The problem of computing the dimension of spaces of splines whose elements are piecewise polynomials of degree d with r continuous derivatives globally has attracted a great deal of attention recently. We contribute to this theory by obtaining dimension formulae for certain spaces of super sphnes, including the case where varying amounts of additional smoothness is enforced at each vertex. We also explicitly construct minimally supported bases for the spaces. The main tool is the Bernstein-B6zier method.
Contiguous relations for hypergeometric series contain an enormous amount of hidden information. Applications of contiguous relations range from the evaluation of hypergeometric series to the derivation of summation and transformation formulas for such series. In this paper, a general formula joining three Gauss functions of the form 2 F 1 [a 1 , a 2 ; a 3 ; z] with arbitrary integer shifts is presented. Our analysis depends on using shifted operators attached to the three parameters a 1 , a 2 and a 3 . We also, discussed the existence condition of our formula.
Two Gauss functions are said to be contiguous if they are alike except for one pair of parameters, and these differ by unity. Contiguous relations are of great use in extending numerical tables of the function. In this paper we will introduce a new method for computing such types of relations.
For i = 0, we have the well known, interesting and useful formula due to Kummer which was independently discovered by Ramanujan. The results are derived with the help of generalizations of Gauss's second summation theorem obtained recently by Rakha et al.. As applications, we also obtained a large number of interesting results closely related to other results of Ramanujan. In the end, using Beta integral method, a large number of new and interesting hypergeometric identities are established. Known results earlier obtained by Choi et al. follow special cases of our main findings.
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