On Some Mixed Polynomial Exponential Diophantine Equation: \(\alpha^n+\beta^n+a(\alpha^s\pm\beta^s)^m+D=r(u^k+v^k+w^k)\) with \(\alpha\) and \(\beta\) Consecutive

Let \(a,\alpha,\beta,r,u,v,w\) and \(D\) be any integers and suppose that \(n,m,s\) and \(k\) are non-negative exponent. In thispaper, the diophantine equation \(\alpha^n+\beta^n+a(\alpha^s\pm\beta^s)^m+D=r(u^k+v^k+w^k)\) is developed and investigated for integer solution and its various polynomial identities. Moreover, the study formulates some conjectures for the title equation.