In this paper, we present new extensions of incomplete gamma, beta, Gauss hypergeometric, confluent hypergeometric function and Appell-Lauricella functions, by using the extended Bessel due to Boudjelkha [?]. Some recurrence relations, transformation formulas, Mellin transform integral representations are obtained for these generalizations. Further, an extension Riemann-Liouville fractional deri...

A short review of the method for the tensor reduction of Feynman integrals based on recurrence relations w.r.t. space-time dimension d is given. A solution of the difference equation w.r.t. d for the n-point one-loop integrals with arbitrary momenta and masses is presented. The result is written as multiple hypergeometric series depending on ratios of Gram determinants. For the 3-point function...

in a domain R+3 ≡ {(x, y, z) : x > 0, y > 0, z > 0}. Here α, β, γ are constants, moreover 0 < 2α, 2β, 2γ < 1. Main result of this paper is a construction of eight fundamental solutions for above-given equation in an explicit form. They are expressed by Lauricella’s hypergeometric functions of three variables. Using expansion of Lauricella’s hypergeometric function by products of Gauss’s hyperge...

Nonpolynomial basic hypergeometric eigenfunctions of the Askey–Wilson second order difference operator are known to be expressible as very-well-poised 8φ7 series. In this paper we use this fact to derive various basic hypergeometric and theta function identities. We relate most of them to identities from the existing literature on basic hypergeometric series. This leads for example to a new der...

The hypergeometric differential equation was found by Euler [Euler, L. (1769) Opera Omnia Ser. 1, 11-13] and was extensively studied by Gauss [Gauss, C. F. (1812) Comm. Soc. Reg. Sci. II 3, 123-162], Kummer [Kummer, E. J. (1836) Riene Ang. Math. 15, 39-83; Kummer, E. J. (1836) Riene Ang. Math. 15, 127-172], and Riemann [Riemann, B. (1857) K. Gess. Wiss. 7, 1-24]. The hypergeometric function kno...

This paper introduces the hypergeo package of R routines for numerical calculation of hypergeometric functions. The package is focussed on efficient and accurate evaluation of the Gauss hypergeometric function over the whole of the complex plane within the constraints of fixed-precision arithmetic. The hypergeometric series is convergent only within the unit circle, so analytic continuation mus...