Let $f={\sum }_{n=1}^{\infty }a\left(n\right)q^n$ and $g={\sum }_{n=1}^{\infty }b\left(n\right)q^n$ be holomorphic common eigenforms of all Hecke operators for the congruence subgroup ${\Gamma }_0\left(N\right)$ of $S{L}_2\left(\mathbf{Z}\right)$ with “Nebentypus” character $\psi$ and $\xi$ and of weight $k$ and $\ell$, respectively. Define the Rankin product of $f$ and $g$ by$${𝒟}_N(s,f,g)=(\sum _{n=1}^{\infty }\psi \xi \left(n\right)n^{k+\ell -2s-2}\left)\right(\sum _{n=1}^{\infty }a\left(n\right)b\left(n\right)n^{-s}).$$Supposing $f$ and $g$ to be ordinary at a prime $p\ge 5$, we shall construct a $p$-adically analytic $L$-function of three variables which interpolate the values $\frac{{𝒟}_N(\ell +m,f,g)}{{\pi }^{\ell +2m+1}\<f,f\>}$ for integers $m$ with $0\le m\<k-1,$ by regarding all the ingredients $m$, $f$ and $g$ as variables. Here $⟨f,f⟩$ is the Petersson self-inner product of $f$.

We construct analogs of the classical Δ-function for quotients of the upper half plane 𝓗 by certain arithmetic triangle groups Γ coming from quaternion division algebras B. We also establish a relative integrality result concerning modular functions of the form Δ(αz)/Δ(z) for α in B⁺. We give two explicit examples at the end.