Sums of Reciprocals of Irreducible Polynomials over Finite Fields

We will revisit a theorem first proved by L. Carlitz in 1935 in which he provided an intriguing formula for sums involving the reciprocals of all monic polynomials of a given degree over a finite field of a specified order. Expanding on this result, we will consider the equally curious case where instead of adding reciprocals all monic polynomials of a given degree, we only consider adding reciprocals of those that are irreducible

Determination of a Type of Permutation Trinomials over Finite Fields

Let $f=a{\tt x} +b{\tt x}^q+{\tt x}^{2q-1}\in\Bbb F_q[{\tt x}]$. 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 $\sum_{c\in\Bbb F_q}({\tt x}+c)^n=g_{n,q}({\tt x}^q-{\tt x})$. We determine all $n$ of the form $n=q^\alpha-q^\beta-1$, $\alpha>\beta\ge 0$, for which $g_{n,q}$ is a permutation polynomial of $\Bbb F_{q^2}$.Comment: 28 page