We consider pairs of automorphisms $(\phi,\sigma)$ acting on fields Laurent or Puiseux series: shift operators $(\phi\colon x\mapsto x+h\_1, \sigma\colon x+h\_2)$, $q$-difference q\_1x$, $\sigma\colon q\_2x)$, and Mahler x^{p\_1},\ x^{p\_2})$. Given a solution $f$ to linear $\phi$-equation $g$ an algebraic $\sigma$-equation, both transcendental, we show that are algebraically independent over t...