Euler’s Beta formula on nonuniform lattices and its applications

In this article, we obtain the analogue of Euler’s Beta formula on nonuniform lattices x ( z ) = c 1 z 2 + c 2 z + c 3 or x ( z ) = c ~ 1 q z + c ~ 2 q - z + c ~ 3 , which has independent significance. Some important applications of Euler Beta formula on nonuniform lattices help us to solve the generalized Abel integral equation and define fractional sum and difference on nonuniform lattices effectively. In addition, the most interesting application is that we can use fractional calculus on nonuniform lattices to get solutions of hypergeometric difference equations on nonuniform lattices.