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Some summations formulae in commutative Leibniz algebras with logarithms

100%

Przeworska-Rolewicz D.

Control and Cybernetics

2007

tom Vol. 36, no 3

841-857

A survey of summation formulae in commutative Leibniz algebras with logarithms is given. New results concerning generic functions and related summation formulae, which generalize well-known properties of the Bessel functions, are demonstrated.

Multidimensional Riemann-Hilbert type problems in Leibniz algebras with logarithms

100%

Przeworska-Rolewicz D.

2002

tom Vol. 35, nr 3

629-644

Riemann-Hilbert type problems in Leibniz algebras with logarithms have been studied in PR[8] (cf. Chapter 14). These problems correspond to such classical problems when the Cauchy transformation is an involution. It was shown that this involution is not multiplicative. On the other hand, in the same book equations with multiplicative involutions were considered. These results can be applied to equations with an involutive transformation of argument, in particular, to equations with transformed argument by means of a function of Carleman type. Riemann-Hilbert type problems with an additional multiplicative involution in commutative Leibniz algebras with logarithms are examined in PR[13]. Results obtained there can be applied not only to problems with a transformation of argument but also to problems with the conjugation (in the complex sense). In the present paper there are considered similar problems in several variables with Riemann- Hilbert condition posed on each variable separately. For instance, these problems correspond in the classical case to problems for polyanalytic functions on polydiscs (cf. HD[1], Ms[1).

True shifts revisited

100%

Przeworska-Rolewicz D.

2001

tom Vol. 34, nr 1

111-122

An algebraic analysis approach to 2-D discrete problems

63%

Przeworska-Rolewicz D. , Wysocki H.

Control and Cybernetics

2001

tom Vol. 30, no 2

149-158

A model of algebraic analysis for the 2-index sequences (of the type 2-D) is considered. For difference operators of the form D{x_m,n}:= {x_m+1,n+1 - A_m,nX_m,n} the right inverses and the corresponding initial operators are constructed. Having already known the initial operators, one can determine solutions of the corresponding initial value problems.