In [1] I.M.Gelfand introduced the conception of hypergeometric functions, associated with the Grassmanian Gk,n of k-dimensional subspaces in C . The class of these functions includes many classical hypergeometric functions as particular cases. Series of consequent papers of I.M.Gelfand and his coathors ([2]–[6]) are devoted to the theory of these functions. In particular, the paper [4] is devot...

Multivariate hypergeometric functions associated with toric varieties were introduced by Gel’fand, Kapranov and Zelevinsky. Singularities of such functions are discriminants, that is, divisors projectively dual to torus orbit closures. We show that most of these potential denominators never appear in rational hypergeometric functions. We conjecture that the denominator of any rational hypergeom...

We give a brief review of the main results of the theory of elliptic hypergeometric functions — a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler’s beta integral, which is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic ...

The purpose of the present paper is to give some characterizations for a Gaussian hypergeometric function to be in various subclasses of starlike and convex functions. We also consider an integral operator related to the hypergeometric function.

The celebrated Zeilberger algorithm which finds holonomic recurrence equations for definite sums of hypergeometric terms F (n, k) is extended to certain nonhypergeometric terms. An expression F (n, k) is called hypergeometric term if both F (n + 1, k)/F (n, k) and F (n, k+1)/F (n, k) are rational functions. Typical examples are ratios of products of exponentials, factorials, Γ function terms, b...

The objective of the present paper is to give some characterizations for a (Gaussian) hypergeometric function to be in various subclasses of starlike and convex functions. We also consider an integral operator related to the hypergeometric function.

Abstract. Motivated mainly by certain interesting recent extensions of the Gamma, Beta and hypergeometric functions, we introduce here new extensions of the Beta function, hypergeometric and confluent hypergeometric functions. We systematically investigate several properties of each of these extended functions, such as their various integral representations, Mellin transforms, derivatives, tran...

The Padé table of 2F1(a, 1; c; z) is normal for c > a > 0 (cf. [4]). For m ≥ n−1 and c / ∈ Z − , the denominator polynomial Qmn(z) in the [m/n] Padé approximant Pmn(z)/Qmn(z) for 2F1(a, 1; c; z) and the remainder term Qmn(z)2F1(a, 1; c; z)−Pmn(z) were explicitly evaluated by Padé (cf. [2], [6] or [9]). We show that for c > a > 0 and m ≥ n−1, the poles of Pmn(z)/Qmn(z) lie on the cut (1,∞). We d...

Hypergeometric functions and their inequalities have found frequent applications in various fields of mathematical sciences. Motivated by the above, we set up certain including extended type Gauss hypergeometric function confluent function, respectively, virtue Hölder integral inequality Chebyshev’s inequality. We also studied monotonicity, log-concavity, log-convexity functions, which are deri...

Abstract The paper aims to generalize Clausen’s identity to the square of any Gauss hypergeometric function. Accordingly, solutions of the related 3rd order linear differential equation are found in terms of certain bivariate series that can reduce to 3F2 series similar to those in Clausen’s identity. The general contiguous variation of Clausen’s identity is found. The related Chaundy’s identit...