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A new \(h\)-discrete fractional operator, fractional power and finite summation of hypergeometric polynomials. (English) Zbl 1504.26016

Summary: In the present paper, we introduce the discrete fractional trapezoidal operators \(T_h^{\alpha}\) for \(\alpha \in (0,1)\) as the fractional power of the classical trapezoidal formula. Consequently, we derive the fractional power of a triangular matrix. As applications, we determine the eigenvectors of \(T_h^{\alpha}\) and a finite summation formula of the product of hypergeometric polynomials.

MSC:

26A33 Fractional derivatives and integrals
15A16 Matrix exponential and similar functions of matrices
33C05 Classical hypergeometric functions, \({}_2F_1\)
39A12 Discrete version of topics in analysis
47B12 Sectorial operators
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