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An introduction to finite fibonomial calculus. (English) Zbl 1160.11308

Let \(P\) the algebra of polynomials over the field \(K\) of characteristic zero. Using the Fibonacci sequence \(\{F_{n}\}_{n\geq 0}\), the linear operator \(\partial_{F}: P \to P\), \(\partial_{F} x^{n}= F_{n}x^{n-1}\) is defined. \(\partial_{F}\) is called the F-derivative. The purpose of the present paper is to develop the classical umbral calculus of Gian-Carlo Rota. Starting with \(\partial_{F}\) on define: the \(F\)-translation operator \(E^{y}(\partial_{F})\), the \(\partial_{F}\)-delta operator, the \(\partial_{F}\)-basic plynomial sequence, the Pincherle F-derivative and the Sheffer F-polynomials. The first and the second F-expansion theorem, the isomorphism theorem, the F-transfer formulas and the F-Rodrigues formula are presented. The results are ilustrated by examples.
Reviewer: Emil Popa (Sibiu)

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
05A40 Umbral calculus
33D99 Basic hypergeometric functions
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Online Encyclopedia of Integer Sequences:

Triangle of Fibonomial coefficients.
Signed Fibonomial triangle.

References:

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