On the Shorey–Tijdeman Diophantine equation involving terms of Lucas sequences

Abstract

Let r≥1 be an integer and U≔{Un}n≥0 be the Lucas sequence given by U0=0,U1=1, and Un+2=rUn+1+Un for n≥0. In this paper, we explain how to find all the solutions of the Diophantine equation, AUn+BUm=CUn1+DUm1, in integers r≥1, 0≤m