Laurent coefficients of Zagier-type zeta function {à} la Ishibashi and arithmetic aspects of extended Ramanujan period function

One of the remarkable contributions of Don Zagier was the Kronecker limit formula for a real quadratic field, where he connects the double series $\mathcal{Z}(s,w,w^{\mathrm{\prime }})$ to the Dedekind zeta function associated to a real quadratic field. Later, Ishibashi determined all the Laurent coefficients of $\mathcal{Z}(s,w,w^{\mathrm{\prime }})$ at $s=1$. Recently, Choie and kumar have studied the analytic behaviour of the analogous double series $\tilde{\mathcal{Z}}(s,w,w^{\mathrm{\prime }})$. In this article, we derive all the Laurent coefficients of $\tilde{\mathcal{Z}}(s,w,w^{\mathrm{\prime }})$, akin to Ishibashi. These Laurent coefficients involve an interesting function ${\mathfrak{F}}_k^0(x)$, which was earlier studied by Dixit et. al. (Ramanujan for $k=1$), where they obtained a beautiful symmetric relation for ${\mathfrak{F}}_k^0(x)$. We establish both the two term and the three term functional equation of ${\mathfrak{F}}_k^0(x)$, derive the action of the period-like Hecke operator on ${\mathfrak{F}}_k^0(x)$ and connect an important integral with ${\mathfrak{F}}_k^0(x)$.