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Fractional sum and fractional difference on non-uniform lattices and analogue of Euler and Cauchy Beta formulas

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Abstract

As is well known, the definitions of fractional sum and fractional difference of f (z) on non-uniform lattices x(z) = c1z2 + c2z + c3 or x(z) = c1qz + c2q−z + c3 are more difficult and complicated. In this article, for the first time we propose the definitions of the fractional sum and fractional difference on non-uniform lattices by two different ways. The analogue of Euler’s Beta formula, Cauchy’ Beta formula on non-uniform lattices are established, and some fundamental theorems of fractional calculas, the solution of the generalized Abel equation on non-uniform lattices are obtained etc.

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Authors and Affiliations

  1. Department of Mathematics, Xiamen University, Xiamen, 361005, China

    Jin-fa Cheng

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  1. Jin-fa Cheng
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Correspondence to Jin-fa Cheng.

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Supported by the National Natural Science Foundation Fujian province of China(2016J01032).

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Cheng, Jf. Fractional sum and fractional difference on non-uniform lattices and analogue of Euler and Cauchy Beta formulas. Appl. Math. J. Chin. Univ. 36, 420–442 (2021). https://doi.org/10.1007/s11766-021-4013-1

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  • DOI: https://doi.org/10.1007/s11766-021-4013-1

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