For triangle groups, the (quasi-)automorphic forms are known just as explicitly as for the modular group PSL(2, Z). We collect these expressions here, and then interpret them using the Halphen differential equation. We study the arithmetic properties of their Fourier coefficients at cusps and Taylor coefficients at elliptic fixedpoints -in both cases integrality is related to the arithmeticity of the triangle group. As an application of our formulas, we provide an explicit modular interpretation of periods of 14 families of Calabi-Yau three-folds over the thrice-punctured sphere.

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We classify all primes appearing in the denominators of the Hauptmodul J and modular forms for non-arithmetic triangle groups with a cusp. These primes have a congruence condition in terms of the order of the generators of the group. As a corollary we show that for the Hecke group of type (2, m, ∞), the prime p does not appear in the denominator of J if and only if p ≡ ±1 (mod m).

Motivated by Schur’s result on computing the Galois groups of the exponential Taylor polynomials, this paper aims to compute the Galois groups of the Taylor polynomials of the elementary functions [Formula: see text] and [Formula: see text]. We first show that the Galois groups of the [Formula: see text]th Taylor polynomials of [Formula: see text] are as large as possible, namely, [Formula: see text] (full symmetric group) or [Formula: see text] (alternating group), depending on the residue of the integer number [Formula: see text] modulo [Formula: see text]. We then compute the Galois groups of the [Formula: see text]th Taylor polynomials of [Formula: see text] and show that these Galois groups essentially coincide with the Coexter groups of type [Formula: see text] (or an index 2 subgroup of the corresponding Coexter group).

In this paper we give a description of the coefficients of the asymptotic expansion of the logarithmic derivative of a family of hypergeometric series. This family plays an important role in the computation of the reduced genus one Gromov–Witten invariants of projective hypersurfaces and the confirmation of Bershadsky, Cecotti, Ooguri, Vafa (BCOV) conjecture for genus one Gromov–Witten invariants of a generic quintic threefold by Zinger.