On a Generalization of Kummer’s Second-Type F 1 1 and F 2 2

The aim of this paper is to establish a general form of Kummer’s second-type summation theory. By defining a new form for the divided of the Pochhammer symbol d + i n / d n , we can develop a general form of Kummer’s second-type summation theorem as e − x / 2 F 2 2 a , i + d ; x b + l , d ; in the form of a sum of e − x / 2 F 1 1 a ; x b + i ; for i , l = 0,1,2 , … , Then, some properties of the generalized Kummer’s second-type summation theorem can yield a number of known and novel results.

Investigation of the k-Analogue of Gauss Hypergeometric Functions Constructed by the Hadamard Product

Traditionally, the special function theory has many applications in various areas of mathematical physics, economics, statistics, engineering, and many other branches of science. Inspired by certain recent extensions of the k-analogue of gamma, the Pochhammer symbol, and hypergeometric functions, this work is devoted to the study of the k-analogue of Gauss hypergeometric functions by the Hadamard product. We give a definition of the Hadamard product of k-Gauss hypergeometric functions (HPkGHF) associated with the fourth numerator and two denominator parameters. In addition, convergence properties are derived from this function. We also discuss interesting properties such as derivative formulae, integral representations, and integral transforms including beta transform and Laplace transform. Furthermore, we investigate some contiguous function relations and differential equations connecting the HPkGHF. The current results are more general than previous ones. Moreover, the proposed results are useful in the theory of k-special functions where the hypergeometric function naturally occurs.

Properties and Applications of a New Extended Gamma Function Involving Confluent Hypergeometric Function

In this paper, a new confluent hypergeometric gamma function and an associated confluent hypergeometric Pochhammer symbol are introduced. We discuss some properties, for instance, their different integral representations, derivative formulas, and generating function relations. Different special cases are also considered.

Shiraishi functor and non-Kerov deformation of Macdonald polynomials

We suggest a further generalization of the hypergeometric-like series due to M. Noumi and J. Shiraishi by substituting the Pochhammer symbol with a nearly arbitrary function. Moreover, this generalization is valid for the entire Shiraishi series, not only for its Noumi–Shiraishi part. The theta function needed in the recently suggested description of the double-elliptic systems [Awata et al. JHEP 2020:150, arXiv:2005.10563, (2020)], 6d N = 2* SYM instanton calculus and the doubly-compactified network models, is a very particular member of this huge family. The series depends on two kinds of variables, $$\vec {x}$$ x → and $$\vec {y}$$ y → , and on a set of parameters, which becomes infinitely large now. Still, one of the parameters, p is distinguished by its role in the series grading. When $$\vec {y}$$ y → are restricted to a discrete subset labeled by Young diagrams, the series multiplied by a monomial factor reduces to a polynomial at any given order in p. All this makes the map from functions to the hypergeometric-like series very promising, and we call it Shiraishi functor despite it remains to be seen, what are exactly the morphisms that it preserves. Generalized Noumi–Shiraishi (GNS) symmetric polynomials inspired by the Shiraishi functor in the leading order in p can be obtained by a triangular transform from the Schur polynomials and possess an interesting grading. They provide a family of deformations of Macdonald polynomials, as rich as the family of Kerov functions, still very different from them, and, in fact, much closer to the Macdonald polynomials. In particular, unlike the Kerov case, these polynomials do not depend on the ordering of Young diagrams in the triangular expansion.

An Extension of the Mittag-Leffler Function and Its Associated Properties

Inspired by certain fascinating ongoing extensions of the special functions such as an extension of the Pochhammer symbol and generalized hypergeometric function, we present a new extension of the generalized Mittag-Leffler (ML) function εa,b;p,vκz1 in terms of the generalized Pochhammer symbol. We then deliberately find certain various properties and integral transformations of the said function εa,b;p,vκz1. Some particular cases and outcomes of the main results are also established.

A new extension of Srivastava’s triple hypergeometric functions and their associated properties

In this paper, we define a new extension of Srivastava’s triple hypergeometric functions by using a new extension of Pochhammer’s symbol that was recently proposed by Srivastava, Rahman and Nisar [H. M. Srivastava, G. Rahman and K. S. Nisar, Some extensions of the Pochhammer symbol and the associated hypergeometric functions, Iran. J. Sci. Technol. Trans. A Sci. 43 2019, 5, 2601–2606]. We present their certain basic properties such as integral representations, derivative formulas, and recurrence relations. Also, certain new special cases have been identified and some known results are recovered from main results.

On Solutions of the Initial Value Problem for the Three-Term Fractional Differential Equation with Caputo Derivatives

In this paper, two forms of an exact solution and an analytical–numerical solution of the three-term fractional differential equation with the Caputo derivatives are presented. The Prabhakar function and an asymptotic expansion are utilized to present the double series solution. Using properties of the Pochhammer symbol, a solution is obtained in the form of an infinite series of generalized hypergeometric functions. As an alternative for the series solutions of the fractional commensurate equation, a solution received by an analytical–numerical method based on the Laplace transform technique is proposed. This solution is obtained in the form of a finite sum of the Mittag-Leffler type functions. Numerical examples and a discussion are presented.

Some properties of generalized hypergeometric Appell polynomials

Let $x^{(n)}$ denotes the Pochhammer symbol (rising factorial) defined by the formulas $x^{(0)}=1$ and $x^{(n)}=x(x+1)(x+2)\cdots (x+n-1)$ for $n\geq 1$. In this paper, we present a new real-valued Appell-type polynomial family $A_n^{(k)}(m,x)$, $n, m \in {\mathbb{N}}_0$, $k \in {\mathbb{N}},$ every member of which is expressed by mean of the generalized hypergeometric function ${}_{p} F_q \begin{bmatrix} \begin{matrix} a_1, a_2, \ldots, a_p \:\\ b_1, b_2, \ldots, b_q \end{matrix} \: \Bigg| \:z \end{bmatrix}= \sum\limits_{k=0}^{\infty} \frac{a_1^{(k)} a_2^{(k)} \ldots a_p^{(k)}}{b_1^{(k)} b_2^{(k)} \ldots b_q^{(k)}} \frac{z^k}{k!}$ as follows $$ A_n^{(k)}(m,x)= x^n{}_{k+p} F_q \begin{bmatrix} \begin{matrix} {a_1}, {a_2}, {\ldots}, {a_p}, {\displaystyle -\frac{n}{k}}, {\displaystyle -\frac{n-1}{k}}, {\ldots}, {\displaystyle-\frac{n-k+1}{k}}\:\\ {b_1}, {b_2}, {\ldots}, {b_q} \end{matrix} \: \Bigg| \: \displaystyle \frac{m}{x^k} \end{bmatrix} $$ and, at the same time, the polynomials from this family are Appell-type polynomials. The generating exponential function of this type of polynomials is firstly discovered and the proof that they are of Appell-type ones is given. We present the differential operator formal power series representation as well as an explicit formula over the standard basis, and establish a new identity for the generalized hypergeometric function. Besides, we derive the addition, the multiplication and some other formulas for this polynomial family.

Extended Mittag-Leffler function and associated fractional calculus operators

Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [H. M. Srivastava, A. Çetinkaya and I. Onur Kıymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 2014, 484–491] by means of the generalized Pochhammer symbol, we introduce here a new extension of the generalized Mittag-Leffler function. We then systematically investigate several properties of the extended Mittag-Leffler function including some basic properties, Mellin, Euler-Beta, Laplace and Whittaker transforms. Furthermore, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag-Leffler function are also investigated. Some interesting special cases of our main results are pointed out.