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In this paper we introduce and study some basic properties of the Fock space (also known as Segal-Bargmann space) in the slice hyperholomorphic setting. We discuss both the case of slice regular functions over quaternions and also the case…
The aim of this paper is to apply generalized operators of fractional integration and differentiation involving Appell's function $F_{3}(:)$ due to Marichev-Saigo-Maeda (MSM), to the Jacobi type orthogonal polynomials. The results are…
A representation-theoretic approach to special functions was developed in the 40-s and 50-s in the works of I.M.Gelfand, M.A.Naimark, N.Ya.Vilenkin, and their collaborators. The essence of this approach is the fact that most classical…
We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine's ${}_2\phi_1$ transformation formula and Sears' ${}_3\phi_2$ transformation formula can be easily obtained by the symmetric property of some parameters in…
In this paper, we introduce a pair of multiplication-like operations, $L_0$ and $L_1$, which derive $k$-regular functions from $(k+1)$-regular functions. The investigation of the inverse problem naturally leads to a deeper study of the…
It is known that many constructions arising in the classical Gaussian infinite dimensional analysis can be extended to the case of more general measures. One such extension can be obtained through biorthogonal systems of Appell polynomials…
This work proposes a unified theory of regularity in one hypercomplex variable: the theory of $T$-regular functions. In the special case of quaternion-valued functions of one quaternionic variable, this unified theory comprises…
The aim of this work is to define a continuous functional calculus in quaternionic Hilbert spaces, starting from basic issues regarding the notion of spherical spectrum of a normal operator. As properties of the spherical spectrum suggest,…
This paper studies variance functions of Cauchy-Stieltjes Kernel families generated by compactly supported centered probability measures. We describe several operations that allow us to construct additional variance functions from known…
Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are…
The polynomial relationship between elementary symmetric functions (Cauchy enumeration formula) is formulated via a ``raising operator" and Fock space construction. A simple graphical proof of this relation is proposed. The new operator…
The affine $su(3)$ modular invariant partition functions in 2d RCFT are associated with a set of generalized Coxeter graphs. These partition functions fall into two classes, the block-diagonal (Type I) and the non block-diagonal (Type II)…
We revisit and generalize our previous algebraic construction of the chiral effective action for Conformal Field Theory on higher genus Riemann surfaces. We show that the action functional can be obtained by evaluating a certain Deligne…
The new global version of the Cauchy-Kovalevskaia theorem presented here is a strengthening and extension of the regularity of similar global solutions obtained earlier by the author. Recently the space-time foam differential algebras of…
Holomorphic functions in several complex variables are generalized to regular functions in several quaternionic variables, and further to monogenic functions of several vector variables, which are annihilated by several Dirac operators on…
In previous work the framework for a hypercomplex function theory in superspace was established and amply investigated. In this paper a Cauchy integral formula is obtained in this new framework by exploiting techniques from orthogonal…
The Cauchy integral formula in Clifford analysis allows us to associate a holomorphic function $\tilde f:L_n\to \C$ on the Lie ball $L_n$ in $\C^n$ with its monogenic counterpart $f:B_1(0)\to \C^{n+1}$ via the formula $\tilde f(z) =…
We present an axiomatic frame (in Prt I of this book) in which many results of the K-theory for C*-algebras are proved. Then we construct an example for this axiomatic theory (in Part II), which generalizes the classical theory for…
As is well known, the common elementary functions defined over the real numbers can be generalized to act not only over the complex number field but also over the skew (non-commuting) field of the quaternions. In this paper, we detail a…
This paper explores the well-posedness of the Cauchy problem for the Fokker-Planck equation associated with the partial differential operator $L$ with low regularity condition. To address uniqueness, we apply a recently developed…