Hypergeometric symbolic calculus. I - Systems of two symbolic hypergeometric equations

We define a new hypergeometric symbolic calculus which allows the determination of the general solutions of two variables hypergeometric partial differential equations. We apply this method to the determination of a basis of the vector space of solutions of the 14 systems of Appell-Kampé de Fériet and Horn, as well as new integral representations for the solutions.     

Comment on a paper of Rao et al., an entry of Ramanujan and a new 3F2(1)

A hypergeometric transformation formula is developed that simultaneously simplifies and generalizes arguments and identities in a previous paper of Rao et al. [An entry of Ramanujan on hypergeometric series in his notebooks, J. Comput. Appl. Math. 173(2) (2005) 239–246].     

Certain summation and transformation formulas for generalized hypergeometric series

We derive summation formulas for generalized hypergeometric series of unit argument, one of which upon specialization reduces to Minton’s summation theorem. As an application we deduce a reduction formula for a certain Kampé de Fériet function that in turn provides a Kummer-type transformation formula for the generalized hypergeometric function _pF_p(x).     

Solving the hypergeometric system of Okubo type in terms of a certain generalized hypergeometric function

We introduce one scalar function f of a complex variable and finitely many parameters, which allows to represent all solutions of the so-called hypergeometric system of Okubo type under the assumption that one of the two coefficient matrices has all distinct eigenvalues. In the simplest non-trivial situation, f is equal to the hypergeometric function, while in other more complicated cases it is related, but not equal, to the generalized hypergeometric functions. In general, however, this function appears to be a new higher transcendental one. The coefficients of the power series of f about the origin can be explicitly given in terms of a generalized version of the classical Pochhammer symbol, involving two square matrices that in general do not commute. The function can also be characterized by a Volterra integral equation, whose kernel is expressed in terms of the solutions of another hypergeometric system of lower dimension.     

Certain values of Hecke L-functions and generalized hypergeometric functions

We compare two calculations due to Bloch and the author of the regulator of an elliptic curve with complex multiplication which is a quotient of a Fermat curve, and express the special value of its L-function at s=0 in terms of special values of generalized hypergeometric functions.     

Orthogonal properties for a typical class of hypergeometric functions

We present three orthogonal properties for a typical class of hypergeometric functions. We employ orthogonal properties to generate a theory concerning infinite series expansions involving our hypergeometric functions.     

Some results involving series representations of hypergeometric functions

In this paper, we prove some generalisations of several theorems given in [K.A. Driver, S.J. Johnston, An integral representation of some hypergeometric functions, Electron. Trans. Numer. Anal. 25 (2006) 115-120] and examine some special cases which correspond to a transformation given by Chaundy in [T.W. Chaundy, An extension of hypergeometric functions, I., Quart. J. Maths. Oxford Ser. 14 (1943) 55-78] and other transformations involving the Riemann zeta function and the beta function.     

Generalizations of Whipple's theorem on the sum of a 3F2

Thirty-eight summation closely related to Whipple's theorem, in the theory of the generalized hypergeometric series, are obtained. Some limiting cases are also considered.     

Combinatorics of Rational Functions and Poincaré- Birchoff-Witt Expansions of the Canonical U ({\frak n}\_)-Valued Differential Form

We study the canonical U -valued differential form, whose projections to different Kac-Moody algebras are key ingredients of the hypergeometric integral solutions of KZ-type differential equations and Bethe ansatz constructions. We explicitly determine the coefficients of the projections in the simple Lie algebras A_r, B_r, C_r, D_r in a conveniently chosen Poincaré- Birchoff-Witt basis.Received June 28, 2004     

Two-variable orthogonal polynomials of big q-Jacobi type

A four-parameter family of orthogonal polynomials in two discrete variables is defined for a weight function of basic hypergeometric type. The polynomials, which are expressed in terms of univariate big q-Jacobi polynomials, form an extension of Dunkl’s bivariate (little) q-Jacobi polynomials [C.F. Dunkl, Orthogonal polynomials in two variables of q-Hahn and q-Jacobi type, SIAM J. Algebr. Discrete Methods 1 (1980) 137-151]. We prove orthogonality property of the new polynomials, and show that they satisfy a three-term relation in a vector-matrix notation, as well as a second-order partial q-difference equation.     

Legendre inversions and balanced hypergeometric series identities

By means of Legendre inverse series relations, we prove two terminating balanced hypergeometric series formulae. Their reversals and linear combinations yield several known and new hypergeometric series identities.     

Euler transformation formula for multiple basic hypergeometric series of type A and some applications

A multiple generalization of the Euler transformation formula for basic hypergeometric series ₂φ₁ is derived. It is obtained from the symmetry of the reproducing kernel for Macdonald polynomials by a method of multiple principal specialization. As applications, elementary proofs of the Pfaff-Saalschutz summation formula and the Gauss summation formula for basic hypergeometric series in U(n+1) due to S.C. Milne are given. Some other multiple transformation and summation formulas for very-well-poised ₁₀φ₉ and ₈φ₇ series, balanced ₄φ₃ series and ₃φ₂ series are also given.     

Bivariate hypergeometric D-modules

We undertake the study of bivariate Horn systems for generic parameters. We prove that these hypergeometric systems are holonomic, and we provide an explicit formula for their holonomic rank as well as bases of their spaces of complex holomorphic solutions. We also obtain analogous results for the generalized hypergeometric systems arising from lattices of any rank.     

More on the Bernoulli*–Taylor Formula for Extended Umbral Calculus

One presents here the *_ψ-Bernoulli-Taylor* formula of a new sort with the rest term of the Cauchy type recently derived by the author in the case of the so called ψ-difference calculus which constitutes the representative for the purpose case of extended umbral calculus. The central importance of such a type formulas is beyond any doubt - and recent publications do confirm this historically established experience. *see below: a historical remark based on Academician N. Y. Sonin article published in Petersburg in 19-th century. We owe this information and the article to Professor O. V. Viscov from Moscow. **see: C. Graves, On the principles which regulate the interchange of symbols in certain symbolic equations, Proc. Royal Irish Academy 6 (1853-1857), 144-152.     

A Fredholm determinant formula for Toeplitz determinants

We prove a formula expressing a generaln byn Toeplitz determinant as a Fredholm determinant of an operator 1 –K acting onl ₂ (n,n+1,...), where the kernelK admits an integral representation in terms of the symbol of the original Toeplitz matrix. The proof is based on the results of one of the authors, see [14], and a formula due to Gessel which expands any Toeplitz determinant into a series of Schur functions. We also consider 3 examples where the kernel involves the Gauss hypergeometric function and its degenerations.     

Determinantal formulas for zonal spherical functions on hyperboloids

Abstract. We construct determinantal expressions for the zonal spherical functions on the hyperboloids with p,q odd (and larger than 1). This gives rise to explicit evaluation formulas for hypergeometric series representing half-integer parameter families of Jacobi functions and (via specialization) Jacobi polynomials. Received November 18, 1999 / Published online October 30, 2000     

On the largest principal angle between random subspaces

Formulas are derived for the probability density function and the probability distribution function of the largest canonical angle between two p-dimensional subspaces of Rⁿ chosen from the uniform distribution on the Grassmann manifold (which is the unique distribution invariant by orthogonal transformations of Rⁿ). The formulas involve the gamma function and the hypergeometric function of a matrix argument.     

Approximations to -, di- and tri-logarithms

We propose hypergeometric constructions of simultaneous approximations to polylogarithms. These approximations suit for computing the values of polylogarithms and satisfy 4-term Apéry-like (polynomial) recursions.