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Antilogarithms of second order in algebras with logarithms and their applications to special functions

Przeworska-Rolewicz D.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2004

Vol. 44, nr spec.

167-191

In the paper [PR5] it was shown that the so-called special functions of Mathematical Physics can be obtained by means of antilogarithms of the second order for the usual differential operator ^j. The same method applied to a right invertible operator D in a commutative Leibniz algebra with logarithms permits to determine eigenvectors of linear equations of order two in D with coefficients in the algebra X under consideration by a reduction to the generalized Sturm-Liouville operator. It seems that, in a sense, the proposed method is an answer for the question of Gian-Carlo Rota concerning a unified approach to special functions (cf. [Rl], problem 4). Section 6 of the present paper is devoted to some summations formulae expressing special functions by means of exponentials. Note that, in general, we do not need any assumption about the Hilbert structure of the algebra X.