On restricted approximation measures of Jacobi’s triple product

Abstract We obtain rational approximations for Jacobi’s triple product $$\begin{aligned} \Pi _q(t):= \prod _{m=1}^\infty (1-q^{2m})(1+q^{2m-1}t)(1+q^{2m-1}t^{-1}), \end{aligned}$$ Π q ( t ) : = ∏ m = 1 ∞ ( 1 - q 2 m ) ( 1 + q 2 m - 1 t ) ( 1 + q 2 m - 1 t - 1 ) , when $$t=a/b\in {\mathbb {Q}}$$ t = a / b ∈ Q is non-zero and $$q=1/d$$ q = 1 / d with $$d\in {\mathbb {Z}}{\setminus }\{0, \pm 1 \}$$ d ∈ Z \ { 0 , ± 1 } . Especially we give effective and restricted approximation for the values of Jacobi’s triple product and for the values of Euler’s infinite product.