Rational Extension of the Newton Diagram for the Positivity of $${}_1F_2$$ Hypergeometric Functions and Askey–Szegö Problem

Abstract The main aim of this paper is to establish some Fejér-type inequalities involving hypergeometric functions in terms of GA-s-convexity. For this purpose, we construct a Hadamard k-fractional identity related to geometrically symmetric mappings. Moreover, we give the upper and lower bounds for the weighted inequalities via products of two different mappings. Some applications of the presented results to special means are also provided.

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Towards Feynman rules for conformal blocks

Journal of High Energy Physics ◽

10.1007/jhep01(2021)005 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Sarah Hoback ◽

Sarthak Parikh

Keyword(s):

Field Theory ◽

Effective Field Theory ◽

Hypergeometric Functions ◽

Power Series Expansion ◽

Effective Field ◽

Conformal Block ◽

Tree Level ◽

Conformal Blocks ◽

Simple Set ◽

Feynman Rules

Abstract We conjecture a simple set of “Feynman rules” for constructing n-point global conformal blocks in any channel in d spacetime dimensions, for external and exchanged scalar operators for arbitrary n and d. The vertex factors are given in terms of Lauricella hypergeometric functions of one, two or three variables, and the Feynman rules furnish an explicit power-series expansion in powers of cross-ratios. These rules are conjectured based on previously known results in the literature, which include four-, five- and six-point examples as well as the n-point comb channel blocks. We prove these rules for all previously known cases, as well as two new ones: the seven-point block in a new topology, and all even-point blocks in the “OPE channel.” The proof relies on holographic methods, notably the Feynman rules for Mellin amplitudes of tree-level AdS diagrams in a scalar effective field theory, and is easily applicable to any particular choice of a conformal block beyond those considered in this paper.

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New Specific and General Linearization Formulas of Some Classes of Jacobi Polynomials

Mathematics ◽

10.3390/math9010074 ◽

2020 ◽

Vol 9 (1) ◽

pp. 74

Author(s):

Waleed Mohamed Abd-Elhameed ◽

Afnan Ali

Keyword(s):

Chebyshev Polynomials ◽

Hypergeometric Functions ◽

Jacobi Polynomials ◽

Algebraic Computation ◽

Ultraspherical Polynomials ◽

Current Article ◽

Linearization Coefficients

The main purpose of the current article is to develop new specific and general linearization formulas of some classes of Jacobi polynomials. The basic idea behind the derivation of these formulas is based on reducing the linearization coefficients which are represented in terms of the Kampé de Fériet function for some particular choices of the involved parameters. In some cases, the required reduction is performed with the aid of some standard reduction formulas for certain hypergeometric functions of unit argument, while, in other cases, the reduction cannot be done via standard formulas, so we resort to certain symbolic algebraic computation, and specifically the algorithms of Zeilberger, Petkovsek, and van Hoeij. Some new linearization formulas of ultraspherical polynomials and third-and fourth-kinds Chebyshev polynomials are established.

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A General Family of q-Hypergeometric Polynomials and Associated Generating Functions

Mathematics ◽

10.3390/math9111161 ◽

2021 ◽

Vol 9 (11) ◽

pp. 1161

Author(s):

Hari Mohan Srivastava ◽

Sama Arjika

Keyword(s):

Generating Function ◽

Generating Functions ◽

Hypergeometric Functions ◽

Additional Parameter ◽

Physical Sciences ◽

General Family ◽

Hypergeometric Polynomials

Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and the basic (or q-) hypergeometric polynomials are studied extensively and widely due mainly to their potential for applications in many areas of mathematical and physical sciences. Here, in this paper, we introduce a general family of q-hypergeometric polynomials and investigate several q-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of q-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized q-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various q-results, which we investigate here, with those in several related earlier works on this subject. We conclude this paper by remarking that it will be a rather trivial and inconsequential exercise to give the so-called (p,q)-variations of the q-results, which we have investigated here, because the additional parameter p is obviously redundant.

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Conjectures on Stably Newton Degenerate Singularities

Arnold Mathematical Journal ◽

10.1007/s40598-021-00178-8 ◽

2021 ◽

Author(s):

Jan Stevens

Keyword(s):

Characteristic Zero ◽

Arbitrary Number ◽

Milnor Number ◽

Plane Curves ◽

Newton Diagram ◽

Finite Characteristic ◽

Vanishing Cycles

AbstractWe discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions.We review the various non-degeneracy concepts in the literature. For finite characteristic, we conjecture that there are no wild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, $$x^p+x^q$$ x p + x q in characteristic p, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate.

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