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Some sufficient conditions for biholomorphic convex mappings of order on the Reinhardt domain in are given; from that, criteria for biholomorphic convex mappings of order with particular form become direct. As applications of these sufficient conditions, some concrete biholomorphic convex mappings of order on are provided.
Computing the matrix elements of the linear operator, which transforms the spherical basis of -representation space into the hyperbolic basis, very recently, Shilin and Choi (2013) presented an integral formula involving the product of two Legendre functions of the first kind expressed in terms of -hypergeometric function and, using the general Mehler-Fock transform, another integral formula for the Legendre function of the first kind. In the sequel, we investigate the pairwise connections between the spherical, hyperbolic, and parabolic bases. Using the above connections, we give an interesting series involving the Gauss hypergeometric functions expressed in terms of the Macdonald function.
Ever since Euler first evaluated and , numerous interesting solutions of the problem of evaluating the have appeared in the mathematical literature. Until now no simple formula analogous to the evaluation of is known for or even for any special case such as . Instead, various rapidly converging series for have been developed by many authors. Here, using Fourier series, we aim mainly at presenting a recurrence formula for rapidly converging series for . In addition, using Fourier series and recalling some indefinite integral formulas, we also give recurrence formulas for evaluations of and , which have been treated in earlier works.