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We find explicit conditions on $a$ and $b$ that are necessary and sufficient for $f$ to be a permutation polynomial of $\\Bbb F_{q^2}$. This result allows us to solve a related problem. Let $g_{n,q}\\in\\Bbb F_p[{\\tt x}]$ ($n\\ge 0$, $p=\\text{char}\\,\\Bbb F_q$) be the polynomial defined by the functional equation $\\sum_{c\\in\\Bbb F_q}({\\tt x}+c)^n=g_{n,q}({\\tt x}^q-{\\tt x})$. 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