Some Remarks on the Location of Non-Asymptotic Zeros of Whittaker and Kummer Hypergeometric Functions

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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General Summation Formulas Contiguous to the q-Kummer Summation Theorems and Their Applications

Symmetry ◽

10.3390/sym13061102 ◽

2021 ◽

Vol 13 (6) ◽

pp. 1102

Author(s):

Yashoverdhan Vyas ◽

Hari M. Srivastava ◽

Shivani Pathak ◽

Kalpana Fatawat

Keyword(s):

Hypergeometric Functions ◽

Theory Theory ◽

Binomial Theorem ◽

Generalized Hypergeometric Functions ◽

Quantum Calculus ◽

The Third ◽

Summation Formulas ◽

Symmetric Quantum ◽

Summation Theorem

This paper provides three classes of q-summation formulas in the form of general contiguous extensions of the first q-Kummer summation theorem. Their derivations are presented by using three methods, which are along the lines of the three types of well-known proofs of the q-Kummer summation theorem with a key role of the q-binomial theorem. In addition to the q-binomial theorem, the first proof makes use of Thomae’s q-integral representation and the second proof needs Heine’s transformation. Whereas the third proof utilizes only the q-binomial theorem. Subsequently, the applications of these summation formulas in obtaining the general contiguous extensions of the second and the third q-Kummer summation theorems are also presented. Furthermore, the investigated results are specialized to give many of the known as well as presumably new q-summation theorems, which are contiguous to the three q-Kummer summation theorems. This work is motivated by the observation that the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) gamma and q-hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas including Number Theory, Theory of Partitions and Combinatorial Analysis as well as in the study of Combinatorial Generating Functions. Just as it is known in the theory of the Gauss, Kummer (or confluent), Clausen and the generalized hypergeometric functions, the parameters in the corresponding basic or quantum (or q-) hypergeometric functions are symmetric in the sense that they remain invariant when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. A case has therefore been made for the symmetry possessed not only by hypergeometric functions and basic or quantum (or q-) hypergeometric functions, which are studied in this paper, but also by the symmetric quantum calculus itself.

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New Approaches to the General Linearization Problem of Jacobi Polynomials Based on Moments and Connection Formulas

Mathematics ◽

10.3390/math9131573 ◽

2021 ◽

Vol 9 (13) ◽

pp. 1573

Author(s):

Waleed Mohamed Abd-Elhameed ◽

Badah Mohamed Badah

Keyword(s):

Differential Equation ◽

Hypergeometric Functions ◽

Jacobi Polynomials ◽

Tau Method ◽

Riccati Differential Equation ◽

Shifted Jacobi Polynomials ◽

Linearization Problem ◽

Linearization Coefficients ◽

New Formula ◽

Connection Formulas

This article deals with the general linearization problem of Jacobi polynomials. We provide two approaches for finding closed analytical forms of the linearization coefficients of these polynomials. The first approach is built on establishing a new formula in which the moments of the shifted Jacobi polynomials are expressed in terms of other shifted Jacobi polynomials. The derived moments formula involves a hypergeometric function of the type 4F3(1), which cannot be summed in general, but for special choices of the involved parameters, it can be summed. The reduced moments formulas lead to establishing new linearization formulas of certain parameters of Jacobi polynomials. Another approach for obtaining other linearization formulas of some Jacobi polynomials depends on making use of the connection formulas between two different Jacobi polynomials. In the two suggested approaches, we utilize some standard reduction formulas for certain hypergeometric functions of the unit argument such as Watson’s and Chu-Vandermonde identities. Furthermore, some symbolic algebraic computations such as the algorithms of Zeilberger, Petkovsek and van Hoeij may be utilized for the same purpose. As an application of some of the derived linearization formulas, we propose a numerical algorithm to solve the non-linear Riccati differential equation based on the application of the spectral tau method.

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Neoteric formulas of the monic orthogonal Chebyshev polynomials of the sixth-kind involving moments and linearization formulas

Advances in Difference Equations ◽

10.1186/s13662-021-03244-9 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Waleed M. Abd-Elhameed ◽

Youssri H. Youssri

Keyword(s):

Orthogonal Polynomials ◽

Chebyshev Polynomials ◽

Hypergeometric Functions ◽

Connection Coefficients ◽

Current Article ◽

In Virtue Of ◽

Connection Formulas

AbstractThe principal aim of the current article is to establish new formulas of Chebyshev polynomials of the sixth-kind. Two different approaches are followed to derive new connection formulas between these polynomials and some other orthogonal polynomials. The connection coefficients are expressed in terms of terminating hypergeometric functions of certain arguments; however, they can be reduced in some cases. New moment formulas of the sixth-kind Chebyshev polynomials are also established, and in virtue of such formulas, linearization formulas of these polynomials are developed.

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