## Fractional sum and fractional difference on non-uniform lattices and analogue of Euler and Cauchy beta formulas.(English)Zbl 07439146

Summary: As is well known, the definitions of fractional sum and fractional difference of $$f (z)$$ on non-uniform lattices $$x(z) = c_1z^2 + c_2z + c_3$$ or $$x(z) = c_1q^z + c_2q^{- z} + c_3$$ 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.

### MSC:

 39A13 Difference equations, scaling ($$q$$-differences) 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) 26A33 Fractional derivatives and integrals 34K37 Functional-differential equations with fractional derivatives
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