The fundamental set of solutions of the generalized hypergeometric differential equation in the neighborhood of unity has been built by Nørlund in 1955. The behavior of the generalized hypergeometric function in the neighborhood of unity has been described in the beginning of 1990s by Bühring, Srivastava and Saigo. In the first part of this paper we review their results rewriting them in terms ...

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 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...

Euler’s two-term transformation formula for the Gauss hypergeometric function 2F1 is extended to hypergeometric functions of higher order. The generalized two-term transformation follows from the theory of Fuchsian differential equations with accessory parameters, although it also has a combinatorial proof. Unusually, it constrains the hypergeometric function parameters algebraically and not li...

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...

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...

Let $$n\ge 3$$ be an integer and p a prime with $$p\equiv 1\pmod {n}$$ . In this paper, we show that $$\begin{aligned} {}_nF_{n-1}\bigg [\begin{array}{llll} \frac{n-1}{n}&{}\frac{n-1}{n}&{}\ldots &{}\frac{n-1}{n}\\ &{}1&{}\ldots &{}1\end{array}\bigg | \, 1\bigg ]_{p-1}\equiv -\Gamma _p\bigg (\frac{1}{n}\bigg )^n\pmod {p^3}, \end{aligned}$$ where the truncated hypergeometric series x_1&{}x_2&{}\...