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Certain results including the successive derivatives of the H-function of one and more variables are established. These remove the limitations of Ławrynowicz's (1969) formulas and as a result extend the results of Skibiński [13] and various other authors. As an application some finite expansion formulas are also established, which reduce to hypergeometric functions of one and more variables that are of common interest.
We define Picard cycles on each smooth three-sheeted Galois
cover C of the Riemann sphere. The moduli space of all these algebraic
curves is a nice Shimura surface, namely a symmetric quotient of the projective
plane uniformized by the complex two-dimensional unit ball. We show that
all Picard cycles on C form a simple orbit of the Picard modular group
of Eisenstein numbers. The proof uses a special surface classification in
connection with the uniformization of a classical Picard-Fuchs system....
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