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At the international congress of mathematicians in 1900, Hilbert claimed thatthe Riemann zeta function $\zeta(s)$ is not the solution of any algebraicordinary differential equations on its region of analyticity. Let $T$ be aninfinite order linear differential operator introduced by Van Gorder in 2015.Recently, Prado and Klinger-Logan (J. Number Theory 217 (2020), 422-442) showedthat the Hurwitz zeta function $\zeta(s,a)$ formally satisfies the followinglinear differential equation $$T\left[\zeta (s,a) - \frac{1}{a^s}\right] =\frac{1}{(s-1)a^{s-1}}. $$ Then in (Abh. Math. Semin. Univ. Hambg. 91 (2021),no.1, 117-135), by defining $T_{p}^{a}$, a $p$-adic analogue of Van Gorder'soperator $T,$ we constructed the following convergent infinite order lineardifferential equation satisfied by the $p$-adic Hurwitz-type Euler zetafunction $\zeta_{p,E}(s,a)$ $$ T_{p}^{a}\left[\zeta_{p,E}(s,a)-\langlea\rangle^{1-s}\right] =\frac{1}{s-1}\left(\langle a-1 \rangle^{1-s}-\langlea\rangle^{1-s}\right). $$ In this paper, we consider this problem in thepositive characteristic case. That is, by introducing$\zeta_{\infty}(s_{0},s,a,z)$, a Hurwitz type refinement of Goss zeta function,and an infinite order linear difference operator $L$, we establish thefollowing difference equation \begin{equation*}L\left[\zeta_{\infty}\left(\frac{1}{T},s,a,0\right)\right]=\sum_{\alpha\in\mathbb{F}_{q}}\frac{1}{\langle a+\alpha\rangle^{s}}. \end{equation*}

Infinite order linear difference equation satisfied by a refinement of Goss zeta function