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A General Theorem on Inversion Problems for Polynomial Sets

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Abstract

The inversion problem for a given polynomial set, \({\{P_n\}_{n\geq0}}\), expanded in a basis \({\{\mathfrak{B}_{n}\}}\) is the problem of finding the coefficients \({I_{k}(n)}\) in the expansion

$${\mathfrak{B}_n(x) } = \sum_{k=0}^{n}I_k(n) P_{k}(x).$$

In this paper, we state a theorem solving this problem for each family of polynomials defined only by its explicit representation. The obtained coefficients are expressed by means of a simple recurrence relation with two indexes. We illustrate the method by producing explicit formulas for some known polynomial sets.

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  1. École Supérieure des Sciences et de Technologie de Hammam Sousse, Sousse University, Sousse, Tunisia

    Neila Ben Romdhane

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  1. Neila Ben Romdhane
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Correspondence to Neila Ben Romdhane.

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Romdhane, N.B. A General Theorem on Inversion Problems for Polynomial Sets. Mediterr. J. Math. 13, 2783–2793 (2016). https://doi.org/10.1007/s00009-015-0654-8

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  • DOI: https://doi.org/10.1007/s00009-015-0654-8

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