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.     

Hypergeometric symbolic calculus. II - Systems of confluent equations

We apply the hypergeometric symbolic calculus introduced in the previous work [A. Debiard, B. Gaveau, Hypergeometric symbolic calculus. I - Systems of two symbolic hypergeometric equations] to the determination of the general solution of degenerate hypergeometric equations in two variables and to the determination of a basis of the vector space of solutions of the 20 confluent systems of Horn.     

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.     

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).     

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.     

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     

Zonal spherical functions on the complex reflection groups and (n+1,m+1)-hypergeometric functions

The pair of groups, complex reflection group G(r,1,n) and symmetric group S_n, is a Gelfand pair. Its zonal spherical functions are expressed in terms of multivariate hypergeometric functions called (n+1,m+1)-hypergeometric functions. Since the zonal spherical functions have orthogonality, they form discrete orthogonal polynomials. Also shown is a relation between monomial symmetric functions and the (n+1,m+1)-hypergeometric functions.     

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.     

Functions averages over the roots of unity,cauchy problems and addition formulas

An averaging operator over the roots of unity is defined on a class of analytic functions and its algebraic and analytic properties are investigated. A Cauchy like integral formula for this is obtained. This operator and its properties are then employed to solve higher order Cauchy problems, to derive addition formulas for hypergeometric functions and to obtain integral representations for special classes of hypergeometric functions.     

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.     

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].     

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.     

A semiclassical perspective on multivariate orthogonal polynomials

Differential properties for orthogonal polynomials in several variables are studied. We consider multivariate orthogonal polynomials whose gradients satisfy some quasi-orthogonality conditions. We obtain several characterizations for these polynomials including the analogues of the semiclassical Pearson differential equation, the structure relation and a differential-difference equation.     

Representations of lie groups and multidimensional special functions

This work is devoted to the construction and investigation of two new classes of special functions, related to representations of groups of motions in the spaces of constant curvature as well as the unitary group of large ranks. These are special functions with matrix indices and some types of orthogonal polynomials in several continuous and discrete variables. The functions introduced generalize a number of classical scalar special functions in one variable.     

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.     

Exponentially small expansions in the asymptotics of the Wright function

We consider exponentially small expansions present in the asymptotics of the generalised hypergeometric function, or Wright function, _pΨ_q(z) for large |z| that have not been considered in the existing theory. Our interest is principally with those functions of this class that possess either a finite algebraic expansion or no such expansion and with parameter values that produce exponentially small expansions in the neighbourhood of the negative real z axis. Numerical examples are presented to demonstrate the presence of these exponentially small expansions.     

A note on a family of two-variable polynomials

The main object of this paper is to construct a two-variable analogue of Jacobi polynomials and to give some properties of these polynomials. We show that these polynomials are orthogonal, then we obtain various recurrence formulas for them. Furthermore, we give some integral representations for these polynomials.     

Zeros of a family of hypergeometric para‐orthogonal polynomials on the unit circle

Para‐orthogonal polynomials derived from orthogonal polynomials on the unit circle are known to have all their zeros on the unit circle. In this note we study the zeros of a family of hypergeometric para‐orthogonal polynomials. As tools to study these polynomials, we obtain new results which can be considered as extensions of certain classical results associated with three term recurrence relations and differential equations satisfied by orthogonal polynomials on the real line. One of these results which might be considered as an extension of the classical Sturm comparison theorem, enables us to obtain monotonicity with respect to the parameters for the zeros of these para‐orthogonal polynomials. Finally, a monotonicity of the zeros of Meixner‐Pollaczek polynomials is proved.     

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.