special functions

We reexamine and continue the work of J. Vosmansky [J. Vosmanský, Zeros of solutions of linear differential equations as continuous functions of the parameter k, in: J. Wiener, J.K. Hale (Eds.), Partial Differential Equations, Proceedings of Conference, Edinburg, TX, 1991, in: Pitman Res. Notes Math. Ser., vol. 273, 1992, pp. 253-257] on the concept of continuous ranking of zeros of certain special functions from the point of view of the transformation theory of second-order linear differential equations. This leads to results on higher monotonicity of such zeros with respect to the rank and to the evaluation of some definite integrals. The applications are to Airy, Bessel and Hermite functions.     

Approximation of bandlimited functions

Many signals encountered in science and engineering are approximated well by bandlimited functions. We provide suitable error bounds for the approximation of bandlimited functions by linear combinations of certain special functions—the prolate spheroidal wave functions of order 0. The coefficients in the approximating linear combinations are given explicitly via appropriate quadrature formulae.     

The Chowla-Selberg formula

The Chowla-Selberg formula is a monomial relation connecting the values of certain automorphic form at special points to the values of Γ functions at rational points. A generalization of this formula is established in the context of CM-fields: the values of a distinguished Hilbert automorphic form at special points are expressed in terms of multiple Γ functions.     

The Generalized Segal–Bargmann Transform and Special Functions

Analysis of function spaces and special functions are closely related to the representation theory of Lie groups. We explain here the connection between the Laguerre functions, the Laguerre polynomials, and the Meixner–Pollacyck polynomials on the one side, and highest weight representations of Hermitian Lie groups on the other side. The representation theory is used to derive differential equations and recursion relations satisfied by those special functions.     

Estimate for initial Maclaurin coefficients of bi-univalent functions for a class defined by fractional derivatives

Estimates for second and third Maclaurin coefficients of certain bi-univalent functions in the open unit disk defined by convolution are determined. Certain special cases are also indicated.     

Erdélyi-type integrals for generalized hypergeometric functions with integral parameter differences

In this paper, we extend one of Erdélyi's classical integrals for the Gauss hypergeometric functions to a special class of generalized hypergeometric functions in which certain pairs of numerator and denomiator parameters differ by positive integers. Our main results are achieved by applying the fractional integration by parts and series manipulation technique. Some special cases are also pointed out which includes a new extension of a Thomae-type transformation.     

New class of generating functions associated with generalized hypergeometric polynomials

In this paper authors prove a general theorem on generating relations for a certain sequence of functions. Many formulas involving the families of generating functions for generalized hypergeometric polynomials are shown here to be special cases of a general class of generating functions involving generalized hypergeometric polynomials and multiple hypergeometric series of several variables. It is then shown how the main result can be applied to derive a large number of generating functions involving hypergeometric functions of Kampé de Fériet, Srivastava, Srivastava-Daoust, Chaundy, Fasenmyer, Cohen, Pasternack, Khandekar, Rainville and other multiple Gaussian hypergeometric polynomials scattered in the literature of special functions.     

On regularity of twisted spherical means and special Hermite expansions

Regularity properties of twisted spherical means are studied in terms of certain Sobolev spaces defined using Laguerre functions. As an application we prove a localisation theorem for special Hermite expansions.     

On the curious series related to the elliptic integrals

By using the theory of the elliptic integrals, a new method of summation is proposed for a certain class of series and their derivatives involving hyperbolic functions. It is based on the termwise differentiation of the series with respect to the elliptic modulus and integral representations of several of the series in terms of the inverse Mellin transforms related to the Riemann zeta function. The relation with the corresponding case of the Voronoi summation formula is exhibited. The involved series are expressed in closed form in terms of complete elliptic integrals of the first and second kind, and some special cases are calculated in terms of particular values of the Euler gamma function.     

A unified approach,using spheroidal functions,for solving the problem of light scattering by axisymmetric particles

A theory that joins three well-known methods is suggested. These methods are the separation of variables, extended boundary conditions and point matching, where the fields are represented by their expansions in (spheroidal) wave functions. Applying similar field expansions, the methods essentially differ in formulation of the problem, and thus they were always discussed in the literature independently. An original approach is employed, in which the fields are divided in two parts with certain properties, and special scalar potentials are selected for each of the parts. The theory allows one to see the similarity and differences of the methods under consideration. Analysis performed earlier shows that the methods significantly supplement each other and the original approach used with a spheroidal basis gives reliable results for particles of high eccentricity for which other techniques do not work. Thus, the suggested theory provides a ground for development of a universal efficient algorithm for calculating optical characteristics of nonspherical scatterers in a very wide region of their parameter values. Bibliography: 21 titles.     

Generalized trace and modified dimension functions on ribbon categories

In this paper, we use topological techniques to construct generalized trace and modified dimension functions on ideals in certain ribbon categories. Examples of such ribbon categories naturally arise in representation theory where the usual trace and dimension functions are zero, but these generalized trace and modified dimension functions are nonzero. Such examples include categories of finite dimensional modules of certain Lie algebras and finite groups over a field of positive characteristic and categories of finite dimensional modules of basic Lie superalgebras over the complex numbers. These modified dimensions can be interpreted categorically and are closely related to some basic notions from representation theory.