Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider an ordered pair of linear transformations A : V → V and A * : V → V which satisfy the following two properties:(i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A * is diagonal.(ii) There exists a basis for V with respect to which the matrix representing A * is irreducible tridiagonal and the matrix representing A is diagonal.We call such a pair a Leonard pair on V . Referring to the above Leonard pair, we show there exists a sequence of scalars β, γ, γ * , ̺, ̺ * , ω, η, η * taken from K such that bothThe sequence is uniquely determined by the Leonard pair provided the dimension of V is at least 4. The equations above are called the Askey-Wilson relations.

A celebrated theorem of Klein implies that any hypergeometric differential equation with algebraic solutions is a pull-back of one of the few standard hypergeometric equations with algebraic solutions. The most interesting cases are hypergeometric equations with tetrahedral, octahedral or icosahedral monodromy groups. We give an algorithm for computing Klein's pull-back coverings in these cases, based on certain explicit expressions (Darboux evaluations) of algebraic hypergeometric functions. The explicit expressions can be computed from a data base (covering the Schwarz table) and using contiguous relations. Klein's pull-back transformations also induce algebraic transformations between hypergeometric solutions and a standard hypergeometric function with the same finite monodromy group.

A complete classification of Belyi functions for transforming certain hypergeometric equations to Heun equations is given. The considered hypergeometric equations have the local exponent differences 1/k, 1/ , 1/m that satisfy k, , m ∈ N and the hyperbolic condition 1/k + 1/ + 1/m < 1. There are 366 Galois orbits of Belyi functions giving the considered (non-parametric) hypergeometric-to-Heun pull-back transformations. Their maximal degree is 60, which is well beyond reach of standard computational methods. To obtain these Belyi functions, we developed two efficient algorithms that exploit the implied pull-back transformations.

We analyze the space of geometrically continuous piecewise polynomial functions, or splines, for rectangular and triangular patches with arbitrary topology and general rational transition maps. To define these spaces of G 1 spline functions, we introduce the concept of topological surface with gluing data attached to the edges shared by faces. The framework does not require manifold constructions and is general enough to allow non-orientable surfaces. We describe compatibility conditions on the transition maps so that the space of differentiable functions is ample and show that these conditions are necessary and sufficient to construct ample spline spaces. We determine the dimension of the space of G 1 spline functions which are of degree k on triangular pieces and of bi-degree (k, k) on rectangular pieces, for k big enough. A separability property on the edges is involved to obtain the dimension formula. An explicit construction of basis functions attached respectively to vertices, edges and faces is proposed; examples of bases of G 1 splines of small degree for topological surfaces with boundary and without boundary are detailed.

Abstract. The hypergeometric and Heun functions are classical special functions. Transformation formulas between them are commonly induced by pull-back transformations of their di¤erential equations, with respect to some coveringsThis gives expressions of Heun functions in terms of better understood hypergeometric functions. This article presents the list of hypergeometric-to-Heun pull-back transformations with a free continuous parameter, and illustrates most of them by a Heunto-hypergeometric reduction formula. In total, 61 parametric transformations exist, of maximal degree 12.