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Evaluation of low degree hypergeometric polynomials to zero defines an algebraic hypersurface in the affine space of the free parameters and the argument. This article investigates the algebraic surfaces 2F1(-N,b;c;z)=0 for N=3 and N=4. As a captivating application, these surfaces parametrize certain families of genus 0 Belyi maps., Comment: Added Remark 6.8, extra references, corrections; 29 pages; 3 figures

We derive algebraic equations on the coefficients of the implicit equation to characterize all Dupin cyclides passing through a fixed circle. The results are applied to solve the basic problems in CAGD about blending of Dupin cyclides along circles., Comment: 19 pages, 10 figures

Dupin cyclides are interesting algebraic surfaces used in geometric design and architecture to join canal surfaces smoothly and to construct model surfaces. Dupin cyclides are special cases of Darboux cyclides, which in turn are rather general surfaces in $\mathbb R^3$ of degree 3 or 4. This article derives the algebraic conditions for recognition of Dupin cyclides among the general implicit form of Darboux cyclides. We aim at practicable sets of algebraic equations on the coefficients of the implicit equation, each such set defining a complete intersection (of codimension 4) locally. Additionally, the article classifies all real surfaces and lower dimensional degenerations defined by the implicit equation for Dupin cyclides., Comment: 24 pages, 2 figures

While teaching a course on integral equations, I noticed that a straightforward combination of Neumann series and Fourier series for the resolvent (or the solution) of an integral equation has good approximation qualities. This short article presents and investigates this combination of approximating series., Comment: 7 pages, 1 figure, 1 table

By definition, a rigid graph in $\mathbb{R}^d$ (or on a sphere) has a finite number of embeddings up to rigid motions for a given set of edge length constraints. These embeddings are related to the real solutions of an algebraic system. Naturally, the complex solutions of such systems extend the notion of rigidity to $\mathbb{C}^d$. A major open problem has been to obtain tight upper bounds on the number of embeddings in $\mathbb{C}^d$, for a given number $|V|$ of vertices, which obviously also bound their number in $\mathbb{R}^d$. Moreover, in most known cases, the maximal numbers of embeddings in $\mathbb{C}^d$ and $\mathbb{R}^d$ coincide. For decades, only the trivial bound of $O(2^{d\cdot |V|})$ was known on the number of embeddings.Recently, matrix permanent bounds have led to a small improvement for $d\geq 5$. This work improves upon the existing upper bounds for the number of embeddings in $\mathbb{R}^d$ and $S^d$, by exploiting outdegree-constrained orientations on a graphical construction, where the proof iteratively eliminates vertices or vertex paths. For the most important cases of $d=2$ and $d=3$, the new bounds are $O(3.7764^{|V|})$ and $O(6.8399^{|V|})$, respectively. In general, the recent asymptotic bound mentioned above is improved by a factor of $1/ \sqrt{2}$. Besides being the first substantial improvement upon a long-standing upper bound, our method is essentially the first general approach relying on combinatorial arguments rather than algebraic root counts.

Algebraic hypergeometric functions can be compactly expressed as radical or dihedral functions on pull-back curves where the monodromy group is much simpler. This article considers the classical 3F2-functions with the projective monodromy group PSL(2,F7) and their pull-back transformations of degree 21 that reduce the projective monodromy to the dihedral group D4 of 8 elements., Comment: 18 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1808.04524

Algebraic hypergeometric functions can be compactly expressed as radical functions on pull-back curves where the monodromy group is simpler, say, a finite cyclic group. These so-called Darboux evaluations were already considered for algebraic 2F1-functions. This article presents Darboux evaluations for the classical case of 3F2-functions with the projective monodromy group PSL(2,F7). As an application, appealing modular evaluations of the same 3F2-functions are derived., Comment: Coordinated with arxiv:1809.01502; 37 pages, 2 tables

A notion of delegated causality is introduced. This subtle kind of causality is dual to interventional causality. Delegated causality elucidates the causal role of dynamical systems at the "edge of chaos", explicates evident cases of downward causation, and relates emergent phenomena to Godel's incompleteness theorem. Apparently rich implications are noticed in biology and Chinese philosophy., Comment: Causal duality is formulated; 18 pages

We present a method of obtaining a Belyi map on an elliptic curve from that on the Riemann sphere. This is done by writing the former as a radical of the latter, which we call a quadratic correspondence, with the radical determining the elliptic curve. With a host of examples of various degrees we demonstrate that the correspondence is an efficient way of obtaining genus one Belyi maps. As applications, we find the Belyi maps for the dessins d'enfant which have arisen as brane-tilings in the physics community, including ones, such as the so-called suspended pinched point, which have been a standing challenge for a number of years., Comment: 27 pages, 29 figures; Some sections expanded and pictures adjusted

Several kinds of differential relations for polynomial components of almost Belyi maps are presented. Saito's theory of free divisors give particularly interesting (yet conjectural) logarithmic action of vector fields. The differential relations implied by Kitaev's construction of algebraic Painleve VI solutions through pull-back transformations are used to compute almost Belyi maps for the pull-backs giving all genus 0 and 1 Painleve VI solutions in the Lisovyy-Tykhyy classification., Comment: Typos corrected; 21 pages; 1 table

Motivated by a demand for explicit genus 1 Belyi maps from theoretical physics, we give an efficient method of explicitly computing genus one Belyi maps by (1) composing covering maps from elliptic curves to the Riemann sphere with simpler (univariate) genus zero Belyi maps, as well as by (2) composing further with isogenies between elliptic curves. This gives many new explicit dessins on the doubly periodic plane, including several which have been realized in the physics literature as so-called brane-tilings in the context of quiver gauge theories., Comment: 35 pages, 28 figures

A holonomic system for the probability density function of the largest eigenvalue of a non-central complex Wishart distribution with identity covariance matrix is derived. Furthermore a new determinantal formula for the probability density function is derived (for m=2,3) or conjectured., Comment: 23 pages

We study well-constrained bilinear algebraic systems in order to formulate their discriminant. We derive a new determinantal formula for the discriminant of a multilinear system that appears in the study of Nash equilibria of multiplayer games with mixed strategies., Comment: 4 pages

With the renewed and growing interest in geometric continuity in mind, this article gives a general definition of geometrically continuous polygonal surfaces and geometrically continuous spline functions on them. Polynomial splines defined by G1 gluing data in terms of rational functions are analyzed further. A general structure for a spline basis is defined, and a dimension formula is proved for spline spaces of bounded degree on polygonal surfaces made up of rectangles and triangles. Lastly, a comprehensive example is presented, and practical perspectives of geometric continuity are discussed. The whole objective of the paper is to put forward a modernized, practicable framework of modeling with geometric continuity., Comment: 46 pages; 12 figures

We analyze the space of geometrically continuous piecewise polynomial functions or splines for quadrangular 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 quadrangular 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 and examples of bases of G 1 splines of small degree for topological surfaces with boundary and without boundary are detailed., Comment: Computer Aided Geometric Design, Elsevier, 2016

The maximal number of totally mixed Nash equilibria in games of several players equals the number of block derangements, as proved by McKelvey and McLennan.On the other hand, counting the derangements is a well studied problem. The numbers are identified as linearization coefficients for Laguerre polynomials. MacMahon derived a generating function for them as an application of his master theorem. This article relates the algebraic, combinatorial and game-theoretic problems that were not connected before. New recurrence relations, hypergeometric formulas and asymptotics for the derangement counts are derived. An upper bound for the total number of all Nash equilibria is given., Comment: 22 pages, 1 table; Theorem 3.3 added

Just as with the Gauss hypergeometric function, particular cases of the local Heun function can be Liouvillian (that is, "elementary") functions. One way to obtain these functions is by pull-back transformations of Gauss hypergeometric equations with Liouvillian solutions. This paper presents the Liouvillian solutions of Heun's equations that are pull-backs of the parametric hypergeometric equations with cyclic or dihedral monodromy groups., Comment: 10 pages; 1 figure; proofs improved, text corrected

A tool package for computing genus 0 Belyi functions is presented, including simplification routines, computation of moduli fields, decompositions, dessins d'enfant. The main algorithm for computing the Belyi functions themselves is based on implied transformations of the hypergeometric differential equation to Fuchsian equations, preferably with few singular points. This gives interesting differential relations between polynomial components of a Belyi function., Comment: 29 pages, 1 fugure

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/l,1/m that satisfy k,l,m in N and the hyperbolic condition 1/k+1/l+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., Comment: 41 pages; ~15 figures, ~15 tables in a more compact form

Pull-back transformations between Heun and Gauss hypergeometric equations give useful expressions of Heun functions in terms of better understood hypergeometric functions. This article classifies, up to Mobius automorphisms, the coverings P1-to-P1 that yield pull-back transformations from hypergeometric to Heun equations with at least one free parameter (excluding the cases when the involved hypergeometric equation has cyclic or dihedral monodromy). In all, 61 parametric hypergeometric-to-Heun transformations are found, of maximal degree 12. Among them, 28 pull-backs are compositions of smaller degree transformations between hypergeometric and Heun functions. The 61 transformations are realized by 48 different Belyi coverings (though 2 coverings should be counted twice as their moduli field is quadratic). The same Belyi coverings appear in several other contexts. For example, 38 of the coverings appear in Herfutner's list of elliptic surfaces over P1 with four singular fibers, as their j-invariants. In passing, we demonstrate an elegant way to show that there are no coverings P1-to-P1 with some branching patterns., Comment: 32 pages, 2 figures, 5 tables

Hypergeometric equations with a dihedral monodromy group can be solved in terms of elementary functions. This paper gives explicit general expressions for quadratic monodromy invariants for these hypergeometric equations, using a generalization of Clausen's formula and terminating double hypergeometric sums. Besides, pull-back transformations for the dihedral hypergeometric equations are presented, including Klein's pullback transformations for the equations with a finite (dihedral) monodromy group., Comment: 29 pages; a split off from arxiv.org/0807.4888

A starting point of this paper is a classification of quadratic polynomial transformations of the monodromy manifold for the 2x2 isomonodromic Fuchsian systems associated to the Painleve VI equation. Up to birational automorphisms of the monodromy manifold, we find three transformations. Two of them are identified as the action of known quadratic or quartic transformations of the Painleve VI equation. The third transformation of the monodromy manifold gives a new transformation of degree 3 of Picard's solutions of Painleve VI., Comment: Added: classification of quadratic transformations of the Monodromy manifold; new cubic (and quartic) transformations for Picard's case. 26 Pages, 3 figures

The hypergeometric and Heun functions are classical special functions. Transformation formulas between them are commonly induced by pull-back transformations of their differential equations, with respect to some coverings P1-to-P1. This 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 Heun-to-hypergeometric reduction formula. In total, 61 parametric transformations exist, of maximal degree 12., Comment: One-parameter transformations added; 35 pages, 6 tables

The paper aims to generalize Clausen's identity to the square of any Gauss hypergeometric function. Accordingly, solutions of the related 3rd order linear differential equation are found in terms of certain bivariate series that can reduce to 3F2 series similar to those in Clausen's identity. The general contiguous variation of Clausen's identity is found. The related Chaundy's identity is generalized without any restriction on the parameters of Gauss hypergeometric function. The special case of dihedral Gauss hypergeometric functions is underscored.

It is tempting to evaluate F2(x,1) and similar univariate specializations of Appell's functions by evaluating the apparent power series at x=0 straight away using the Gauss formula for 2F1(1). But this kind of naive evaluation can lead to errors as the 2F1(1) coefficients might eventually diverge; then the actual power series at x=0 might involve branching terms. This paper demonstrates these complications on concrete examples., Comment: 10 pages

The paper classifies algebraic transformations of Gauss hypergeometric functions with the local exponent differences $(1/2,1/4,1/4)$, $(1/2,1/3,1/6)$ and $(1/3,1/3,1/3)$. These form a special class of algebraic transformations of Gauss hypergeometric functions, of arbitrary high degree. The Gauss hypergeometric functions can be identified as elliptic integrals on the genus 1 curves $y=x^3-x$ or $y=x^3-1$. Especially interesting are algebraic transformations of the hypergeometric functions into themselves; these transformations come from isogenies of the respective elliptic curves., Comment: 18 pages

Various Schlesinger transformations can be combined with a direct pull-back of a hypergeometric 2x2 system to obtain $RS^2_4$-pullback transformations to isomonodromic 2x2 Fuchsian systems with 4 singularities. The corresponding Painleve VI solutions are algebraic functions, possibly in different orbits under Okamoto transformations. This paper demonstrates a direct computation of Schlesinger transformations acting on several apparent singular points, and presents an algebraic procedure (via syzygies) of computing algebraic Painleve VI solutions without deriving full RS-pullback transformations., Comment: 26 pages

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 using contiguous relations and a data base of simplest Darboux evaluations (covering the Schwarz table). Klein's pull-back transformations also induce algebraic transformations between hypergeometric solutions and a standard hypergeometric function with the same finite monodromy group., Comment: 25 pages

Gauss hypergeometric functions with a dihedral monodromy group can be expressed as elementary functions, since their hypergeometric equations can be transformed to Fuchsian equations with cyclic monodromy groups by a quadratic change of the argument variable. The paper presents general elementary expressions of these dihedral hypergeometric functions, involving finite bivariate sums expressible as terminating Appell's F2 or F3 series. Additionally, trigonometric expressions for the dihedral functions are presented, and degenerate cases (logarithmic, or with the monodromy group Z/2Z) are considered., Comment: 28 pages; trigonometric expressions added; transformations and invariants moved to arxiv.org/1101.3688

Univariate specializations of Appell's hypergeometric functions F1, F2, F3, F4 satisfy ordinary Fuchsian equations of order at most 4. In special cases, these differential equations are of order 2, and could be simple (pullback) transformations of Euler's differential equation for the Gauss hypergeometric function. The paper classifies these cases, and presents corresponding relations between univariate specializations of Appell's functions and univariate hypergeometric functions. The computational aspect and interesting identities are discussed., Comment: 30 pages; A few wrong evaluations of the published version are rectified (see arXiv:0906.1861)

An almost Belyi covering is an algebraic covering of the projective line, such that all ramified points except one simple ramified point lie above a set of 3 points of the projective line. In general, there are 1-dimensional families of these coverings with a fixed ramification pattern. (That is, Hurwitz spaces for these coverings are curves.) In this paper, three almost Belyi coverings of degrees 11, 12, and 20 are explicitly constructed. We demonstrate how these coverings can be used for computation of several algebraic solutions of the sixth Painleve equation., Comment: 26 pages

Algebraic solutions of the sixth Painleve equation can be computed using pullback transformations of hypergeometric equations with respect to specially ramified rational coverings. In particular, as was noticed by the second author and Doran, some algebraic solutions can be constructed from a rational covering alone, without computation of the pullbacked Fuchsian equation. But the same covering can be used to pullback different hypergeometric equations, resulting in different algebraic Painleve VI solutions. This paper presents computations of explicit RS-pullback transformations, and derivation of algebraic Painleve VI solutions from them., Comment: 18 pages; 7 figures; Schlesinger transformations (and some examples) are brought to the new paper http://arxiv.org/abs/0705.2963

The considered problem is uniform convergence of sequences of hypergeometric series. We give necessary and sufficient conditions for uniformly dominated convergence of infinite sums of proper bivariate hypergeometric terms. These conditions can be checked algorithmically. Hence the results can be applied in Zeilberger type algorithms for nonterminating hypergeometric series., Comment: 42 pages, 2 Figures. A more compact form of hypergeometric terms is used. Examples added

It is known that if $(A,B)$ is a Leonard pair, then the linear transformations $A$, $B$ satisfy the Askey-Wilson relations A^2 B - b A B A + B A^2 - g (A B+B A) - r B = h A^2 + w A + e I, B^2 A - b B A B + A B^2 - h (A B+B A) - s A = g B^2 + w B + f I, for some scalars $b,g,h,r,s,w,e,f$. The problem of this paper is the following: given a pair of Askey-Wilson relations as above, how many Leonard pairs are there that satisfy those relations? It turns out that the answer is 5 in general. We give the generic number of Leonard pairs for each Askey-Wilson type of Askey-Wilson relations., Comment: 22 pages; corrected version; the example of Section 2 has the normalization consistent with the rest of the paper

In 1991, one of the authors showed the existence of quadratic transformations between the Painleve' VI equations with local monodromy differences $(1/2,a,b,\pm 1/2)$ and $(a,a,b,b)$. In the present paper we give concise forms of these transformations. %, up to fractional-linear transformations. They are related to the %better known quadratic transformations obtained by Manin and Ramani-Grammaticos-Tamizhmani via Okamoto transformations. To avoid cumbersome expressions with differentiation, we use contiguous relations instead of the Okamoto transformations. The 1991 transformation is particularly important as it can be realized as a quadratic-pull back transformation of isomonodromic Fuchsian equations. The new formulas are illustrated by derivation of explicit expressions for several complicated algebraic Painleve' VI functions., Comment: 27 pages, Submitted to Mathematische Nachrichten

Let $V$ denote a vector space with finite positive dimension, and let $(A,B)$ denote a Leonard pair on $V$. As is known, the linear transformations $A,B$ satisfy the Askey-Wilson relations A^2B -bABA +BA^2 -g(AB+BA) -rB = hA^2 +wA +eI, B^2A -bBAB +AB^2 -h(AB+BA) -sA = gB^2 +wB +fI, for some scalars $b,g,h,r,s,w,e,f$. The scalar sequence is unique if the dimension of $V$ is at least 4. If $c,c*,t,t*$ are scalars and $t,t*$ are not zero, then $(tA+c,t*B+c*)$ is a Leonard pair on $V$ as well. These affine transformations can be used to bring the Leonard pair or its Askey-Wilson relations into a convenient form. This paper presents convenient normalizations of Leonard pairs by the affine transformations, and exhibits explicit Askey-Wilson relations satisfied by them., Comment: 22 pages; corrected version, with improved presentation of Section 9

This paper presents explicit expressions for algebraic Gauss hypergeometric functions. We consider solutions of hypergeometric equations with the tetrahedral, octahedral and icosahedral monodromy groups. Conceptually, we pull-back such a hypergeometric equation onto its Darboux curve so that the pull-backed equation has a cyclic monodromy group. Minimal degree of the pull-back coverings is 4, 6 or 12 (for the three monodromy groups, respectively). In explicit terms, we replace the independent variable by a rational function of degree 4, 6 or 12, and transform hypergeometric functions to radical functions., Comment: The list of seed hypergeometric evaluations (in Section 2) reduced by half; uniqueness claims explained; 34 pages; Kyushu Journal of Mathematics, 2013

This article gives a classification scheme of algebraic transformations of Gauss hypergeometric functions, or pull-back transformations between hypergeometric differential equations. The classification recovers the classical transformations of degree 2, 3, 4, 6, and finds other transformations of some special classes of the Gauss hypergeometric function. The other transformations are considered more thoroughly in a series of supplementing articles., Comment: 29 pages; 3 tables; Uniqueness claims and Remark 7.1 clarified by footnotes; formulas (28), (29) corrected

This is a study of terminating and ill-defined Gauss hypergeometric functions. Corresponding hypergeometric equations have a degenerate set of of 24 Kummer's solutions. We describe those solutions and relations between them., Comment: 22 pages

This paper presents expressions for gamma values at rational points with the denominator dividing 24 or 60. These gamma values are expressed in terms of 10 distinct gamma values and rational powers of $\pi$ and a few real algebraic numbers. Our elementary list of formulas can be conveniently used to evaluate, for example, algebraic Gauss hypergeometric functions by the Gauss identity. Also, algebraic independence of gamma values and their relation to the elliptic K-function are briefly discussed., Comment: 14 pages

This paper presents explicit algebraic transformations of some Gauss hypergeometric functions. Specifically, the transformations considered apply to hypergeometric solutions of hypergeometric differential equations with the local exponent differences $1/K,1/L,1/M$ such that $K,L,M$ are positive integers and $1/K+1/L+1/M<1$. All algebraic transformations of these Gauss hypergeometric functions are considered. We show that apart from classical transformations of degree 2, 3, 4, 6 there are several other transformations of degree 6, 8, 9, 10, 12, 18, 24. Besides, we present an algorithm to compute relevant Belyi functions explicitly., Comment: 16 pages; the accepted text for publication (in the OPSFA 7 proceedings)

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 $A:V\to V$ and $A^*:V\to V$ 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 $\beta,\gamma,\gamma^*, \varrho,\varrho^*,\omega, \eta, \eta^*$ taken from K such that both (i) A^2 A^*-\beta A A^*A+A^*A^2-\gamma (AA^*+A^*A) -\varrho A^* =\gamma^*A^2+\omega A+\etaI; (ii) A^{*2}A-\beta A^*AA^*+AA^{*2}-\gamma^*(A^*A+AA^*) -\varrho^*A =\gamma A^{*2}+\omega A^*+\eta^*I. The 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., Comment: 17 pages

Several asymptotic expansions of parabolic cylinder functions are discussed and error bounds for remainders in the expansions are presented. In particular Poincar{\'e}-type expansions for large values of the argument $z$ and uniform expansions for large values of the parameter are considered. The expansions are based on those given in Olver(1959) and on modifications of these expansions given in Temme(2000). Computer algebra techniques are used for obtaining representations of the bounds and for numerical computations., Comment: 22 pages, 4 figures

The 15 Gauss contiguous relations for ${}_2F_1$ hypergeometric series imply that any three ${}_2F_1$ series whose corresponding parameters differ by integers are linearly related (over the field of rational functions in the parameters). We prove several properties of coefficients of these general contiguous relations, and use the results to propose effective ways to compute contiguous relations. We also discuss contiguous relations of generalized and basic hypergeometric functions, and several applications of them., Comment: 12 pages; full bibliography added. This is the published text, with corrected formulas (24)-(25)