Quantum canonical transformations are used to derive the integral representations and Kummer solutions of the confluent hypergeometric and hypergeometric equations. Integral representations of the solutions of the non-periodic three body Toda equation are also found. The derivation of these representations motivate the form of a two-dimensional generalized hypergeometric equation which contains...

HYPERDIRE is a project devoted to the creation of a set of Mathematica based programs for the differential reduction of hypergeometric functions. The current version includes two parts: the first one, FdFunction, for manipulations with Appell hypergeometric functions FD of r variables; and the second one, FsFunction, for manipulations with Lauricella-Saran hypergeometric functions FS of three v...

We deduce an explicit representation for the coefficients in a finite expansion of a certain class of generalized hypergeometric functions that contain multiple pairs of numeratorial and denominatorial parameters differing by positive integers. The expansion alluded to is given in terms of these coefficients and hypergeometric functions of lower order. Applications to Euler and Kummer-type tran...

In recent years, several interesting families of generating functions for various classes of hypergeometric and generalized hypergeometric functions in one, two and more variables were investigated systematically. Here, in this sequel, we aim at establishing several (presumably) new generating relations for the generalized Gauss type hypergeometric functions which are introduced by means of som...

We show a connection formula for the q-confluent hypergeometric functions 2φ1(a, b; 0; q, x). Combining our connection formula with Zhang’s connection formula for 2φ0(a, b;−; q, x), we obtain the connection formula for the q-confluent hypergeometric equation in the matrix form. Also we obtain the connection formula of Kummer’s confluent hypergeometric functions by taking the limit q → 1− of our...

We start from an interpretation of the BC2-symmetric “Type I” (elliptic Dixon) elliptic hypergeometric integral evaluation as a formula for a Casoratian of the elliptic hypergeometric equation, and give an extension to higher-dimensional integrals and higher-order hypergeometric functions. This allows us to prove the corresponding elliptic beta integral and transformation formula in a new way, ...

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...

In this paper, using similar symbolical method of Burchnall and Chaundy formulas of expansion for the generalized hypergeometric function were constructed. By means of the found formulas of expansion the formulas of an analytic continuation for hypergeometric function of Clausen is defined. The obtained formulas of an analytic continuation express known hypergeometric Appell function F2 (a; b1,...

In this article we find connections between the values of Gauss hypergeometric functions and the dimension of the vector space of Hodge cycles of four dimensional cubic hypersurfaces. Since the Hodge conjecture is well-known for those varieties we calculate values of Hypergeometric series on certain CM points. Our methods is based on the calculation of the Picard-Fuchs equations in higher dimen...