Supercongruences Occurred to Rigid Hypergeometric Type Calabi–Yau Threefolds

In this paper, we provide a formal proof of the existence of a polynomial solution of fixed degree for a second-order divided-difference equation of the hypergeometric type on non-uniform lattices, generalizing therefore previous work proving existence of the polynomial solution for second-order differential, difference or q-difference equation of hypergeometric type. This is achieved by studying the properties of the mean operator and the divided-difference operator as well as by defining explicitly, the right and the “left” inverse for the second operator. The method constructed to provide this formal proof is likely to play an important role in the characterization of orthogonal polynomials on non-uniform lattices and might also be used to provide hypergeometric representation (when it does exist) of the second solution—non polynomial solution—of a second-order divided-difference equation of hypergeometric type.

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DIFFERENCE EQUATION OF THE COLORED JONES POLYNOMIAL FOR TORUS KNOT

International Journal of Mathematics ◽

10.1142/s0129167x04002582 ◽

2004 ◽

Vol 15 (09) ◽

pp. 959-965 ◽

Cited By ~ 22

Author(s):

KAZUHIRO HIKAMI

Keyword(s):

Difference Equation ◽

Jones Polynomial ◽

Order Difference ◽

Colored Jones Polynomial ◽

First Order ◽

Torus Knot ◽

Second Order Difference Equation ◽

Order Difference Equation ◽

Hypergeometric Type ◽

The Difference

We prove that the N-colored Jones polynomial for the torus knot [Formula: see text] satisfies the second order difference equation, which reduces to the first order difference equation for a case of [Formula: see text]. We show that the A-polynomial of the torus knot can be derived from the difference equation. Also constructed is a q-hypergeometric type expression of the colored Jones polynomial for [Formula: see text].

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The analysis of point quadrat data

Australian Journal of Botany ◽

10.1071/bt9560167 ◽

1956 ◽

Vol 4 (2) ◽

pp. 167 ◽

Cited By ~ 19

Author(s):

CD Kemp ◽

AW Kemp

Keyword(s):

Optimum Number ◽

Percentage Cover ◽

Hypergeometric Type ◽

Point Quadrat ◽

Theoretical Considerations ◽

Angular Transformation

The type of distribution followed by point quadrat percentage-cover data has been studied. The hypergeometric type IIA distribution with constant n has been deduced from theoretical considerations and found to give a good fit to data published by Goodall in 1952. The parameters of this distribution assist the study not only of the overall percentage cover but also of the patchiness of the vegetation. The assumption of this type of distribution does not preclude the use of the angular transformation; this remains the appropriate transformation for stabilizing the variance. Theoretical considerations have been put forward in support of the policy of reducing the number of pins per frame. However, if this is done, the number of necessary locations is increased; only by practical experimentation can the optimum number of pins per frame be determined.

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Some characterization and distortion theorems involving fractional calculus, generalized hypergeometric functions, Hadamard products, linear operators, and certain subclasses of analytic functions

Nagoya Mathematical Journal ◽

10.1017/s0027763000000854 ◽

1987 ◽

Vol 106 ◽

pp. 1-28 ◽

Cited By ~ 48

Author(s):

H. M. Srivastava ◽

Shigeyoshi Owa

Keyword(s):

Fractional Calculus ◽

Analytic Functions ◽

Hypergeometric Functions ◽

Linear Operators ◽

Coefficient Estimates ◽

Hadamard Products ◽

Generalized Hypergeometric Functions ◽

Hypergeometric Type ◽

Distortion Theorems ◽

Characterization Theorems

By using a certain linear operator defined by a Hadamard product or convolution, several interesting subclasses of analytic functions in the unit disk are introduced and studied systematically. The various results presented here include, for example, a number of coefficient estimates and distortion theorems for functions belonging to these subclasses, some interesting relationships between these subclasses, and a wide variety of characterization theorems involving a certain functional, some general functions of hypergeometric type, and operators of fractional calculus. Some of the coefficient estimates obtained here are fruitfully applied in the investigation of certain subclasses of analytic functions with fixed finitely many coefficients.

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