Recently, J. A. Tirao [Proc. Nat. Acad. Sci. 100 (14) (2003), 8138–8141] considered a matrix-valued analogue of the 2F1 Gauß hypergeometric function and showed that it is the unique solution of a matrix-valued hypergeometric equation analytic at z = 0 with value I, the identity matrix, at z = 0. We give an independent proof of Tirao’s result, extended to the more general setting of hypergeometr...

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...

I will describe three sets of results in which computer search and computer algebra played a large role in the discovery/and or proof of results each ultimately relating to hypergeometric functions. 1. Ap ery: A generating function for (4n+3). 2. Ramanujan: Integrals, means and transformations of the hypergeometric function 2F1(1=3;2=3;1; 1 x): 3. Euler: Multivalued {function values and a produ...

Gauss’s 2F1 ( 1 2 1 2 1 | x ) hypergeometric function gives periods of elliptic curves in Legendre normal form. Certain truncations of this hypergeometric function give the Hasse invariants for these curves. Here we study another form, which we call the Clausen form, and we prove that certain truncations of 3F2 ( 1 2 1 2 1 2 1 1 | x ) and 2F1 ( 1 4 3 4 1 | x ) in Fp[x] are related to the charac...

Explicit closed forms are derived for the probability density function of the stable distribution Sα(β, γ, μ), α ∈ (1, 2]. Consequent asymptotic expansions are given. The expressions involve the Srivastava-Daoust generalized Kampé de Fériet hypergeometric S-function, the Fox-Wright generalized hypergeometric Ψ-function, and the Gauss hypergeometric function 2F1. 2000 Mathematics subject classif...

By employing two well-known Euler’s transformations for the hypergeometric function 2F1, Liu and Wang established numerous general transformation reduction formulas Kampé de Fériet deduced many new summation with aid of classical theorems 2F1 due to Kummer, Gauss Bailey. Here, by making a fundamental use above-mentioned formulas, we aim establish 32 help generalizations above-referred Relevant ...

We give a new proof of the classical Watson theorem for the summation of a 3F2 hypergeometric series of unit argument. The proof relies on the two well-known Gauss summation theorems for the 2F1 function. Mathematics Subject Classification: 33C20

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 1Ψ0-function, while the latter is ex...

The special case of the hypergeometric function 2F1 represents the binomial series (1 + x) = ∑∞ n=0 ( α n ) xn that always converges when |x| < 1. Convergence of the series at the endpoints, x = ±1, depends on the values of α and needs to be checked in every concrete case. In this note, using new approach, we reprove the convergence of the hypergeometric series for |x| < 1 and obtain new result...

The 15 Gauss contiguous relations for 2F1 hypergeometric series imply that any three 2F1 series whose corresponding parameters differ by integers are linearly related (over the field of rational functions in the parameters). We prove several properties of coefficients of these general contiguous relations, and use the results to propose effective ways to compute contiguous relations. We also di...