54 research outputs found
Expressions for values of the gamma function
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 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 page
Degenerate Gauss hypergeometric functions
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 page
Dihedral Gauss hypergeometric functions
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/1101.368
Algebraic transformations of Gauss hypergeometric functions
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) correcte
Darboux evaluations of algebraic Gauss hypergeometric functions
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,
201
A generalization of Kummer's identity
The well-known Kummer's formula evaluates the hypergeometric series
2F1(A,B;C;-1) when the relation B-A+C=1 holds. This paper deals with evaluation
of 2F1(-1) series in the case when C-A+B is an integer. Such a series is
expressed as a sum of two \Gamma-terms multiplied by terminating 3F2(1) series.
A few such formulas were essentially known to Whipple in 1920's. Here we give a
simpler and more complete overview of this type of evaluations. Additionally,
algorithmic aspects of evaluating hypergeometric series are considered. We
illustrate Zeilberger's method and discuss its applicability to non-terminating
series, and present a couple of similar generalizations of other known
formulas.Comment: 13 pages; classical proofs simplified, possible transformations
reviewed; in the algoritmic part similar evaluations of other series adde
Counting derangements and Nash equilibria
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 adde
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