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In this paper, which is a direct continuation and generalization of the recent works by the authors [17, 35], we show the validity of the generic multiplicity-induced-dominancy property for a general class of linear functional differential equations with a single delay, including the retarded as well as the neutral cases. The result is based on an appropriate integral representation of the corresponding characteristic quasipolynomial functions involving some appropriate degenerate hypergeometric functions. [ABSTRACT FROM AUTHOR]

We study the maximum value of the confluent hypergeometric function with oscillatory conditions of parameters. As a consequence, we obtain new inequalities for the Gauss hypergeometric function. [ABSTRACT FROM AUTHOR]

In this paper, we present two characterizations of the sequences of Kummer hypergeometric polynomials B a , b , n (x) and Kummer hypergeometric polynomials of the second kind K a , b , n (x) , which are respectively defined by the exponential generating functions: e x t M (a , a + b ; t) = ∑ n = 0 ∞ B a , b , n (x) t n n ! and e x t U (a , a + b ; t) = ∑ n = 0 ∞ K a , b , n (x) t n n ! with M (a , b ; t) = ∑ n = 0 ∞ (a) n (b) n t n n ! , where U (a , a + b ; t) is the Kummer hypergeometric function of the second kind. First we construct Gauss–Weierstrass-type convolution operators T w a , b with a well-chosen kernel (density) function for each sequence of Kummer hypergeometric polynomials and for Kummer hypergeometric polynomials of the second kind. Then we characterize Kummer hypergeometric polynomials as the only Appell polynomials having a weighted-integral mean equal to zero. Our approach is inspired by the Gauss–Weierstrass convolution transform for Hermite polynomials and the Kummer integral representation for confluent hypergeometric functions. [ABSTRACT FROM AUTHOR]

In this work, we study some special hypergeometric functions in positive characteristic, introduced by D. S. Thakur. We shall establish functional relationships among them, and deduce, with the help of the function field version of the Schneider–Lang theorem obtained by J. Yu, that at least one of their values at nonzero algebraic arguments is transcendental. [ABSTRACT FROM AUTHOR]

We introduce the ( p , q ) -extended τ -hypergeometric and confluent hypergeometric functions along with their integral representations. We also present closed integral expressions for the Mathieu-type a -series and for the associated alternating versions whose terms contain the ( p , q ) -extended τ -hypergeometric functions with related contiguous functional relations. [ABSTRACT FROM AUTHOR]

Recently, an extension of the Pochhammer symbol was used in order to introduce and investigate a family of generalized hypergeometric functions [Srivastava et al. (2014) [11] ]. The main object of this paper is to present an extension of the τ -Gauss hypergeometric functions R 1 τ 2 ( z ) and investigate its several properties, including, for example, its integral representations, derivative formulas, Mellin transforms and fractional calculus operators. Some interesting special cases of our main results are also pointed out. [ABSTRACT FROM AUTHOR]

Abstract: In this work we study the fractional hypergeometric functions for function fields, introduced by D.S. Thakur. We shall characterize algebraic functions among them, and show the transcendence of special values at some nonzero algebraic arguments, in the case when they are entire transcendental functions. [Copyright &y& Elsevier]

Abstract: Let Γ be a complex algebraic curve of degree d, non-degenerate, reduced, and irreducible, in the complex projective space . Denoting by the dimension of the vector space of homogeneous polynomials of degree h with respect to n variables, let be the integer (⩾1) such that . The curve Γ is said to be ordinary if it has the following property: the set of algebraic hypersurfaces of a “generic” hyperplane H of which have degree h and contain the hyperplane section , is empty if , and is a projective space of dimension if . Equivalently when , the associated web in of such a curve is “ordinary” in the sense of Cavalier and Lehmann (2012) . The arithmetic genus of an ordinary curve is upper-bounded by the number , and this bound is reached for these ordinary curves which are arithmetically Cohen–Macaulay. For and any , the family of the ordinary curves of degree d which are arithmetically Cohen–Macaulay is non-empty, and is an irreducible component of the Hilbert scheme . By contrast, the complete intersection of algebraic hypersurfaces is never ordinary if . [Copyright &y& Elsevier]

Abstract: In this Note we consider the Cauchy problems with the modified conditions for the Euler–Poisson–Darboux equations. We give the explicit solutions in terms of the Gauss and Appell hypergeometric functions with applications to the classical and radial wave equation. [Copyright &y& Elsevier]

Abstract: For the first time, an explicit closed form expression is derived for the characteristic function of the generalized normal distribution (GND). Also derived is an expression for the correlation coefficient between variate-values and their ranks in samples from the GND. The expression for the former involves the Fox–Wright generalized confluent hypergeometric -function, while the latter is expressed via the Gaussian hypergeometric . [Copyright &y& Elsevier]

Abstract: We study hypergeometric functions for , and show in the entire (non-polynomial) case the transcendence of their special values at nonzero algebraic arguments which generate extension of the rational function field with less than q places at infinity. We also characterize in the balanced case the algebraicity of hypergeometric functions. To cite this article: D.S. Thakur et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009). [Copyright &y& Elsevier]

Abstract: The aim of this Note is to establish some relations between Markov chains in Dirichlet Environments on directed graphs and certain hypergeometric integrals associated with a particular arrangement of hyperplanes. We deduce some new relations on important functionals of the Markov chain. From these relations and the computation of the connexion obtained by moving one hyperplane of the arrangement. To cite this article: C. Sabot, C. R. Acad. Sci. Paris, Ser. I 342 (2006). [Copyright &y& Elsevier]

Abstract: Recent papers by Professor T. Pham-Gia derived distributions of sums, differences, products and ratios of independent beta random variables. In this Note we extend Professor Pham-Gia''s results when and are independent random variables distributed according to the confluent and Gauss hypergeometric distributions (which are generalizations of the beta distribution). For each of these distributions, we derive exact expressions for the densities of , , , and . The expressions turn out to involve the hypergeometric functions of one and two variables. To cite this article: S. Nadarajah, C. R. Acad. Sci. Paris, Ser. I 341 (2005). [Copyright &y& Elsevier]