Abstract:
The general properties of two-dimensional generalized Bessel functions are discussed. Various asymptotic approximations are derived and applied to analyze the basic structure of the two-dimensional Bessel functions as well as their nodal lines.

Abstract:
We study a mixed problem with an integral two-space-variables condition for parabolic equation with the Bessel operator. The existence and uniqueness of the solution in functional weighted Sobolev space are proved. The proof is based on a priori estimate “energy inequality” and the density of the range of the operator generated by the problem considered. 1. Introduction The importance of boundary value problems with integral boundary condition has been pointed out by Samarski？ [1] and problems with integral conditions for parabolic equations were treated by Kamynin [2], Ionkin [3], Yurchuk [4], Benouar and Yurchuk [5], Bouziani [6], Bouziani and Benouar [7, 8], and Mesloub and Bouziani [9]. Other parabolic problems also arise in plasma physics by Samarski？ [1], heat conduction by Cannon [10], Ionkin [3], dynamics of ground waters by Nakhushev [11], Vodakhova [12], Kartynnik [13], and Lin [14]. Regular case of this problem is studied in [15]. The problem where the equation contains an operator of the form , instead of Bessel operator, is treated in [16]. Similar problems for second-order parabolic equations are investegated by the potential method in [17]. Later problems with an integral two-space-variables condition for parabolic equations were treated by Marhoune [18], Marhoune and Lakhal [19]. Motivated by this, we study a mixed problem with an integral two-space-variables condition for parabolic equation with the Bessel operator. 2. Setting of the Problem In the rectangular domain , with , we consider the following equation: with the initial data Neumann boundary condition and the integral condition where is a known function. We shall assume that the functions and satisfy the compatibility condition with (4), that is, The presence of integral terms in boundary condition can, in general, greatly complicate the application of standard functional or numerical techniques, specially the integral two-space-variables condition. Then to avoid this difficulty, we introduce a technique for transfering this problem to another classically less complicated one that does not contain integral conditions. For that, we establish the following lemma. Lemma 1. Problem (1)–(4) is equivalent to the following problem : Proof. Let be a solution of (1)–(4), we prove that So, multiplying (1) by and integrating with respect to over and and taking into account (4) and (6), we obtain Then, from (3), we obtain Let now be a solution of , we are bound to prove that So, multiplying (1) by and integrating with respect to over and and taking into account we obtain combining the two

Abstract:
We consider the space of Henstock integrable functions of two variables. Equipped with the Alexiewicz norm the space is proved to be barrelled. We give a partial description of its dual. We show by an example that the dual can't be described in a manner analogous to the one-dimensional case, since in two variables there exist functions whose distributional partials are measures and which are not multipliers for Henstock integrable functions.

Abstract:
We prove a noncommutative analogue of the fact that every symmetric analytic function of $(z,w)$ in the bidisc $\D^2$ can be expressed as an analytic function of the variables $z+w$ and $zw$. We construct an analytic nc-map $S$ from the biball to an infinite-dimensional nc-domain $\Omega$ with the property that, for every bounded symmetric function $\ph$ of two noncommuting variables that is analytic on the biball, there exists a bounded analytic nc-function $\Phi$ on $\Omega$ such that $\ph=\Phi\circ S$. We also establish a realization formula for $\Phi$, and hence for $\ph$, in terms of operators on Hilbert space.

Abstract:
We consider Szasz-Mirakyan operators in polynomial and exponential weighted spaces of functions of two variables. We give Voronowskaya type theorem and theorem on convergence of certain sequences.

Abstract:
We prove the analogue of an identity of Huard, Ou, Spearman and Williams and apply it to evaluate a variety of sums involving divisor functions in two variables. It turns out that these sums count representations of positive integers involving radicals.

Abstract:
In the recent development in a various disciplines of physics, it is noted the need for including the deformed versions of the exponential functions. In this paper, we consider the deformations which have two purposes: to have them like special cases, and, even more, to acquit their inauguration from mathematical point of view. Really, we will show interesting differential and difference properties of our deformations which are important in forming and explanation of continuous and discrete models of numerous phenomena.

Abstract:
In this note the authors derive three transformations for the basic hypergeometric functions of two variables by making use of certain known summations formula. As q → 1, known transformations for ordinary hypergeometric functions giver earlier by Carlitz [4], Jackson [7] and Slater [11] are obtained.