Advances in Hypergeometric Series, Orthogonal Polynomials and Their Natural Extensions

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 31 October 2024 | Viewed by 2090

Special Issue Editor

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Guest Editor
Quantitative Methods Department, Universidad Loyola Andalucia, Seville, Andalucía, Spain
Interests: hypergeometric functions; basic hypergeometric functions; special functions and orthogonal polynomials
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Special Issue Information

Dear Colleagues,

One-variable hypergeometric functions, one of the oldest branches of real and complex analysis, have been exploited by Leonhard  Euler, Carl Friedrich Gauss, Bernhard Riemann, and Ernst  Kummer. Their integral representations were studied by Ernest William and Hjalmar Mellin, and their special properties by Schwarz and Goursat, among others. 

One natural extension of the hypergeometric series is the basic hypergeometric series, which was first considered by Eduard Heine. Moreover, both basic hypergeometric series and hypergeometric series appear naturally within the theory of orthogonal polynomials and special functions. 

In the recent past, using many diverse methods, new special functions and orthogonal polynomials have been introduced and explored. 

In this Special Issue of Axioms, we wish to continue exploring and developing new algebraic and analytic properties of the well-known hypergeometric, or any of its natural extensions, in one or several variables. 

Our goal is to gather experts, as well as young researchers focused on the same task, in order to promote and exchange knowledge and improve communication and applications. We invite research papers as well as review articles.

Dr. Roberto S. Costas-Santos
Guest Editor

Manuscript Submission Information

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.


  • hypergeometric series
  • basic hypergeometric series
  • orthogonal polynomials
  • zeros
  • recurrence relations
  • inner product
  • integral equations
  • generating functions
  • asymptotics
  • applications of special functions and orthogonal polynomials
  • inequalities
  • mathematical knowledge management of special functions and orthogonal polynomials

Published Papers (1 paper)

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15 pages, 390 KiB  
Complex Generalized Representation of Gamma Function Leading to the Distributional Solution of a Singular Fractional Integral Equation
by Asifa Tassaddiq, Rekha Srivastava, Ruhaila Md Kasmani and Rabab Alharbi
Axioms 2023, 12(11), 1046; - 10 Nov 2023
Cited by 1 | Viewed by 1000
Firstly, a basic question to find the Laplace transform using the classical representation of gamma function makes no sense because the singularity at the origin nurtures so rapidly that Γzesz cannot be integrated over positive real numbers. Secondly, [...] Read more.
Firstly, a basic question to find the Laplace transform using the classical representation of gamma function makes no sense because the singularity at the origin nurtures so rapidly that Γzesz cannot be integrated over positive real numbers. Secondly, Dirac delta function is a linear functional under which every function f is mapped to f(0). This article combines both functions to solve the problems that have remained unsolved for many years. For instance, it has been demonstrated that the power law feature is ubiquitous in theory but challenging to observe in practice. Since the fractional derivatives of the delta function are proportional to the power law, we express the gamma function as a complex series of fractional derivatives of the delta function. Therefore, a unified approach is used to obtain a large class of ordinary, fractional derivatives and integral transforms. All kinds of q-derivatives of these transforms are also computed. The most general form of the fractional kinetic integrodifferential equation available in the literature is solved using this particular representation. We extend the models that were valid only for a class of locally integrable functions to a class of singular (generalized) functions. Furthermore, we solve a singular fractional integral equation whose coefficients have infinite number of singularities, being the poles of gamma function. It is interesting to note that new solutions were obtained using generalized functions with complex coefficients. Full article
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