On q-integrals over order polytopes (extended abstract)

International audience A q-integral over an order polytope coming from a poset is interpreted as a generating function of linear extensions of the poset. As an application, theq-beta integral and aq-analog of Dirichlet’s integral are computed. A combinatorial interpretation of aq-Selberg integral is also obtained.

Generalizations of a formula due to Kummer with applications

The aim of this research paper is to obtain explicit expressions of 2F1 [a,b 1/2(a+b ? ?+1); 1+x/2] in the most general case for any ? = 0, 1, 2, ... For ? = 0, we have the well known, interesting and useful formula due to Kummer which was proved independently by Ramanujan. The results presented here are obtained with the help of known generalizations of Gauss?s second summation theorem for the series 2F1(1/2), which were given earlier by Rakha and Rathie [Integral Transforms Spec. Func. 22 (11) (2011), 823-840]. The results are further utilized to obtain new hypergeometric identities by using beta integral method developed by Krattenthaler & Rao [J. Comput. Appl. Math. 160 (2003), 159-173]. Several interesting results due to Ramanujan, Choi, et. al. and Krattenthaler & Rao follow special cases of our main findings.

Further results on multiple q-Eulerian integrals for various q-hypergeometric functions

We continue the study of single and multiple q-Eulerian integrals in the spirit of Exton, Driver, Johnston, Pandey, Saran and Erd?lyi. The method of proof is often the q-beta integral method with the correct q-power together with the q-binomial theorem. By the Totov method we can prove summation theorems as special cases of multiple q-Eulerian integrals. The Srivastava ? notation for q-hypergeometric functions is used to enable the shortest possible form of the long formulas. The various q-Eulerian integrals are in fact meromorphic continuations of the various multiple q-functions, suitable for numerical computations. In the end of the paper a generalization of the q-binomial theorem is used to find q-analogues of a multiple integral formulas for q-Kamp? de F?riet functions.

A Further Extension for Ramanujan’s Beta Integral and Applications

In 1915, Ramanujan stated the following formula ∫ 0 ∞ t x - 1 ( - a t ; q ) ∞ ( - t ; q ) ∞ d t = π sin π x ( q 1 - x , a ; q ) ∞ ( q , a q - x ; q ) ∞ , where 0 < q < 1 , x > 0 , and 0 < a < q x . The above formula is called Ramanujan’s beta integral. In this paper, by using q-exponential operator, we further extend Ramanujan’s beta integral. As some applications, we obtain some new integral formulas of Ramanujan and also show some new representation with gamma functions and q-gamma functions.

On the extended Appell-Lauricella hypergeometric functions and their applications

The main object of this paper is to present a systematic introduction to the theory and applications of the extended Appell-Lauricella hypergeometric functions defined by means of the extended beta function and extended Dirichlet?s beta integral. Their connections with the Laguerre polynomials, the ordinary Lauricella functions and the Srivastava-Daoust generalized Lauricella functions are established for some specific paramters. Furthermore, by applying the various methods and known formulas (such as fractional integral technique; some results of the Lagrange polynomials), we also derive some elegant generating functions for these new functions.