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The beta integral method proved itself as a simple but nonetheless powerful method for generating hypergeometric identities at a fixed argument. In this paper, we propose a generalization by substituting the beta density with a particular type of Meijer's G function. By the application of our method to known transformation formulas, we derive about forty hypergeometric identities, the majority of which are believed to be new. [ABSTRACT FROM AUTHOR]

A derivation (of order 1) satisfies the reduction formula f (x k) = k x k - 1 f (x) for any integer k. In this article we find corresponding reduction formulas for derivations of higher order on commutative rings. More precisely, for every derivation f of order n and every positive integer k we find an explicit formula for f (x k) as a linear combination of x k - 1 f (x) , x k - 2 f (x 2) , ... , x k - n f (x n) . The proof hinges on the hypergeometric identity ∑ k ≥ 0 (- 1) k n k n + d + 1 - k n - j d + j - k j = n j for any positive integer n, nonnegative integer d, and integer j satisfying 0 ≤ j ≤ n . We prove this identity via the WZ-method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove a conjecture on a common region of a convergence of Padé iterations for the matrix sector function. For this purpose, we show that all Padé approximants to a special case of hypergeometric function have a power series expansion with positive coefficients. Using a sharpened version of Schwarz’s lemma, we also demonstrate a better estimate of the convergence speed. Our results are also applicable to a family of rational iterations for computing the matrix th root. [Copyright &y& Elsevier]

The main object of this paper is to consider the asymptotic distribution of the zeros of certain classes of the Gauss hypergeometric polynomials. Some classical analytic methods and techniques are used here to analyze the behavior of the zeros of the Gauss hypergeometric polynomials, 2F1(-n, a;-n + b; z), where n is a nonnegative integer. Owing to the connection between the classical Jacobi polynomials and the Gauss hypergeometric polynomials, we prove a special case of a conjecture made by Martínez-Finkelshtein, Martínez-González and Orive. Numerical evidence and graphical illustrations of the clustering of the zeros on certain curves are generated by Mathematica (Version 4.0). [ABSTRACT FROM AUTHOR]

Abstract: A general transformation involving generalized hypergeometric functions has been recently found by Rathie and Rakha using simple arguments and exploiting Gauss’s summation theorem. In this sequel to the work of Rathie and Rakha, a new hypergeometric transformation formula is derived by their method and by appealing to Gauss’s second summation theorem. In addition, it is shown that the method fails to give similar hypergeometric transformations in the cases of the classical summation theorems of Kummer, Bailey, Watson and Dixon. [ABSTRACT FROM AUTHOR]

Abstract: We consider the fractionally integrated ARFIMA Processes with seasonality s, denoted by SARFIMA (0, D, 0) s . This work presents a closed formula for the Durbin-Levinson''s algorithm relating the partial autocorrelation and the autocorrelation functions of these processes. In order to obtain the closed formula we show a hypergeometric identity, namely , for any nonnegative integer l and for any D ∈ (−0.5, 0.5). Any recursive algorithm that requires the use of the left-hand side of the above expression will have smaller error under the use of the right-hand side formula. The Durbin-Levinson''s algorithm is fully calculated for the SARFIMA (0, D, 0) s processes. [Copyright &y& Elsevier]

Abstract: The reductions of the Heun equation to the hypergeometric equation by polynomial transformations of its independent variable are enumerated and classified. Heun-to-hypergeometric reductions are similar to classical hypergeometric identities, but the conditions for the existence of a reduction involve features of the Heun equation that the hypergeometric equation does not possess; namely, its cross-ratio and accessory parameters. The reductions include quadratic and cubic transformations, which may be performed only if the singular points of the Heun equation form a harmonic or an equianharmonic quadruple, respectively; and several higher-degree transformations. This result corrects and extends a theorem in a previous paper, which found only the quadratic transformations. (SIAM J. Math. Anal. 10 (3) (1979) 655). [Copyright &y& Elsevier]