In this paper, we completely classify the rational solutions of the Sasano system of type D₃⁽²⁾. This system of ordinary differential equations is given by the coupled P_II system, and has the affine Weyl group symmetry of type D₃⁽²⁾. The rational solutions are classified to one type by the Bäcklund transformations.

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We investigate the values of Dirichlet L-functions L(s, χ_p) at s = 1 as p runs through the primes in an arithmetic progression, where χ_p denotes the character given by Legendre' s symbol (•/p). We show that the numbers h_Q(√-p)/√p exist densely in the positive real numbers, where h_Q(√-p) is the class number of the quadratic field Q(√-p).We also give a quantitative result for the problem of Ayoub, Chowla and Walum [ACW] about the character sum Σ^{p -1}_n=1n^k(n/p).

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Monodromy representations on the space of solutions of the Gauss hypergeometric equation are studied by using integrals of a multivalued function. We first establish the fact that any solution of the Gauss hypergeometric equation is expressed by the integral of a multivalued function. Second, we give a necessary and sufficient condition for the irreducibility. Third, we realize the representations in the reducible cases. In the reducible cases, the exponents of the integrand do not satisfy the genericity condition and chains which are not necessarily cycles are needed to give a basis of the solution space. Finally, we give a complete list of finite reducible representations.

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Monodromy representations on the solution space of Lauricella' s system of differential equations E_D and the Jordan-Pochhammer differential equation E_JP are studied by using integrals of a multivalued function. We first establish the fact that any solution of E_D and any solution of E_JP are both expressed by the integrals of a multivalued function. Second, a necessary and sufficient condition for the irreducibility is given.

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Monodromy representations on the solution space of Appell' s system of differential equations E₁ are studied by using integrals of a multivalued function. In particular, we realize the representations in the reducible cases and give a complete list of finite reducible representations.

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We consider a certain multiple Dirichlet series that is a generalization of that introduced by Masri 2005, and we prove the meromorphic continuation to the whole space. Also, using certain functional relations and the technique of changing variables, we prove that ‘the possible singularities’ are indeed ‘the true singularities’.

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The orthosymplectic Lie superalgebra osp_2,2 is a classical simple Lie superalgebra. Its Lie part forms a copy of the direct sum of sl₂ ⊕ so₂. In this paper, we first study its structure in detail. Based on the well-known description of irreducible representations of sl₂, we obtain the classification of the irreducible representations of osp_2,2 subject to some technical restrictions. These irreducible representations actually appear in the natural correspondence of (O_p,q, osp_2,2) acting on the space S(Rⁿ, Λ^∗ ((Rⁿ) ^∗ )) of Schwartz class differential forms on Rⁿ.

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Hypergeometric equations with a dihedral monodromy group can be solved in terms of elementary functions. This paper gives explicit general expressions for quadratic monodromy invariants for these hypergeometric equations, using a generalization of Clausen' s formula and terminating double hypergeometric sums. Pull-back transformations for the dihedral hypergeometric equations are also presented, including Klein' s pullback transformations for the equations with a finite (dihedral) monodromy group.

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We are concerned with two types of h transform of one-dimensional generalized diffusion operators treated by Maeno (2006) and by Tomisaki (2007). We show that these two types of h transform are in inverse relation to each other in some sense. Further, we show that a recurrent one-dimensional generalized diffusion operator cannot be represented as an h transform of another one-dimensional generalized diffusion operator different from the original one. We also consider a spectral representation of elementary solutions corresponding to h transformed one-dimensional generalized diffusion operators.

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In this paper, we study the period mappings for the families of K3 surfaces derived from the three-dimensional reflexive polytopes with five vertices. We determine the lattice structures, the period differential equations and the projective monodromy groups. Moreover, we show that one of our period differential equations coincides with the uniformizing differential equation of the Hilbert modular orbifold for the field Q(√5).

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