THE DUAL OF A FORMULA OF VISKOV

Abstract

This minipaper offers a formula which is dual to that of Viskov [5]. While Viskov's can be thought of as a rising formula for Laguerre polynomials, ours is precisely the lowering one. Besides documenting the formula, which seems to be missing, we want to provide a (rather elementary) operator theory argument instead of making crude calculations. In other words, the annihilation and creation operators are confronted with lowering and rising formulae; they are often failed to be distinguished.

Keywords

• Laguerre polynomials;
• raising and lowering formulae;
• weighted shift operator

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References

1. Delft University of Technology, Report of the Department of Technical Mathematics and Informatics no. 98-17 The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue R.Koekoek;R.F.Swarttouw
2. Acta Sci. Math. v.37 An identity for Laguerre polynomials L.B.Redei
3. Univ. Iagel. Acta Math. v.28 A few assorted questions about unbounded subnormal operators Jan Stochel;F.H.Szafraniec
4. Proc. Segovia (Spain), 1986 Monogr. Acad. Cienc. Zaragoza R.-C. Plalacios v.1 Orthogonal polynomials and subnormality of related shift operators, "Orthogonal polynomials and their applications" F.H.Szafraniec
5. Acta. Sci. Math. v.39 L. B. Redei identity for Laguerre polynomials O.V.Viskov