The umbral calculus on logarithmic algebras |
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Authors: | Alexander Nickolaevich Kholodov |
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Institution: | (1) Institute of Problems in Cybernetics, Academy of Sciences of the U.S.S.R., Vavilov St. 37, 117312 Moscow, U.S.S.R. |
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Abstract: | D. E. Loeb and G.-C. Rota, using the operator of differentiation D, constructed the logarithmic algebra that is the generalization of the algebra of formal Laurent series. They also introduced Appell graded logarithmic sequences and binomial (basic) graded logarithmic sequences as sequences of elements of the logarithmic algebra and extended the main results of the classical umbral calculus on such sequences. We construct an algebra by an operator d that is defined by the formula (1.1). This algebra is an analog of the logarithmic algebra. Then we define sequences analogous to Boas-Buck polynomial sequences and extend the main results of the nonclassical umbral calculus on such sequences. The basic logarithmic algebra constructed by the operator of q-differentiation is considered. The analog of the q-Stirling formula is obtained. |
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Keywords: | 05A40 33A35 33A70 33A65 |
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